Finite Group Databasebeta

← #34029524

Subgroups — #34029524

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 620 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#22626033C211yesmaximal, characteristic, complemented
4#22626028C211yesmaximal, characteristic, complemented
5#22626005C211yesmaximal, characteristic
6#22626005C211yesmaximal, characteristic
7#22626001C211yesmaximal, characteristic
8#22626001C211yesmaximal, characteristic
9#22625988C211yesmaximal, characteristic, complemented
10#22625988C211yesmaximal, characteristic, complemented
11#22625987C211yesmaximal, characteristic, complemented
12#22625984C211yesmaximal, characteristic, complemented
13#22625822C211yesmaximal, characteristic
14#22625785C211yesmaximal, characteristic
15C22⋊Q8×Dic73C211yesmaximal, characteristic
16C2×#14556658C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×C4⋊#10596833C2211yescharacteristic
19C2×C4⋊#10596833C2211yescharacteristic
20C2×C4⋊#10596833C2211yescharacteristic
21C2×C4⋊#10596833C2211yescharacteristic
22C2×C22⋊#10596833C2211yescharacteristic
23C2×C22⋊#10596833C2211yescharacteristic
24C2×C22⋊#10596833C2211yescharacteristic
25C2×C22⋊#10596833C2211yescharacteristic
26C2×#14556646C2211yescharacteristic
27C2×#14556646C2211yescharacteristic
28C2×C4×#10596833C2211yescharacteristic
29C2×C4×#10596833C2211yescharacteristic
30#17989453C2211yescharacteristic
31#17989453C2211yescharacteristic
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
45C22×C4×Dic73C2211yescharacteristic
46#17988494C411yescharacteristic
47#17988486C411yescharacteristic
48#17988473C411yescharacteristic
49#17988473C411yescharacteristic
50#17988462C411yescharacteristic
51#17988462C411yescharacteristic
52#17988447C411yescharacteristic
53C4⋊C4×Dic73C2211yescharacteristic
54C4⋊C4×Dic73C2211yescharacteristic
55C4⋊C4×Dic73C2211yescharacteristic
56C22⋊C4×Dic73C2211yescharacteristic
57C22⋊C4×Dic73C2211yescharacteristic
58C2×C22⋊Q8×C73C411yescharacteristic
59C22×#10596833C2311yescharacteristic
60C22×#10596833C2311yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67#14556658—41no—
68#14556658—41no—
69C4⋊#10596833—41no—
70C4⋊#10596833—41no—
71C4⋊#10596833—41no—
72C4⋊#10596833—41no—
73C22⋊#10596833—41no—
74C22⋊#10596833—41no—
75C22⋊#10596833—41no—
76C22⋊#10596833—41no—
77#14556646—41no—
78#14556646—41no—
79C4×#10596833—41no—
80C4×#10596833—41no—
81C23×Dic73C2×C411yescharacteristic
82C23×Dic73C2311yescharacteristic
83Q8×Dic73—41no—
84Q8×Dic73—41no—
85C2×C4⋊Dic73C2×C411yescharacteristic
86C2×C4⋊Dic73C2×C411yescharacteristic
87C2×C4⋊Dic73C2×C411yescharacteristic
88C2×C4⋊Dic73C2×C411yescharacteristic
89C2×C4⋊Dic73C2×C411yescharacteristic
90C2×C4⋊Dic73C2×C411yescharacteristic
91C2×C22⋊Dic73C2×C411yescharacteristic
92C2×C22⋊Dic73C2×C411yescharacteristic
93C2×C4×Dic73—41no—
94C2×C4×Dic73—41no—
95C2×C4×Dic73C2×C411yescharacteristic
96C2×C4×Dic73C2×C411yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C2×C4×Dic73C2311yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113C22×Q8×C73C2×C411yescharacteristic
114C23×C4×C73C2×C411yescharacteristic
115C22⋊Q8×C73—24no—
116C2×C4⋊C4×C73C2×C411yescharacteristic
117C2×C4⋊C4×C73C2×C411yescharacteristic
118C2×C4⋊C4×C73C2×C411yescharacteristic
119C2×C22⋊C4×C73C2×C411yescharacteristic
120C2×C22⋊C4×C73C2×C411yescharacteristic
121C2×#10596833—21no—
122C2×#10596833—21no—
123C2×#10596833—21no—
124C2×#10596833—21no—
125C2×#10596833—21no—
126C2×#10596833—21no—
127C2×#10596833—21no—
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—
145C22×Dic73—22no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—41no—
153C22×Dic73—41no—
