Finite Group Databasebeta

← #34026009

Subgroups — #34026009

Order 18688.

768 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626105C211yesmaximal, characteristic, complemented
2#22626100C211yesmaximal, characteristic, complemented
3#22626005C211yesmaximal, characteristic, complemented
4#22626005C211yesmaximal, characteristic, complemented
5#22626004C211yesmaximal, characteristic, complemented
6#22626000C211yesmaximal, characteristic, complemented
7#22625908C211yesmaximal, characteristic
8#22625904C211yesmaximal, characteristic
9#22625878C211yesmaximal, characteristic, complemented
10#22625874C211yesmaximal, characteristic, complemented
11#22625859C211yesmaximal, characteristic, complemented
12#22625717C211yesmaximal, characteristic, complemented
13#22625673C211yesmaximal, characteristic, complemented
14#22625668C211yesmaximal, characteristic, complemented
15#124×Dic73C211yesmaximal, characteristic
16C22×#11924037C2211yescharacteristic
17C23×#10596833C2211yescharacteristic
18C2×#14556658C2211yescharacteristic
19C2×C4⋊#10596833C2211yescharacteristic
20C2×C4⋊#10596833C2211yescharacteristic
21C2×C4⋊#10596833C2211yescharacteristic
22C2×C22⋊#10596833C2211yescharacteristic
23C2×C22⋊#10596833C2211yescharacteristic
24C2×C22⋊#10596833C2211yescharacteristic
25C2×C22⋊#10596833C2211yescharacteristic
26C2×C22⋊#10596833C2211yescharacteristic
27C2×C22⋊#10596833C2211yescharacteristic
28C2×#14556646C2211yescharacteristic
29C2×#14556646C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×C4×#10596833C2211yescharacteristic
32#17989453C2211yescharacteristic
33#17989453C2211yescharacteristic
34#17989453C2211yescharacteristic
35#17989453C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44C2×Q8×Dic73C2211yescharacteristic
45C2×D4×Dic73C2211yescharacteristic
46#17988494C411yescharacteristic
47#17988490C411yescharacteristic
48#17988480C411yescharacteristic
49#17988480C411yescharacteristic
50#17988480C411yescharacteristic
51#17988480C411yescharacteristic
52#17988453C411yescharacteristic
53C22⋊C4×Dic73C2211yescharacteristic
54C22⋊C4×Dic73C2211yescharacteristic
55C22⋊C4×Dic73C2211yescharacteristic
56C22⋊C4×Dic73C2211yescharacteristic
57C42×Dic73C2211yescharacteristic
58C2×#124×C73C411yescharacteristic
59C2×#11924037—22no—
60C2×#11924037—41no—
61C2×#11924037—41no—
62C22×#10596833—22no—
63C22×#10596833—41no—
64C22×#10596833—41no—
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70C22×#10596833C2311yescharacteristic
71C22×#10596833C2311yescharacteristic
72C22×#10596833C2311yescharacteristic
73#14556658—41no—
74C4⋊#10596833—22no—
75C4⋊#10596833—22no—
76C4⋊#10596833—41no—
77C22⋊#10596833—22no—
78C22⋊#10596833—22no—
79C22⋊#10596833—41no—
80C22⋊#10596833—41no—
81C22⋊#10596833—41no—
82C22⋊#10596833—41no—
83#14556646—41no—
84#14556646—41no—
85#14556646—41no—
86C4×#10596833—41no—
87C23×Dic73C2311yescharacteristic
88C23×Dic73C2311yescharacteristic
89Q8×Dic73—41no—
90Q8×Dic73—41no—
91D4×Dic73—41no—
92D4×Dic73—41no—
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×C4⋊Dic73C2×C411yescharacteristic
95C2×C4⋊Dic73C2×C411yescharacteristic
96C2×C4⋊Dic73C2×C411yescharacteristic
97C2×C22⋊Dic73C2×C411yescharacteristic
98C2×C22⋊Dic73C2×C411yescharacteristic
99C2×C22⋊Dic73C2×C411yescharacteristic
100C2×C22⋊Dic73C2×C411yescharacteristic
101C2×C22⋊Dic73C2×C411yescharacteristic
102C2×C22⋊Dic73C2×C411yescharacteristic
103C2×C4×Dic73C2×C411yescharacteristic
104C2×C4×Dic73C2×C411yescharacteristic
105C2×C4×Dic73C2×C411yescharacteristic
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2311yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113#14556498C2×C411yescharacteristic
114#14556498C2×C411yescharacteristic
115#14556498C2×C411yescharacteristic
116#14556498C2×C411yescharacteristic
117#14556498C2×C411yescharacteristic
118#14556498C2×C411yescharacteristic
119C22×Q8×C73C2×C411yescharacteristic
120C22×D4×C73C2×C411yescharacteristic
121#124×C73—24no—
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124C2×C22⋊C4×C73C2×C411yescharacteristic
125C2×C22⋊C4×C73C2×C411yescharacteristic
