Finite Group Databasebeta

← #33056882

Subgroups — #33056882

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619937C211yesmaximal, characteristic
2#22618879C211yesmaximal, characteristic, complemented
3D4×#11924011C211yesmaximal, characteristic
4#22608821C211yesmaximal, characteristic, complemented
5#22608820C211yesmaximal, characteristic, complemented
6#22608776C211yesmaximal, characteristic, complemented
7#22608771C211yesmaximal, characteristic, complemented
8#22608764C211yesmaximal, characteristic, complemented
9SD16×#10596831C211yesmaximal, characteristic, complemented
10C8×#11924016C211yesmaximal, characteristic, complemented
11#22608700C211yesmaximal, characteristic, complemented
12#22608384C211yesmaximal, characteristic, complemented
13D4⋊C8×D73C211yesmaximal, characteristic
14#22608210C211yesmaximal, characteristic, complemented
15C73×#1582C211yesmaximal, characteristic, complemented
16C22×#11924011—21no—
17#17989141C2211yescharacteristic
18C4×D4×D73C2211yescharacteristic
19#17988919C411yescharacteristic
20#17988916C411yescharacteristic
21#17988876C411yescharacteristic
22#17988875C411yescharacteristic
23#17988791C411yescharacteristic
24#17988790C411yescharacteristic
25#17988739C411yescharacteristic
26#17988736C411yescharacteristic
27#17988715—21no—
28#17988714—21no—
29#17988714C2211yescharacteristic, complemented
30Q8×#10596831C2211yescharacteristic, complemented
31C8×Dic146C2211yescharacteristic, complemented
32C4×#11924017C2211yescharacteristic
33C4×#11924015C2211yescharacteristic
34C4×#11924016C2211yescharacteristic
35#17988666C2211yescharacteristic
36#17988664C2211yescharacteristic
37#17988658—21no—
38#17988655—21no—
39D4×#10596831C2211yescharacteristic
40#17988638C2211yescharacteristic
41#17988612—21no—
42#17988609C2211yescharacteristic
43C4⋊C8×D73C2211yescharacteristic
44#17988549—21no—
45#17988544—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic, complemented
49#17988337C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52Q8⋊#10596831C2211yescharacteristic, complemented
53#17988317C2211yescharacteristic, complemented
54#17988316C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58C8×#10596831C2211yescharacteristic, complemented
59C8×#10596831C2211yescharacteristic, complemented
60#446×C73C2211yescharacteristic, complemented
61C4×SD16×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic, complemented
64Q8⋊C8×C73C2211yescharacteristic, complemented
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924015C2×C411yescharacteristic
72C2×#11924011—21no—
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011—44no—
76C2×#11924011D411yescharacteristic
77C2×#11924011D411yescharacteristic
78C22×#10596831—21no—
79C2×C8×D73—21no—
80C2×C8×D73—21no—
81#14556590C2×C411yescharacteristic
82#14556589C2×C411yescharacteristic
83#14556581—21no—
84#14556571C2×C411yescharacteristic
85#14556569—21no—
86#14556567C2×C411yescharacteristic
87C4×Dic146—21no—
88C4×Dic146C2311yescharacteristic
89C4×D292C2311yescharacteristic
90D4×Dic73C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92C4×C22⋊D73—21no—
93#14556551—21no—
94#14556548—21no—
95C4⋊C4×D73C2×C411yescharacteristic
96C22⋊C4×D73—21no—
97C42×D73C2311yescharacteristic
98#14556523C2×C411yescharacteristic
99#14556522C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101C4⋊#10596831—21no—
102C4⋊#10596831C2311yescharacteristic
103C4⋊#10596831C2311yescharacteristic
