Finite Group Databasebeta

← #34029402

Subgroups — #34029402

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626055C211yesmaximal, characteristic
2#22626032C211yesmaximal, characteristic, complemented
3#22626029C211yesmaximal, characteristic, complemented
4#22626005C211yesmaximal, characteristic, complemented
5#22626005C211yesmaximal, characteristic, complemented
6#22626000C211yesmaximal, characteristic, complemented
7#22626000C211yesmaximal, characteristic, complemented
8#22625989C211yesmaximal, characteristic, complemented
9#22625988C211yesmaximal, characteristic, complemented
10#22625908C211yesmaximal, characteristic
11#22625878C211yesmaximal, characteristic, complemented
12#22625878C211yesmaximal, characteristic, complemented
13#22625789C211yesmaximal, characteristic, complemented
14#22625789C211yesmaximal, characteristic, complemented
15#22613187C211yesmaximal, characteristic
16C22×#11924037C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×#14556658C2211yescharacteristic
19C2×#14556658C2211yescharacteristic
20C2×C4⋊#10596833C2211yescharacteristic
21C2×C4⋊#10596833C2211yescharacteristic
22C2×C4⋊#10596833C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×C22⋊#10596833C2211yescharacteristic
25C2×C22⋊#10596833C2211yescharacteristic
26C2×C22⋊#10596833C2211yescharacteristic
27C2×C22⋊#10596833C2211yescharacteristic
28C2×C22⋊#10596833C2211yescharacteristic
29C2×C22⋊#10596833C2211yescharacteristic
30C2×C22⋊#10596833C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×#14556646C2211yescharacteristic
33C2×C4×#10596833C2211yescharacteristic
34#17989453C2211yescharacteristic
35#17989453C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44C2×D4×Dic73C2211yescharacteristic
45C2×#14556547C2211yescharacteristic
46#17988529C411yescharacteristic
47#17988523C411yescharacteristic
48#17988520C411yescharacteristic
49#17988515C411yescharacteristic
50#17988515C411yescharacteristic
51#17988512C411yescharacteristic
52#17988501C411yescharacteristic
53C4⋊C4×Dic73C2211yescharacteristic
54#17988438C2211yescharacteristic
55C22⋊C4×Dic73C2211yescharacteristic
56C22⋊C4×Dic73C2211yescharacteristic
57C22⋊C4×Dic73C2211yescharacteristic
58#394×C73C411yescharacteristic
59C2×#11924037—41no—
60C2×#11924037—41no—
61C2×#11924037—41no—
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70#14556658—41no—
71#14556658—41no—
72#14556658—41no—
73C4⋊#10596833—41no—
74C4⋊#10596833—41no—
75C4⋊#10596833—41no—
76C4⋊#10596833—41no—
77C22⋊#10596833—41no—
78C22⋊#10596833—41no—
79C22⋊#10596833—41no—
80C22⋊#10596833—41no—
81C22⋊#10596833—41no—
82C22⋊#10596833—41no—
83C22⋊#10596833—41no—
84#14556646—41no—
85#14556646—41no—
86C4×#10596833—41no—
87C23×Dic73C2311yescharacteristic
88C23×Dic73C2311yescharacteristic
89D4×Dic73—41no—
90D4×Dic73—41no—
91#14556547—41no—
92#14556547—41no—
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×C4⋊Dic73C2×C411yescharacteristic
95C2×#11924007C2×C411yescharacteristic
96C2×#11924007C2×C411yescharacteristic
97C2×#11924007C2×C411yescharacteristic
98C2×#11924007C2×C411yescharacteristic
99C2×C22⋊Dic73C2×C411yescharacteristic
100C2×C22⋊Dic73C2×C411yescharacteristic
101C2×C22⋊Dic73C2×C411yescharacteristic
102C2×C22⋊Dic73C2×C411yescharacteristic
103C2×C22⋊Dic73C2×C411yescharacteristic
104C2×C22⋊Dic73C2×C411yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2311yescharacteristic
110#14556498C2×C411yescharacteristic
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×D4×C73C2×C411yescharacteristic
120C2×C4⋊C4×C73C2×C411yescharacteristic
121C2×C22⋊C4×C73C2×C411yescharacteristic
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124C2×C22⋊C4×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
126#11924037—44no—
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—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