126C2×C42×C73C2×C411yescharacteristic
127#11924037—44no—
128C2×#10596833—21no—
129C2×#10596833—21no—
130C2×#10596833—21no—
131C2×#10596833—21no—
132C2×#10596833—21no—
133C2×#10596833—21no—
134C2×#10596833—21no—
135C2×#10596833—21no—
136C2×#10596833—21no—
137C2×#10596833—21no—
138C2×#10596833—21no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—44no—
151C2×#10596833#4112yes—
152C2×#10596833#4112yes—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C22×C411yescharacteristic
167C22×Dic73C22×C411yescharacteristic
168C22×Dic73C22×C411yescharacteristic
169C22×Dic73C22×C411yescharacteristic
170C22×Dic73C22×C411yescharacteristic
171C22×Dic73C22×C411yescharacteristic
172C22×Dic73C2411yescharacteristic
173C4⋊Dic73—41no—
174C4⋊Dic73—41no—
175C4⋊Dic73—41no—
176C4⋊Dic73—41no—
177C22⋊Dic73—41no—
178C22⋊Dic73—41no—
179C22⋊Dic73—41no—
180C22⋊Dic73—41no—
181C22⋊Dic73—41no—
182C22⋊Dic73—41no—
183C4×Dic73—22no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—41no—
195C4×Dic73—41no—
196C24×C73C22×C411yescharacteristic
197C24×C73C22×C411yescharacteristic
198C2×Q8×C73—22no—
199C2×Q8×C73—22no—
200C2×Q8×C73—41no—
201C2×D4×C73—22no—
202C2×D4×C73—22no—
203C2×D4×C73—41no—
204C22×C4×C73C22×C411yescharacteristic
205C22×C4×C73C22×C411yescharacteristic
206C22×C4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208C22×C4×C73C22×C411yescharacteristic
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—22no—
212C22⋊C4×C73M1614yescomplemented
213C42×C73—22no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73C2×M1612yes—
228C23×C73C23×C411yescharacteristic
229Q8×C73—44no—
230D4×C73—44no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73C2×M1612yes—
244C2×C4×C73C2×M1612yes—
245C2×C4×C73#13112yes—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73C2×#4111yescharacteristic
276C2×Dic73C2×#4111yescharacteristic
277C2×Dic73H(5,𝔽2)11yescharacteristic
278C2×Dic73H(5,𝔽2)11yescharacteristic
279C2×Dic73H(5,𝔽2)11yescharacteristic
280C2×Dic732−1+411yescharacteristic
281#10596833—22no—
282#10596833—22no—
283#10596833—41no—
284#10596833—41no—
285#10596833—41no—
286#10596833—41no—
287#10596833—41no—
288#10596833—41no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73C4×#4111yescharacteristic
304C22×C73#51811yescharacteristic
305C22×C73#52011yescharacteristic
306C22×C73C22×M1611yescharacteristic
307C22×C73C2×#13111yescharacteristic
308C22×C73#56811yescharacteristic
309C22×C73#56811yescharacteristic
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—22no—
313C4×C73—41no—
314C4×C73—41no—
315Dic73—41no—
316Dic73—41no—
317Dic73—41no—
318Dic73—41no—
319Dic73—41no—
320Dic73—41no—
321Dic73—41no—
322Dic73#53811yescharacteristic
323Dic73#53811yescharacteristic
324Dic73#54211yescharacteristic
325Dic73#54211yescharacteristic
326#22391—731nomaximal, Sylow
327C2×C73—42no—
328C2×C73—81no—
329C2×C73#233711yescharacteristic
330C2×C73#296911yescharacteristic
331C2×C73#297011yescharacteristic
332C2×C73#297411yescharacteristic
333C2×C73#298011yescharacteristic
334C2×C73#298911yescharacteristic
335C2×C73#299311yescharacteristic
336C4×#124—731no—
337#1985—731no—
338#1984—731no—
339#1924—731no—
340#1924—731no—
341#1924—731no—
342#1923—731no—
343#1870—731no—
344#1869—731no—
345#1851—731no—
346#1850—731no—
347#1839—731no—
348#1767—731no—
349#1753—731no—
350#1752—731no—
351C73#2239111yescharacteristic, complemented, Sylow
352C22×M16—731no—
353C23×C8—731no—
354C2×#124#6440811yescharacteristic
355C2×C4×Q8—731no—
356C2×C4×D4—731no—
357C2×C4⋊C8—731no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×C4⋊C8—731no—
361C2×C22⋊C8—731no—
362C2×C22⋊C8—731no—
363C2×C22⋊C8—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×C22⋊C8—731no—
367C2×#97—731no—
368C2×#97—731no—
369C2×#97—731no—
370C2×C4×C8—731no—
371#391—731no—
372#390—731no—
373#388—731no—
374#388—731no—
375#388—731no—
376#388—731no—
377#382—731no—
378C4×C22⋊C4—731no—
379C4×C22⋊C4—731no—
380C4×C22⋊C4—731no—
381C4×C22⋊C4—731no—
382C43—731no—
383#336—731no—
384#336—731no—
385#336—731no—
386#336—731no—
387#336—731no—
388#336—731no—
389#336—731no—
390#336—731no—
391#336—731no—
