Finite Group Databasebeta

← #34019512

Subgroups — #34019512

Order 18688.

674 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#22626863C211yesmaximal, characteristic, complemented
2#22626792C211yesmaximal, characteristic, complemented
3#22625579C211yesmaximal, characteristic, complemented
4#22625562C211yesmaximal, characteristic, complemented
5#22625558C211yesmaximal, characteristic, complemented
6#22625553C211yesmaximal, characteristic, complemented
7#22625538C211yesmaximal, characteristic, complemented
8#22625533C211yesmaximal, characteristic, complemented
9#22625521C211yesmaximal, characteristic, complemented
10#22625515C211yesmaximal, characteristic, complemented
11#22625495C211yesmaximal, characteristic, complemented
12#22625262C211yesmaximal, characteristic, complemented
13#22625258C211yesmaximal, characteristic, complemented
14#22625150C211yesmaximal, characteristic, complemented
15#22618744C211yesmaximal, characteristic
16Q8×#10596833C2211yescharacteristic, complemented
17Q8×#10596833C2211yescharacteristic, complemented
18#17989622—21no—
19#17989622—21no—
20#17989622C2211yescharacteristic, complemented
21#17989622C2211yescharacteristic, complemented
22#17989621C2211yescharacteristic
23#17989621C2211yescharacteristic
24#17989617—21no—
25#17989617—21no—
26#17989615C2211yescharacteristic
27#17989615C2211yescharacteristic
28#17989608C2211yescharacteristic
29#17989604—21no—
30#17989604C2211yescharacteristic
31#17989604C2211yescharacteristic
32#17989590—21no—
33#17989590C2211yescharacteristic
34#17989558—21no—
35#17989555—21no—
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40Q8⋊#10596833C2211yescharacteristic, complemented
41Q8⋊#10596833C2211yescharacteristic, complemented
42#17989435C2211yescharacteristic
43#17989435C2211yescharacteristic
44#17989435C2211yescharacteristic
45#17989435C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50#17989428C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52C2×#14556616C411yescharacteristic
53#41×Dic73C2211yescharacteristic
54C2×#14556553C2211yescharacteristic
55#17988696C411yescharacteristic
56#17988696C411yescharacteristic
57#17988690C411yescharacteristic
58#17988690C411yescharacteristic
59#17988672C2211yescharacteristic
60#17988672C2211yescharacteristic
61#17988667C2211yescharacteristic
62#17988667C2211yescharacteristic
63#17988627C411yescharacteristic
64#17988593C411yescharacteristic
65M16×Dic73C2211yescharacteristic
66#417×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924037—21no—
70C2×#11924037—21no—
71C2×#11924037—21no—
72C2×#11924036—21no—
73C2×#11924036—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658D411yescharacteristic
77#14556658D411yescharacteristic
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556648—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833D411yescharacteristic
93C22⋊#10596833D411yescharacteristic
94#14556646—21no—
95C4×#10596833—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C22×D292C2×C411yescharacteristic
102C22×C4×D73—21no—
103#14556616—24no—
104C2×#11924015C2×C411yescharacteristic
105C2×#11924015C2×C411yescharacteristic
106C2×#11924013C2×C411yescharacteristic
107C2×#11924013C2×C411yescharacteristic
108C2×#11924012C2×C411yescharacteristic
109Q8×Dic73C2×C411yescharacteristic
110Q8×Dic73C2311yescharacteristic
111D4×Dic73—21no—
112D4×Dic73—21no—
113D4×Dic73C2×C411yescharacteristic
114D4×Dic73C2311yescharacteristic
115#14556553—21no—
116#14556553—21no—
117#14556553C2×C411yescharacteristic