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520C2×C411yescharacteristic
109#14556520C2×C411yescharacteristic
110#14556516C2×C411yescharacteristic
111#14556515C2×C411yescharacteristic
112#14556514C2×C411yescharacteristic
113#14556513C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116C22⋊#10596831—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556501—21no—
122#14556501D411yescharacteristic
123#14556501D411yescharacteristic
124C4×#10596831—21no—
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C8×Dic73—21no—
129C8×Dic73—21no—
130C2×SD16×C73C2×C411yescharacteristic
131C22×C8×C73—21no—
132C4×Q8×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#106×C73C2×C411yescharacteristic
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141#97×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C23×D73—21no—
146C2×Dic146—21no—
147C2×Dic146C22×C411yescharacteristic
148C2×D292C22×C411yescharacteristic
149C2×C22⋊D73—21no—
150D4×D73—22no—
151D4×D73—41no—
152C22×Dic73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73C22×C411yescharacteristic
159C2×C4×D73C2×D411yescharacteristic
160C2×C4×D73#4111yescharacteristic
161#11924017—41no—
162#11924016—22no—
163#11924015—41no—
164#11924011—22no—
165#11924011—42no—
166#11924011—42no—
167#11924011—42no—
168#11924011SD1614yescomplemented
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—42no—
173C2×#10596831C22×C411yescharacteristic
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C8×D73—42no—
178C8×D73—42no—
179C4⋊Dic73—21no—
180C4⋊Dic73C22×C411yescharacteristic
181C4⋊Dic73C22×C411yescharacteristic
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188C22⋊Dic73—21no—
189#11924005—21no—
190#11924005—21no—
191#11924005—21no—
192#11924005—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C2×Q8×C73C22×C411yescharacteristic
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200SD16×C73—22no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—42no—
205C2×C8×C73C22×C411yescharacteristic
206C2×C8×C73C22×C411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C4⋊C4×C73—21no—
210C4⋊C4×C73C22×C411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C22⋊C4×C73—21no—
213C42×C73—21no—
214C42×C73C2411yescharacteristic
215C23×C73—21no—
216C22×D73—21no—
217C22×D73—42no—
218C22×D73—42no—
219C22×D73—81no—
220C22×D73C4×D411yescharacteristic
221Q8×C73—21no—
222Q8×C73—21no—
223D4×C73—21no—
224D4×C73#13112yes—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C22⋊D73—42no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—42no—
238C2×Dic73C4×D411yescharacteristic
239C2×Dic73C4×D411yescharacteristic
240C2×Dic73C4×D411yescharacteristic
241D292—21no—
242D292—21no—
243C4×D73—21no—
244C4×D73—21no—
245C4×D73—41no—
246C4×D73—42no—
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—42no—
250C4×D73C2×SD1612yes—
251C4×D73#13512yes—
252Dic146—21no—
253Dic146—21no—
254Dic146—81no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—21no—
258C8×C73—41no—
259C8×C73—41no—
260C8×C73—41no—
261C8×C73C2×SD1612yes—
262#10596831—22no—
263#10596831—21no—
264#10596831—21no—
265#10596831—21no—
266#10596831—41no—
267#10596831—41no—
268#10596831C2×SD1612yes—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73C2×C4×D411yescharacteristic
273C2×D73—21no—
274C2×D73—21no—
275C2×D73—41no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73—84no—
279C2×D73C4×SD1612yes—
280C4×C73—21no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—41no—