392#336—731no—
393#336—731no—
394#336—731no—
395C22×Q8C2×#6440811yescharacteristic
396C22×D4C2×#6440811yescharacteristic
397C23×C4—731no—
398C23×C4—731no—
399C2×M16—2921no—
400C2×M16—2921no—
401C2×M16—1462no—
402C22×C8—2921no—
403C22×C8—2921no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—1462no—
413#124—24no—
414C4×Q8—2921no—
415C4×Q8—2921no—
416C4×D4—2921no—
417C4×D4—2921no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C22⋊C4—731no—
428C2×C22⋊C4C2×#6440811yescharacteristic
429C2×C22⋊C4C2×#6440811yescharacteristic
430C2×C22⋊C4C2×#6440811yescharacteristic
431C2×C22⋊C4C2×#6440811yescharacteristic
432C2×C42—731no—
433C2×C42—731no—
434C2×C42—731no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—731no—
439C2×C42—731no—
440C2×C42C2×#6440811yescharacteristic
441C4⋊C8—2921no—
442C4⋊C8—2921no—
443C4⋊C8—1462no—
444C4⋊C8—1462no—
445C22⋊C8—2921no—
446C22⋊C8—2921no—
447C22⋊C8—2921no—
448C22⋊C8—2921no—
449C22⋊C8—1462no—
450C22⋊C8—1462no—
451#97—2921no—
452#97—2921no—
453#97—2921no—
454C4×C8—2921no—
455#95—731no—
456#95—731no—
457#95—731no—
458#95—731no—
459#95—731no—
460#95—731no—
461#95—731no—
462#95—731no—
463C24C22×#6440811yescharacteristic
464C24C22×#6440811yescharacteristic
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—41no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4C22×#6440811yescharacteristic
492C22×C4C22×#6440811yescharacteristic
493C22×C4C22×#6440811yescharacteristic
494C22×C4C22×#6440811yescharacteristic
495C22×C4C22×#6440811yescharacteristic
496M16—2924no—
497C2×C8—2924no—
498C2×C8—732no—
499C2×C8—732no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C4⋊C4—2921no—
523C4⋊C4—2921no—
524C4⋊C4—2921no—
525C4⋊C4—2921no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—2921no—
530C22⋊C4—2921no—
531C22⋊C4—2921no—
532C22⋊C4—2921no—
533C22⋊C4—2921no—
534C22⋊C4—2921no—
535C22⋊C4#1192403714yescomplemented
536C42—22no—
537C42—2921no—
538C42—2921no—
539C42—1461no—
540C42—1461no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C42—1462no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23C2×#1192403712yes—
564C23C23×#6440811yescharacteristic
565Q8—44no—
566D4—44no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—731no—
588C2×C4—731no—
589C2×C4—731no—
590C2×C4—731no—
591C2×C4—731no—
592C2×C4—731no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4C2×#1192403712yes—
615C2×C4C2×#1192403712yes—
616C2×C4#1455667812yes—
617C8—2921no—
618C8—2921no—
619C8—2921no—
620C8—2921no—
621C8—2921no—
622C8—2921no—
623C8—1462no—
624C8—1462no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22#41×#6440811yescharacteristic
640C22#1798965411yescharacteristic
641C22#1798965611yescharacteristic
642C22C22×#1192403711yescharacteristic
643C22C2×#1455667811yescharacteristic
644C22#1798966611yescharacteristic
645C22#1798966611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—2921no—
652C4—2921no—
653C4—2921no—
654C4—2921no—
655C4—2921no—
656C4—2921no—
657C4—2921no—
658C4—731no—
659C4—731no—
660C4—731no—
661C4—731no—
662C2—42no—
663C2—81no—
664C2#2262643211yescharacteristic
665C2#2262676311yescharacteristic
666C2#2262676811yescharacteristic
667C2#2262678711yescharacteristic
668C2#2262681711yescharacteristic
669C2#2262685011yescharacteristic
670C2#2262687411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×#124×C73C4
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic73C24
hypercenterC2×#124#64408
hypocenterC73#22391
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22626105C2semidirect
2. #22626100C2semidirect
3. #22626005C2semidirect
4. #22626005C2semidirect
5. #22626004C2semidirect
6. #22626000C2semidirect
9. #22625878C2semidirect
10. #22625874C2semidirect
11. #22625859C2semidirect
12. #22625717C2semidirect
13. #22625673C2semidirect
14. #22625668C2semidirect
212. C22⋊C4×C73M16semidirect
351. C73#22391semidirect
535. C22⋊C4#11924037semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226261052
#226261002
#226260052
#226260042
#226260002
#226259082
#226259042
#226258782
#226258742
#226258592
#226257172
#226256732
#226256682
#124×Dic732
#2239173