154C22×Dic73C22×C411yescharacteristic
155C22×Dic73C22×C411yescharacteristic
156C22×Dic73C22×C411yescharacteristic
157C22×Dic73C22×C411yescharacteristic
158C22×Dic73C22×C411yescharacteristic
159C22×Dic73C22×C411yescharacteristic
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C2411yescharacteristic
162C4⋊Dic73—22no—
163C4⋊Dic73—41no—
164C4⋊Dic73—41no—
165C4⋊Dic73—41no—
166C4⋊Dic73—41no—
167C4⋊Dic73—41no—
168C22⋊Dic73—22no—
169C22⋊Dic73—22no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—41no—
183C4×Dic73—41no—
184C24×C73C22×C411yescharacteristic
185C2×Q8×C73—22no—
186C2×Q8×C73—22no—
187C2×Q8×C73—41no—
188C22×C4×C73—22no—
189C22×C4×C73—22no—
190C22×C4×C73—41no—
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C22×C411yescharacteristic
194C22×C4×C73C22×C411yescharacteristic
195C22×C4×C73C22×C411yescharacteristic
196C22×C4×C73C22×C411yescharacteristic
197C4⋊C4×C73—22no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73M1614yes—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73#13112yes—
209C23×C73C23×C411yescharacteristic
210Q8×C73—44no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—44no—
226C2×C4×C73C2×M1612yes—
227C2×C4×C73C2×M1612yes—
228C2×C4×C73C2×M1612yes—
229C2×C4×C73#13112yes—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—44no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73H(5,𝔽2)11yescharacteristic
257C2×Dic73H(5,𝔽2)11yescharacteristic
258C2×Dic73H(5,𝔽2)11yescharacteristic
259C2×Dic732−1+411yescharacteristic
260C2×Dic732−1+411yescharacteristic
261C2×Dic732−1+411yescharacteristic
262#10596833—41no—
263#10596833—41no—
264#10596833—41no—
265#10596833—41no—
266#10596833—41no—
267#10596833—41no—
268#10596833—41no—
269#10596833—41no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73#51811yescharacteristic
278C22×C73#51911yescharacteristic
279C22×C73#52011yescharacteristic
280C22×C73C22×M1611yescharacteristic
281C22×C73C2×#13111yescharacteristic
282C22×C73C2×#13111yescharacteristic
283C22×C73#56811yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290Dic73—41no—
291Dic73—41no—
292Dic73—41no—
293Dic73—41no—
294Dic73—41no—
295Dic73—41no—
296Dic73—41no—
297Dic73#56211yescharacteristic
298Dic73#56211yescharacteristic
299Dic73#56311yescharacteristic
300Dic73#56311yescharacteristic
301#23339—731nomaximal, Sylow
302C2×C73—42no—
303C2×C73#237811yescharacteristic
304C2×C73#297411yescharacteristic
305C2×C73#297611yescharacteristic
306C2×C73#298411yescharacteristic
307C2×C73#298511yescharacteristic
308C2×C73#298911yescharacteristic
309C2×C73#299411yescharacteristic
310C4×C22⋊Q8—731no—
311#1985—731no—
312#1984—731no—
313#1940—731no—
314#1939—731no—
315#1924—731no—
316#1924—731no—
317#1923—731no—
318#1923—731no—
319#1917—731no—
320#1917—731no—
321#1916—731no—
322#1916—731no—
323#1817—731no—
324#1797—731no—
325C73#2333911yescharacteristic, complemented, Sylow
326C2×C22⋊Q8#6440811yescharacteristic
327C2×C4×Q8—731no—
328C22×C42—731no—
329C2×C4⋊C8—731no—
330C2×C4⋊C8—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×C22⋊C8—731no—
336C2×C22⋊C8—731no—
337C2×C22⋊C8—731no—
338C2×C22⋊C8—731no—
339C2×#97—731no—
340C2×#97—731no—
341C2×C4×C8—731no—
342C2×C4×C8—731no—
343#391—731no—
344#389—731no—
345#387—731no—
346#387—731no—
347#385—731no—
348#385—731no—
349#380—731no—
350C4×C4⋊C4—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C22⋊C4—731no—
354C4×C22⋊C4—731no—
355#336—731no—
356#336—731no—
357#336—731no—
358#336—731no—
359#336—731no—
360#336—731no—
361#336—731no—
362#336—731no—
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369C22×Q8C2×#6440811yescharacteristic
