Finite Group Databasebeta

← #34027615

Subgroups — #34027615

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626055C211yesmaximal, characteristic
2#22626052C211yesmaximal, characteristic
3#22626005C211yesmaximal, characteristic, complemented
4#22626005C211yesmaximal, characteristic, complemented
5#22626004C211yesmaximal, characteristic, complemented
6#22626001C211yesmaximal, characteristic, complemented
7#22625849C211yesmaximal, characteristic, complemented
8#22625824C211yesmaximal, characteristic, complemented
9#22625821C211yesmaximal, characteristic, complemented
10#22625730C211yesmaximal, characteristic, complemented
11#22625726C211yesmaximal, characteristic, complemented
12#22625719C211yesmaximal, characteristic, complemented
13#22625661C211yesmaximal, characteristic, complemented
14#22625657C211yesmaximal, characteristic, complemented
15#22612371C211yesmaximal, characteristic
16C2×#14556658C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×C4⋊#10596833C2211yescharacteristic
19C2×C4⋊#10596833C2211yescharacteristic
20C2×C22⋊#10596833C2211yescharacteristic
21C2×C22⋊#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×#14556646C2211yescharacteristic
32C2×C4×#10596833C2211yescharacteristic
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
46C22×C4×Dic73C2211yescharacteristic
47#17988481C411yescharacteristic
48#17988475C411yescharacteristic
49#17988472C411yescharacteristic
50#17988470C411yescharacteristic
51#17988467C411yescharacteristic
52#17988460C411yescharacteristic
53#17988459C411yescharacteristic
54#17988438C2211yescharacteristic
55C22⋊C4×Dic73C2211yescharacteristic
56C22⋊C4×Dic73C2211yescharacteristic
57#17988432C2211yescharacteristic
58#386×C73C411yescharacteristic
59C22×#10596833C2311yescharacteristic
60C22×#10596833C2311yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67#14556658—22no—
68#14556658—22no—
69C4⋊#10596833—22no—
70C4⋊#10596833—22no—
71C22⋊#10596833—22no—
72C22⋊#10596833—22no—
73C22⋊#10596833—22no—
74C22⋊#10596833—22no—
75C22⋊#10596833—41no—
76C22⋊#10596833—41no—
77C22⋊#10596833—41no—
78C22⋊#10596833—41no—
79#14556646—22no—
80#14556646—41no—
81#14556646—41no—
82#14556646—41no—
83C4×#10596833—22no—
84C4×#10596833—41no—
85C23×Dic73C2×C411yescharacteristic
86C23×Dic73C2311yescharacteristic
87C23×Dic73C2311yescharacteristic
88D4×Dic73—22no—
89D4×Dic73—22no—
90#14556547—41no—
91#14556547—41no—
92C2×C4⋊Dic73C2×C411yescharacteristic
93C2×#11924007C2×C411yescharacteristic
94C2×#11924007C2×C411yescharacteristic
95C2×#11924007C2×C411yescharacteristic
96C2×#11924007C2×C411yescharacteristic
97C2×C22⋊Dic73C2×C411yescharacteristic
98C2×C22⋊Dic73C2×C411yescharacteristic
99C2×C22⋊Dic73C2×C411yescharacteristic
100C2×C22⋊Dic73C2×C411yescharacteristic
101C2×C22⋊Dic73C2×C411yescharacteristic
102C2×C4×Dic73—41no—
103C2×C4×Dic73—41no—
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×D4×C73C2×C411yescharacteristic
120C23×C4×C73C2×C411yescharacteristic
121C2×C22⋊C4×C73C2×C411yescharacteristic
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124#95×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
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—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833#4112yes—
149C2×#10596833#4112yes—
150C22×Dic73—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—41no—
163C22×Dic73—41no—
164C22×Dic73—41no—
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—22no—
174#11924007—41no—
175#11924007—41no—
176#11924007—41no—
177#11924007—41no—
178C22⋊Dic73—22no—
179C22⋊Dic73—22no—
180C22⋊Dic73—22no—
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—41no—
192C4×Dic73—41no—