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×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C22×C411yescharacteristic
167C22×Dic73C22×C411yescharacteristic
168C22×Dic73C22×C411yescharacteristic
169C22×Dic73C22×C411yescharacteristic
170C22×Dic73C2411yescharacteristic
171C4⋊Dic73—22no—
172C4⋊Dic73—41no—
173#11924007—41no—
174#11924007—41no—
175#11924007—41no—
176#11924007—41no—
177C22⋊Dic73—41no—
178C22⋊Dic73—41no—
179C22⋊Dic73—41no—
180C22⋊Dic73—41no—
181C22⋊Dic73—41no—
182C22⋊Dic73—41no—
183C4×Dic73—21no—
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—
193C24×C73C22×C411yescharacteristic
194C24×C73C22×C411yescharacteristic
195C2×D4×C73—41no—
196C2×D4×C73—41no—
197C2×D4×C73—41no—
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C411yescharacteristic
200C22×C4×C73C22×C411yescharacteristic
201C22×C4×C73C22×C411yescharacteristic
202C22×C4×C73C22×C411yescharacteristic
203C4⋊C4×C73—41no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—22no—
206C22⋊C4×C73—41no—
207C22⋊C4×C73—41no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
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×C73C23×C411yescharacteristic
223D4×C73—44no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
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#13112yes—
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—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—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73H(5,𝔽2)11yescharacteristic
269C2×Dic73H(5,𝔽2)11yescharacteristic
270C2×Dic73H(5,𝔽2)11yescharacteristic
271C2×Dic73H(5,𝔽2)11yescharacteristic
272C2×Dic732−1+411yescharacteristic
273C2×Dic732−1+411yescharacteristic
274#10596833—41no—
275#10596833—41no—
276#10596833—41no—
277#10596833—41no—
278#10596833—41no—
279#10596833—41no—
280#10596833—41no—
281#10596833—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73#51811yescharacteristic
297C22×C73#52011yescharacteristic
298C22×C73#52011yescharacteristic
299C22×C73C2×#13111yescharacteristic
300C22×C73#56811yescharacteristic
301C22×C73#56811yescharacteristic
302C22×C73#56811yescharacteristic
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312Dic73—41no—
313Dic73—41no—
314Dic73—41no—
315Dic73#56011yescharacteristic
316Dic73#56211yescharacteristic
317Dic73#56211yescharacteristic
318Dic73#56311yescharacteristic
319#23321—731nomaximal, Sylow
320C2×C73—81no—
321C2×C73—81no—
322C2×C73#237611yescharacteristic
323C2×C73#297811yescharacteristic
324C2×C73#298111yescharacteristic
325C2×C73#299211yescharacteristic
326C2×C73#299311yescharacteristic
327C2×C73#299511yescharacteristic
328C2×C73#299511yescharacteristic
329#2395—731no—
330#1955—731no—
331#1940—731no—
332#1939—731no—
333#1924—731no—
334#1924—731no—
335#1923—731no—
336#1923—731no—
337#1917—731no—
338#1917—731no—
339#1870—731no—
340#1851—731no—
341#1851—731no—
342#1798—731no—
343#1798—731no—
344C73#2332111yescharacteristic, complemented, Sylow
345C22×M16—731no—
346C2×C4×D4—731no—
347C2×#117—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8—731no—
350C2×C4⋊C8—731no—
351C2×C4⋊C8—731no—
352C2×C4⋊C8—731no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×C22⋊C8—731no—
356C2×C22⋊C8—731no—
357C2×C22⋊C8—731no—
358C2×C22⋊C8—731no—
359C2×C22⋊C8—731no—
360C2×C22⋊C8—731no—
361C2×C22⋊C8—731no—
362C2×#97—731no—
363C2×#97—731no—
364C2×C4×C8—731no—
365#400—731no—
366#399—731no—
367#398—731no—
368#397—731no—
369#397—731no—
370#397—731no—
371#394—731no—
372#394#6440811yescharacteristic
373C4×C4⋊C4—731no—
374C4×C4⋊C4—731no—
375C4×C22⋊C4—731no—
376C4×C22⋊C4—731no—
377C4×C22⋊C4—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383#336—731no—
384#336—731no—
385#336—731no—
386#336—731no—
387#336—731no—
388C22×D4C2×#6440811yescharacteristic
389C23×C4—731no—
390C23×C4—731no—
391C2×M16—2921no—
392C2×M16—2921no—
393C2×M16—2921no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—731no—