284C4×C73C2×#13111yescharacteristic
285C4×C73#56811yescharacteristic
286C4×C73C22×SD1611yescharacteristic
287C4×C73C2×#13511yescharacteristic
288Dic73—21no—
289Dic73—21no—
290Dic73—41no—
291Dic73—41no—
292Dic73—41no—
293Dic73C4×SD1612yes—
294#17995—731nomaximal, Sylow
295C2×C73—42no—
296C2×C73#293311yescharacteristic
297C2×C73C2×C4×SD1611yescharacteristic
298C2×C73#295611yescharacteristic
299D73—22no—
300D73—81no—
301#3101—731no—
302#2962—731no—
303D4×M16—731no—
304#1590—731no—
305#1590—731no—
306#1583—731no—
307#1583—731no—
308#1582—731no—
309#1582D7311yescharacteristic, complemented
310C8×SD16—731no—
311C8×SD16—731no—
312#1570—731no—
313#1493—731no—
314C2×D4⋊C8—731no—
315C8×M16—731no—
316C73#1799511yescharacteristic, complemented, Sylow
317C22×M16—1461no—
318#533—731no—
319C2×C4×D4—731no—
320#483—731no—
321#483—731no—
322#476—731no—
323#476—731no—
324#461—731no—
325#461—731no—
326#450—731no—
327#450—731no—
328#446—731no—
329#446—1461no—
330#446—1461no—
331#446C2×D7311yescharacteristic, complemented
332C8×Q8—731no—
333C8×Q8—731no—
334C4×SD16—731no—
335C4×SD16—731no—
336C4×SD16—731no—
337C4×SD16C2×D7311yescharacteristic
338#436—731no—
339#436—731no—
340#435—1461no—
341#435—1461no—
342C8×D4—731no—
343C8×D4C2×D7311yescharacteristic
344#431—731no—
345#424—1461no—
346#423—731no—
347C2×C4⋊C8—731no—
348#407—1461no—
349#405—1461no—
350C4×M16—731no—
351C2×C4×C8—731no—
352#334—731no—
353#334—731no—
354#334—731no—
355#334C2×D7311yescharacteristic, complemented
356Q8⋊C8—731no—
357Q8⋊C8—731no—
358Q8⋊C8—731no—
359Q8⋊C8C2×D7311yescharacteristic, complemented
360D4⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8C2×D7311yescharacteristic
364#322—731no—
365#322C2×D7311yescharacteristic, complemented
366C82—731no—
367C82—731no—
368C22×D4—731no—
369C23×C4—1461no—
370C2×SD16—731no—
371C2×SD16—731no—
372C2×SD16—731no—
373C2×SD16C4×D7311yescharacteristic
374C2×M16—2924no—
375C2×M16—731no—
376C2×M16—731no—
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16—1461no—
380C22×C8—21no—
381C22×C8—1461no—
382C22×C8—1461no—
383C22×C8—1461no—
384C4⋊Q8—731no—
385C4⋊Q8—731no—
386#125—1461no—
387C22⋊Q8—731no—
388C22⋊Q8—731no—
389C22⋊Q8—1461no—
390C4×Q8—731no—
391C4×Q8—1461no—
392C4×Q8C22×D7311yescharacteristic
393C4×D4—731no—
394C4×D4—731no—
395C4×D4—731no—
396C4×D4—1461no—
397C4×D4—1461no—
398C4×D4C22×D7311yescharacteristic
399#117—1461no—
400C2×C4⋊C4—731no—
401C2×C22⋊C4—1461no—
402C2×C42—731no—
403#106—731no—
404#106—731no—
405#106—731no—
406#106C4×D7311yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8—1461no—
417C4⋊C8C22×D7311yescharacteristic
418C4⋊C8C22×D7311yescharacteristic
419#103—731no—
420#103—731no—
421#103—731no—
422#103C4×D7311yescharacteristic
423#102—731no—
424#102—731no—
425#102—731no—
426#102C4×D7311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—731no—
429C22⋊C8—731no—
430C22⋊C8—1461no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433#97—21no—
434#97—731no—
435#97—731no—
436#97—1461no—
437C4×C8—731no—
438C4×C8—731no—
439C4×C8—731no—
440C4×C8—1461no—
441C4×C8—1461no—
442C4×C8—1461no—
443C4×C8C22×D7311yescharacteristic
444C4×C8C22×D7311yescharacteristic
445C4×C8C22×D7311yescharacteristic
446C24—1461no—
447C2×Q8—731no—
448C2×Q8—1461no—
449C2×Q8C2×C4×D7311yescharacteristic
450C2×D4—2921no—
451C2×D4—731no—
452C2×D4—1461no—
453C2×D4—1462no—
454C2×D4C2×C4×D7311yescharacteristic