193C4×Dic73#4112yes—
194C4×Dic73#4112yes—
195C24×C73C22×C411yescharacteristic
196C24×C73C22×C411yescharacteristic
197C2×D4×C73—22no—
198C2×D4×C73—22no—
199C2×D4×C73—41no—
200C22×C4×C73—22no—
201C22×C4×C73—41no—
202C22×C4×C73—41no—
203C22×C4×C73C22×C411yescharacteristic
204C22×C4×C73C22×C411yescharacteristic
205C22×C4×C73C22×C411yescharacteristic
206C22×C4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208C22⋊C4×C73—41no—
209C22⋊C4×C73—41no—
210C22⋊C4×C73—41no—
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×C73—21no—
223C23×C73—21no—
224C23×C73#13112yes—
225C23×C73C23×C411yescharacteristic
226D4×C73—44no—
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—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—44no—
242C2×C4×C73#11712yes—
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—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—44no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73#11712yes—
273C2×Dic73C2×#4111yescharacteristic
274C2×Dic73C2×#4111yescharacteristic
275C2×Dic73C2×#4111yescharacteristic
276C2×Dic73C2×#4111yescharacteristic
277C2×Dic73H(5,𝔽2)11yescharacteristic
278C2×Dic73H(5,𝔽2)11yescharacteristic
279#10596833—22no—
280#10596833—22no—
281#10596833—41no—
282#10596833—41no—
283#10596833—41no—
284#10596833—41no—
285#10596833—41no—
286#10596833—41no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
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×C73C2×#11711yescharacteristic
302C22×C73C4×#4111yescharacteristic
303C22×C73#51811yescharacteristic
304C22×C73C2×#13111yescharacteristic
305C22×C73#56811yescharacteristic
306C22×C73#56811yescharacteristic
307C22×C73#56811yescharacteristic
308C4×C73—22no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—41no—
312C4×C73—41no—
313Dic73—22no—
314Dic73—41no—
315Dic73—41no—
316Dic73—41no—
317Dic73—41no—
318Dic73—41no—
319Dic73—41no—
320Dic73#55311yescharacteristic
321Dic73#55311yescharacteristic
322Dic73#55911yescharacteristic
323Dic73#55911yescharacteristic
324#22841—731nomaximal, Sylow
325C2×C73—42no—
326C2×C73—81no—
327C2×C73#235711yescharacteristic
328C2×C73#292311yescharacteristic
329C2×C73#292411yescharacteristic
330C2×C73#296711yescharacteristic
331C2×C73#296811yescharacteristic
332C2×C73#297611yescharacteristic
333C2×C73#297811yescharacteristic
334#2290—731no—
335#1955—731no—
336#1954—731no—
337#1924—731no—
338#1924—731no—
339#1924—731no—
340#1923—731no—
341#1832—731no—
342#1818—731no—
343#1817—731no—
344#1771—731no—
345#1770—731no—
346#1768—731no—
347#1750—731no—
348#1749—731no—
349C73#2284111yescharacteristic, complemented, Sylow
350C2×C4×D4—731no—
351C2×#117—731no—
352C22×C42—731no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×C4⋊C8—731no—
356C2×C4⋊C8—731no—
357C2×C22⋊C8—731no—
358C2×C22⋊C8—731no—
359C2×C22⋊C8—731no—
360C2×C22⋊C8—731no—
361C2×C22⋊C8—731no—
362C2×C22⋊C8—731no—
363C2×C22⋊C8—731no—
364C2×C22⋊C8—731no—
365C2×#97—731no—
366C2×#97—731no—
367C2×#97—731no—
368C2×#97—731no—
369C2×C4×C8—731no—
370C2×C4×C8—731no—
371#388—731no—
372#388—731no—
373#387—731no—
374#387—731no—
375#386—731no—
376#386#6440811yescharacteristic
377#385—731no—
378#385—731no—
379C4×C4⋊C4—731no—
380C4×C22⋊C4—731no—
381C4×C22⋊C4—731no—
382#376—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—
393C22×D4C2×#6440811yescharacteristic
394C23×C4—731no—
395C23×C4—731no—
396C23×C4—731no—
397C23×C4C2×#6440811yescharacteristic
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C4×D4—1462no—
407C4×D4—1462no—
408#117—2921no—
409#117—2921no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—731no—
417C2×C22⋊C4—731no—
418C2×C22⋊C4—731no—
419C2×C22⋊C4—731no—