118#14556553C2×C411yescharacteristic
119#14556553C2311yescharacteristic
120#14556553C2311yescharacteristic
121#14556550—21no—
122#14556550C2×C411yescharacteristic
123C2×#11924005—21no—
124C2×C4×Dic73—21no—
125C2×C4×Dic73C2311yescharacteristic
126#14556524C2×C411yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556522C2×C411yescharacteristic
129#14556522C2×C411yescharacteristic
130#14556513C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556513C2×C411yescharacteristic
133#14556513C2×C411yescharacteristic
134#14556501C2×C411yescharacteristic
135#14556501C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C2×#41×C73C2×C411yescharacteristic
139C2×M16×C73C2×C411yescharacteristic
140C2×C4⋊C4×C73C2×C411yescharacteristic
141#103×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#102×C73C2×C411yescharacteristic
145#11924038—81no—
146#11924038—81no—
147#11924037—81no—
148#11924037—81no—
149#11924037—81no—
150#11924036—81no—
151#11924036—81no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C23×D73—21no—
163C2×D292—22no—
164C2×D292—21no—
165C2×D292—41no—
166C2×D292C22×C411yescharacteristic
167C2×D292C22×C411yescharacteristic
168C22×Dic73—21no—
169C22×Dic73—21no—
170C22×Dic73C2×D411yescharacteristic
171C22×Dic73#4111yescharacteristic
172C2×C4×D73—21no—
173C2×C4×D73—21no—
174C2×C4×D73—21no—
175C2×C4×D73—21no—
176C2×C4×D73—81no—
177C2×C4×D73—81no—
178#11924015—22no—
179#11924015M1614yescomplemented
180#11924013—22no—
181#11924013—22no—
182#11924012—22no—
183C2×#10596831C22×C411yescharacteristic
184C2×#10596831C22×C411yescharacteristic
185C4⋊Dic73—21no—
186C4⋊Dic73—21no—
187C4⋊Dic73C22×C411yescharacteristic
188C4⋊Dic73C22×C411yescharacteristic
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192#11924005—21no—
193#11924005—21no—
194#11924005—21no—
195#11924005—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73C22×C411yescharacteristic
200C4×Dic73C2×D411yescharacteristic
201C4×Dic73#4111yescharacteristic
202C4×Dic73C2411yescharacteristic
203#41×C73—22no—
204#41×C73—41no—
205C2×Q8×C73C22×C411yescharacteristic
206C2×D4×C73—21no—
207C2×D4×C73C22×C411yescharacteristic
208C22×C4×C73—21no—
209C22×C4×C73—21no—
210C22×C4×C73C22×C411yescharacteristic
211M16×C73—41no—
212C2×C8×C73C22×C411yescharacteristic
213C2×C8×C73C22×C411yescharacteristic
214C4⋊C4×C73—21no—
215C4⋊C4×C73C22×C411yescharacteristic
216C4⋊C4×C73C22×C411yescharacteristic
217C23×C73—21no—
218C23×C73C4×D411yescharacteristic
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—42no—
222C22×D73—42no—
223C22×D73—81no—
224Q8×C73—21no—
225Q8×C73C2×M1612yes—
226D4×C73—21no—
227D4×C73—21no—
228D4×C73—42no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—42no—
235C2×C4×C73—81no—
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C23×C411yescharacteristic
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—41no—
243C2×Dic73—42no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73C4×D411yescharacteristic
247C2×Dic73C4×D411yescharacteristic
248C2×Dic73C22×D411yescharacteristic
249C2×Dic73C2×#4111yescharacteristic
250D292—21no—
251D292—21no—
252D292—21no—
253D292—42no—
254D292C2×M1612yes—
255C4×D73—41no—
256C4×D73—41no—
257C4×D73—41no—
258C4×D73—41no—
259C4×D73—81no—
260C4×D73—81no—
261#10596833—21no—
262#10596833—21no—
263#10596833—21no—
264#10596833—21no—
265#10596833—41no—
266#10596833—41no—
267#10596833—41no—
268#10596833—41no—
269#10596833—41no—
270#10596833—41no—
271C8×C73—21no—
272C8×C73—21no—
273#10596831—21no—
274#10596831C2×M1612yes—
275C22×C73—21no—
276C22×C73—21no—