402C4×D4—2921no—
403C4×D4—2921no—
404#117—2921no—
405#117—2921no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4C2×#6440811yescharacteristic
413C2×C22⋊C4—731no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—731no—
417C2×C22⋊C4—731no—
418C2×C22⋊C4—731no—
419C2×C22⋊C4C2×#6440811yescharacteristic
420C2×C22⋊C4C2×#6440811yescharacteristic
421C2×C22⋊C4C2×#6440811yescharacteristic
422C2×C22⋊C4C2×#6440811yescharacteristic
423C2×C42—731no—
424C2×C42—731no—
425C2×C42—731no—
426C2×C42—731no—
427C2×C42—731no—
428C4⋊C8—2921no—
429C4⋊C8—2921no—
430C4⋊C8—2921no—
431C4⋊C8—2921no—
432C4⋊C8—2921no—
433C4⋊C8—2921no—
434C4⋊C8—2921no—
435C22⋊C8—2921no—
436C22⋊C8—2921no—
437C22⋊C8—2921no—
438C22⋊C8—2921no—
439C22⋊C8—2921no—
440C22⋊C8—2921no—
441C22⋊C8—2921no—
442#97—2921no—
443#97—2921no—
444C4×C8—2921no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—731no—
449#95—731no—
450#95—731no—
451#95—731no—
452#95—731no—
453#95—731no—
454#95C2×#6440811yescharacteristic
455C24C22×#6440811yescharacteristic
456C24C22×#6440811yescharacteristic
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4C22×#6440811yescharacteristic
481C22×C4C22×#6440811yescharacteristic
482C22×C4C22×#6440811yescharacteristic
483C22×C4C22×#6440811yescharacteristic
484C22×C4C22×#6440811yescharacteristic
485M16—2924no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
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—
510C4⋊C4—41no—
511C4⋊C4—2921no—
512C4⋊C4—2921no—
513C4⋊C4—2921no—
514C4⋊C4—2921no—
515C4⋊C4—2921no—
516C4⋊C4—1462no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—2921no—
522C22⋊C4—2921no—
523C22⋊C4—2921no—
524C22⋊C4—2921no—
525C22⋊C4—2921no—
526C22⋊C4—2921no—
527C42—1461no—
528C42—1461no—
529C42—1461no—
530C42—1461no—
531C42—1461no—
532C42—1461no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C42—1461no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C23×#6440811yescharacteristic
552D4—44no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—731no—
576C2×C4—731no—
577C2×C4—731no—
578C2×C4—731no—
579C2×C4—731no—
580C2×C4—731no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
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#1455667812yes—
603C8—2921no—
604C8—2921no—
605C8—2921no—
606C8—2921no—
607C8—2921no—
608C8—2921no—
609C8—2921no—
610C8—2921no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22#1798965411yescharacteristic
626C22#1798965611yescharacteristic
627C22#1798965611yescharacteristic
628C22C2×#1455667811yescharacteristic
629C22#1798966511yescharacteristic
630C22#1798966511yescharacteristic
631C22#1798966611yescharacteristic
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—2921no—
638C4—2921no—
639C4—2921no—
640C4—2921no—
641C4—2921no—
642C4—2921no—
643C4—2921no—
644C4—731no—
645C4—731no—
646C4—731no—
647C4—731no—
648C2—81no—
649C2—81no—
650C2#2262647111yescharacteristic
651C2#2262680711yescharacteristic
652C2#2262682111yescharacteristic
653C2#2262686711yescharacteristic
654C2#2262687311yescharacteristic
655C2#2262688511yescharacteristic
656C2#2262688711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#394×C73C4
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic73C24
hypercenter#394#64408
hypocenterC73#23321
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. #22626032C2semidirect
3. #22626029C2semidirect
4. #22626005C2semidirect
5. #22626005C2semidirect
6. #22626000C2semidirect
7. #22626000C2semidirect
8. #22625989C2semidirect
9. #22625988C2semidirect
11. #22625878C2semidirect
12. #22625878C2semidirect
13. #22625789C2semidirect
14. #22625789C2semidirect
344. C73#23321semidirect

Maximal subgroups

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

subgroupindex
#226260552
#226260322
#226260292
#226260052
#226260002
#226259892
#226259882
#226259082
#226258782
#226257892
#226131872
#2332173