420C2×C22⋊C4C2×#6440811yescharacteristic
421C2×C22⋊C4C2×#6440811yescharacteristic
422C2×C22⋊C4C2×#6440811yescharacteristic
423C2×C42—2921no—
424C2×C42—2921no—
425C2×C42—731no—
426C2×C42—731no—
427C2×C42—731no—
428C2×C42—731no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—731no—
432C4⋊C8—1462no—
433C4⋊C8—1462no—
434C4⋊C8—1462no—
435C4⋊C8—1462no—
436C22⋊C8—2921no—
437C22⋊C8—2921no—
438C22⋊C8—2921no—
439C22⋊C8—2921no—
440C22⋊C8—1462no—
441C22⋊C8—1462no—
442C22⋊C8—1462no—
443C22⋊C8—1462no—
444#97—2921no—
445#97—2921no—
446#97—2921no—
447#97—1462no—
448C4×C8—2921no—
449C4×C8—1462no—
450#95—731no—
451#95—731no—
452#95—731no—
453#95—731no—
454#95—731no—
455#95—731no—
456#95—731no—
457#95—731no—
458#95C2×#6440811yescharacteristic
459#95C2×#6440811yescharacteristic
460C24C22×#6440811yescharacteristic
461C24C22×#6440811yescharacteristic
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—2921no—
469C22×C4—2921no—
470C22×C4—2921no—
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
496C2×C8—732no—
497C2×C8—732no—
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—
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—
520C4⋊C4—2921no—
521C4⋊C4—2921no—
522C4⋊C4—2921no—
523C4⋊C4—2921no—
524C4⋊C4—1462no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—2921no—
529C22⋊C4—2921no—
530C22⋊C4—1462no—
531C22⋊C4—1462no—
532C22⋊C4—1462no—
533C42—2921no—
534C42—2921no—
535C42—732no—
536C42—732no—
537C42—1461no—
538C42—1461no—
539C42—1461no—
540C42—1461no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23#1455667812yes—
559C23C23×#6440811yescharacteristic
560D4—44no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
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—44no—
576C2×C4—2924no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—731no—
586C2×C4—731no—
587C2×C4—731no—
588C2×C4—731no—
589C2×C4—731no—
590C2×C4—731no—
591C2×C4—732no—
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—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#1455667012yes—
613C8—2921no—
614C8—2921no—
615C8—2921no—
616C8—2921no—
617C8—2921no—
618C8—2921no—
619C8—1462no—
620C8—1462no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1455667011yescharacteristic
636C22#41×#6440811yescharacteristic
637C22#1798965411yescharacteristic
638C22C2×#1455667811yescharacteristic
639C22#1798966511yescharacteristic
640C22#1798966511yescharacteristic
641C22#1798966611yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—2921no—
648C4—2921no—
649C4—2921no—
650C4—2921no—
651C4—2921no—
652C4—2921no—
653C4—731no—
654C4—731no—
655C4—731no—
656C4—731no—
657C4—1462no—
658C2—42no—
659C2—81no—
660C2#2262645211yescharacteristic
661C2#2262661811yescharacteristic
662C2#2262662011yescharacteristic
663C2#2262675011yescharacteristic
664C2#2262675811yescharacteristic
665C2#2262679711yescharacteristic
666C2#2262680811yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#386×C73C4
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic73C24
hypercenter#386#64408
hypocenterC73#22841
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #22626005C2semidirect
4. #22626005C2semidirect
5. #22626004C2semidirect
6. #22626001C2semidirect
7. #22625849C2semidirect
8. #22625824C2semidirect
9. #22625821C2semidirect
10. #22625730C2semidirect
11. #22625726C2semidirect
12. #22625719C2semidirect
13. #22625661C2semidirect
14. #22625657C2semidirect
349. C73#22841semidirect

Maximal subgroups

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

subgroupindex
#226260552
#226260522
#226260052
#226260042
#226260012
#226258492
#226258242
#226258212
#226257302
#226257262
#226257192
#226256612
#226256572
#226123712
#2284173