277C22×C73—21no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73C2×C4×D411yescharacteristic
282C2×D73—22no—
283C2×D73—41no—
284C2×D73—41no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—84no—
289C4×C73—21no—
290C4×C73—22no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73C22×M1611yescharacteristic
294C4×C73#56811yescharacteristic
295Dic73—21no—
296Dic73—21no—
297Dic73—21no—
298Dic73—41no—
299Dic73—41no—
300Dic73#57611yescharacteristic
301Dic73#57711yescharacteristic
302#18052—731nomaximal, Sylow
303C2×C73—41no—
304C2×C73—81no—
305C2×C73D4×M1611yescharacteristic
306C2×C73#294611yescharacteristic
307C2×C73#295511yescharacteristic
308D73—42no—
309D73—81no—
310#2991—731no—
311#2974—731no—
312C4×AGL(1,ℤ/8)—731no—
313#1594—731no—
314#1589—731no—
315#1588—731no—
316#1586—731no—
317#1583—731no—
318#1581—731no—
319SD16⋊C8—731no—
320C8×SD16—731no—
321#1569—731no—
322#1488—731no—
323#1486—731no—
324#1453—731no—
325C73#1805211yescharacteristic, complemented, Sylow
326C2×AGL(1,ℤ/8)—731no—
327C4×#41—731no—
328C2×C4×D4—731no—
329C8×Q8—731no—
330C8×Q8—731no—
331#442—731no—
332#442—731no—
333#440—731no—
334#440—731no—
335C4×SD16—731no—
336C4×SD16—731no—
337C4×D8—731no—
338C4×D8—731no—
339#436—731no—
340#436—731no—
341#436—731no—
342#436—731no—
343#436—1461no—
344#436—1461no—
345#435—1461no—
346#435—1461no—
347C8×D4—731no—
348C8×D4—731no—
349#432—731no—
350#431—731no—
351#431—731no—
352#431—1461no—
353#428—731no—
354#423—731no—
355#423—1461no—
356#418—731no—
357#417#6440811yescharacteristic
358#407—1461no—
359C2×C22⋊C8—1461no—
360C4×M16—731no—
361#335—731no—
362#335—731no—
363#334—731no—
364#334—731no—
365Q8⋊C8—731no—
366Q8⋊C8—731no—
367D4⋊C8—731no—
368D4⋊C8—731no—
369D4⋊C8—731no—
370D4⋊C8—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373#322—731no—
374#322—731no—
375#322—731no—
376C82—731no—
377C2×#41C2×#6440811yescharacteristic
378C22×D4—731no—
379C23×C4—1461no—
380AGL(1,ℤ/8)—1464no—
381C2×SD16—731no—
382C2×SD16—731no—
383C2×D8—731no—
384C2×D8—731no—
385C2×M16—731no—
386C2×M16—1461no—
387C2×M16—1461no—
388C2×M16—1461no—
389C2×M16—1461no—
390C2×M16—1461no—
391C2×M16C2×#6440811yescharacteristic
392C22×C8—1461no—
393C22×C8—1461no—
394C4×Q8—731no—
395C4×Q8—731no—
396C4×D4—731no—
397C4×D4—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—731no—
402C4×D4—1461no—
403C4×D4—1461no—
404C4×D4—1461no—
405C4×D4—1461no—
406#117—731no—
407#117—1461no—
408C2×C4⋊C4C2×#6440811yescharacteristic
409C2×C22⋊C4—1461no—
410C2×C42—731no—
411C2×C42—1461no—
412#107—731no—
413#107—731no—
414#106—731no—
415#106—731no—
416C4⋊C8—731no—
417C4⋊C8—731no—
418C4⋊C8—731no—
419C4⋊C8—731no—
420C4⋊C8—731no—
421C4⋊C8—731no—
422C4⋊C8—1461no—
423C4⋊C8—1461no—
424C4⋊C8—1461no—
425C4⋊C8—1461no—
426#103C2×#6440811yescharacteristic
427#103C2×#6440811yescharacteristic
428#102—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102C2×#6440811yescharacteristic
433#102C2×#6440811yescharacteristic
434C22⋊C8—731no—
435C22⋊C8—731no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439C22⋊C8—1461no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444#97—731no—
445#97—731no—
446#97—1461no—
447C4×C8—731no—
448C4×C8—731no—
449C4×C8—731no—
450C4×C8—731no—
451C4×C8—731no—
452C4×C8—731no—
453C4×C8—1461no—
454C4×C8—1461no—
455C24—1461no—
456#41—22no—
457#41—41no—
458C2×Q8C22×#6440811yescharacteristic
459C2×D4—21no—
460C2×D4—2921no—
461C2×D4—731no—
462C2×D4—731no—
463C2×D4—1462no—
464C2×D4—1461no—
465C2×D4C22×#6440811yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—5841no—