455C22×C4—21no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465SD16—22no—
466SD16—2921no—
467SD16—2921no—
468SD16—1462no—
469M16—2922no—
470M16—2922no—
471M16—2922no—
472M16—734no—
473M16—1462no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—2922no—
479C2×C8—2922no—
480C2×C8—2922no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8C2×C4×D7311yescharacteristic
489C2×C8C2×C4×D7311yescharacteristic
490C2×C8D4×D7311yescharacteristic
491C2×C8D4×D7311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—731no—
494C4⋊C4—731no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4C2×C4×D7311yescharacteristic
503C4⋊C4C2×C4×D7311yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C42—21no—
511C42—731no—
512C42—731no—
513C42—731no—
514C42—1461no—
515C42C23×D7311yescharacteristic
516C23—21no—
517C23—2922no—
518C23—2922no—
519C23—5841no—
520C23—731no—
521C23—1461no—
522Q8—21no—
523Q8—21no—
524Q8—5841no—
525Q8—1461no—
526Q8—1461no—
527D4—21no—
528D4—2922no—
529D4—1461no—
530D4—1461no—
531D4#1455659912yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—2921no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—731no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—731no—
547C2×C4—731no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4C22×C4×D7311yescharacteristic
554C2×C4C2×D4×D7311yescharacteristic
555C2×C4#41×D7311yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—2921no—
563C8—2921no—
564C8—732no—
565C8—1462no—
566C8—1461no—
567C8—1461no—
568C8—1461no—
569C8SD16×D7312yes—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—2921no—
574C22—2921no—
575C22—2921no—
576C22—5844no—
577C22—732no—
578C22—1461no—
579C22—1461no—
580C22C4×D4×D7311yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—2921no—
586C4—2921no—
587C4—2921no—
588C4—732no—
589C4—1461no—
590C4—1461no—
591C4C2×#1455659911yescharacteristic
592C4#1798935211yescharacteristic
593C4C2×SD16×D7311yescharacteristic
594C4#135×D7311yescharacteristic
595C2—42no—
596C2—5841no—
597C2—1462no—
598C2#2261865711yescharacteristic
599C2C4×SD16×D7311yescharacteristic
600C2#2261883911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1582C2
socleC22×C73C2×C4×D4
prime radicalC2×D4×D73C2×C4
cosocleC42×C73C24
hypercenter#1582D73
hypocenterC73#17995
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #22618879C2semidirect
4. #22608821C2semidirect
5. #22608820C2semidirect
6. #22608776C2semidirect
7. #22608771C2semidirect
8. #22608764C2semidirect
9. SD16×#10596831C2semidirect
10. C8×#11924016C2semidirect
11. #22608700C2semidirect
12. #22608384C2semidirect
14. #22608210C2semidirect
15. C73×#1582C2semidirect
29. #17988714C22semidirect
30. Q8×#10596831C22semidirect
31. C8×Dic146C22semidirect
48. #17988338C22semidirect
49. #17988337C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. Q8⋊#10596831C22semidirect
53. #17988317C22semidirect
57. #17988309C22semidirect
58. C8×#10596831C22semidirect
59. C8×#10596831C22semidirect
60. #446×C73C22semidirect
63. #334×C73C22semidirect
64. Q8⋊C8×C73C22semidirect
66. #322×C73C22semidirect
168. #11924011SD16semidirect
309. #1582D73semidirect
316. C73#17995semidirect
331. #446C2×D73semidirect
355. #334C2×D73semidirect
359. Q8⋊C8C2×D73semidirect
365. #322C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226199372
#226188792
D4×#119240112
#226088212
#226088202
#226087762
#226087712
#226087642
SD16×#105968312
C8×#119240162
#226087002
#226083842
D4⋊C8×D732
#226082102
C73×#15822
#1799573