370C23×C4—731no—
371C23×C4—731no—
372C23×C4C2×#6440811yescharacteristic
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—731no—
381C22⋊Q8—24no—
382C4×Q8—2921no—
383C4×Q8—2921no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4C2×#6440811yescharacteristic
391C2×C4⋊C4C2×#6440811yescharacteristic
392C2×C4⋊C4C2×#6440811yescharacteristic
393C2×C22⋊C4—731no—
394C2×C22⋊C4—731no—
395C2×C22⋊C4C2×#6440811yescharacteristic
396C2×C22⋊C4C2×#6440811yescharacteristic
397C2×C42—2921no—
398C2×C42—2921no—
399C2×C42—731no—
400C2×C42—731no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C4⋊C8—2921no—
408C4⋊C8—2921no—
409C4⋊C8—2921no—
410C4⋊C8—2921no—
411C4⋊C8—2921no—
412C4⋊C8—2921no—
413C22⋊C8—2921no—
414C22⋊C8—2921no—
415C22⋊C8—2921no—
416C22⋊C8—2921no—
417#97—2921no—
418#97—2921no—
419C4×C8—2921no—
420C4×C8—2921no—
421#95—731no—
422#95—731no—
423#95—731no—
424#95—731no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—731no—
429#95—731no—
430#95—731no—
431C24C22×#6440811yescharacteristic
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—2921no—
439C22×C4—2921no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1462no—
455C22×C4C22×#6440811yescharacteristic
456C22×C4C22×#6440811yescharacteristic
457C22×C4C22×#6440811yescharacteristic
458C22×C4C22×#6440811yescharacteristic
459C22×C4C22×#6440811yescharacteristic
460C22×C4C22×#6440811yescharacteristic
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—2921no—
488C4⋊C4—2921no—
489C4⋊C4—2921no—
490C4⋊C4—2921no—
491C4⋊C4—2921no—
492C4⋊C4—1462no—
493C4⋊C4#1192403714yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—1462no—
497C22⋊C4—1462no—
498C42—2921no—
499C42—2921no—
500C42—1461no—
501C42—1461no—
502C42—1461no—
503C42—1461no—
504C42—1461no—
505C42—1461no—
506C42—1461no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23#1455667912yes—
519C23C23×#6440811yescharacteristic
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—2924no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—731no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4C2×#1192403712yes—
569C2×C4C2×#1192403712yes—
570C2×C4C2×#1192403712yes—
571C2×C4#1455667812yes—
572C8—2921no—
573C8—2921no—
574C8—2921no—
575C8—2921no—
576C8—2921no—
577C8—2921no—
578C8—2921no—
579C8—2921no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#1798965411yescharacteristic
588C22#1798965511yescharacteristic
589C22#1798965611yescharacteristic
590C22C22×#1192403711yescharacteristic
591C22C2×#1455667811yescharacteristic
592C22C2×#1455667911yescharacteristic
593C22#1798966611yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—2921no—
601C4—2921no—
602C4—2921no—
603C4—2921no—
604C4—2921no—
605C4—2921no—
606C4—2921no—
607C4—731no—
608C4—731no—
609C4—731no—
610C4—731no—
611C2—42no—
612C2#2262647311yescharacteristic
613C2#2262678711yescharacteristic
614C2#2262680111yescharacteristic
615C2#2262683411yescharacteristic
616C2#2262683511yescharacteristic
617C2#2262685011yescharacteristic
618C2#2262688211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C22⋊Q8×C73C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#23339
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626105C2semidirect
2. #22626100C2semidirect
3. #22626033C2semidirect
4. #22626028C2semidirect
9. #22625988C2semidirect
10. #22625988C2semidirect
11. #22625987C2semidirect
12. #22625984C2semidirect
325. C73#23339semidirect

Maximal subgroups

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

subgroupindex
#226261052
#226261002
#226260332
#226260282
#226260052
#226260012
#226259882
#226259872
#226259842
#226258222
#226257852
C22⋊Q8×Dic732
#2333973