469C22×C4—5841no—
470C22×C4—731no—
471C22×C4—731no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4C22×#6440811yescharacteristic
479SD16—734no—
480SD16—1462no—
481D8—1462no—
482D8—1462no—
483M16—41no—
484M16—5841no—
485M16—5841no—
486M16—5841no—
487M16—5841no—
488M16—5841no—
489M16—1462no—
490C2×C8—5841no—
491C2×C8—5841no—
492C2×C8—731no—
493C2×C8—731no—
494C2×C8—731no—
495C2×C8—731no—
496C2×C8—731no—
497C2×C8—731no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8C22×#6440811yescharacteristic
505C2×C8C22×#6440811yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—731no—
508C4⋊C4—731no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4C22×#6440811yescharacteristic
512C4⋊C4C22×#6440811yescharacteristic
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C22⋊C4—1461no—
516C22⋊C4—1461no—
517C22⋊C4—1461no—
518C22⋊C4—1461no—
519C22⋊C4—1461no—
520C42—731no—
521C42—731no—
522C42—731no—
523C42—731no—
524C42—1461no—
525C42—1461no—
526C42—1461no—
527C23—21no—
528C23—2922no—
529C23—2922no—
530C23—5841no—
531C23—1461no—
532C23—1461no—
533C23D4×#6440811yescharacteristic
534Q8—21no—
535Q8D73⋊M1612yes—
536D4—21no—
537D4—21no—
538D4—42no—
539D4—2922no—
540D4—732no—
541D4—1461no—
542D4—1461no—
543D4—1461no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—81no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2922no—
558C2×C4—2921no—
559C2×C4—5841no—
560C2×C4—5841no—
561C2×C4—731no—
562C2×C4—731no—
563C2×C4—731no—
564C2×C4—731no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4D4×#6440811yescharacteristic
570C2×C4C23×#6440811yescharacteristic
571C8—21no—
572C8—21no—
573C8—2921no—
574C8—2921no—
575C8—2921no—
576C8—2921no—
577C8—2921no—
578C8—2921no—
579C8—732no—
580C8—1461no—
581C8—1461no—
582C8—1461no—
583C8—1461no—
584C8—1461no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—2921no—
592C22—2921no—
593C22—2921no—
594C22—2921no—
595C22—2921no—
596C22—5844no—
597C22—1462no—
598C22C2×D4×#6440811yescharacteristic
599C4—21no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—2921no—
604C4—2921no—
605C4—731no—
606C4—731no—
607C4—1461no—
608C4—1461no—
609C4—1461no—
610C4C2×D73⋊M1611yescharacteristic
611C4#1798966511yescharacteristic
612C2—41no—
613C2—81no—
614C2—2922no—
615C2—5841no—
616C2#2262666911yescharacteristic
617C2#2262668811yescharacteristic
618C2#2262670411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#417×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×#14556616C4
cosocleC4×Dic73C24
hypercenter#417#64408
hypocenterC73#18052
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22626863C2semidirect
2. #22626792C2semidirect
3. #22625579C2semidirect
4. #22625562C2semidirect
5. #22625558C2semidirect
6. #22625553C2semidirect
7. #22625538C2semidirect
8. #22625533C2semidirect
9. #22625521C2semidirect
10. #22625515C2semidirect
11. #22625495C2semidirect
12. #22625262C2semidirect
13. #22625258C2semidirect
14. #22625150C2semidirect
16. Q8×#10596833C22semidirect
17. Q8×#10596833C22semidirect
20. #17989622C22semidirect
21. #17989622C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. Q8⋊#10596833C22semidirect
41. Q8⋊#10596833C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
179. #11924015M16semidirect
325. C73#18052semidirect

Maximal subgroups

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

subgroupindex
#226268632
#226267922
#226255792
#226255622
#226255582
#226255532
#226255382
#226255332
#226255212
#226255152
#226254952
#226252622
#226252582
#226251502
#226187442
#1805273