Finite Group Databasebeta

← #10829907

Subgroups — #10829907

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic, complemented
3#83314C211yesmaximal, characteristic
4#82456C211yesmaximal, characteristic, complemented
5#71868C212yesmaximal, complemented
6#71863C212yesmaximal, complemented
7#71840C211yesmaximal, characteristic, complemented
8#71816C211yesmaximal, characteristic
9#71666C212yesmaximal, complemented
10#71665C212yesmaximal, complemented
11#71637C211yesmaximal, characteristic, complemented
12#24183—31nomaximal, Sylow
13C3×#559—21no—
14C3×#557—21no—
15C3×#556—21no—
16C3×#556—22no—
17C3×#554—21no—
18C3×#552—22no—
19C3×#539—22no—
20C3×#536C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#529—22no—
24C3×#520C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×#951C2212yes—
27C2×#944C2211yescharacteristic
28C2×C4×Dic6C2211yescharacteristic
29C2×#837C2212yes—
30C2×#837C2212yes—
31#5232C2212yes—
32#5229C2211yescharacteristic
33C2×#836C2212yes—
34#5094—21no—
35#5091—21no—
36#5090—21no—
37#5075—21no—
38#5063—22no—
39#5054—22no—
40#5004C2212yes—
41#4995C2212yes—
42#4978—22no—
43#4977C2211yescharacteristic
44#4973C2211yescharacteristic
45#4933—22no—
46#4932C2211yescharacteristic
47#4924C2211yescharacteristic
48#4924C2211yescharacteristic
49#4532C2212yes—
50#4532C2212yes—
51#4530C2212yes—
52#3507D311yescharacteristic
53#3078—31no—
54#3037—31no—
55#2327—31no—
56#2038—32no—
57#2005—32no—
58#1995—31no—
59#1963—31no—
60#1882—32no—
61#1860—32no—
62#1801—31no—
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×Dic6C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D411yescharacteristic
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×#127D411yescharacteristic
71C3×#126—24no—
72C3×#126—24no—
73C3×#125—22no—
74C3×#125—24no—
75C3×#125D412yes—
76C3×#124—22no—
77C3×#124—22no—
78C3×#124D412yes—
79C3×#123—24no—
80C3×#123—24no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×#121—21no—
87C3×#121—21no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—24no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×#117—24no—
102C3×#117—22no—
103C3×#117C2312yes—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4D412yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C42D412yes—
108C2×C3×C42C2311yescharacteristic
109#951—24no—
110#951—42no—
111#944—22no—
112#944—22no—
113C2×#216—22no—
114C2×#214—22no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—22no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×#208C2312yes—
122C2×#208C2312yes—
123C4×Dic6—21no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127#837—24no—
128#837—42no—
129#836—42no—
130#835—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—22no—
135C3⋊#95—22no—
136C3⋊#95—22no—
137#810—22no—
138#810—22no—
139#808—22no—
140#807—22no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C2×#137—32no—
146#559—21no—
147#557—21no—
148#556—21no—
149#556—22no—
150#554—21no—
151#552—22no—
152#539—22no—
153#536C2×D312yes—
154#533C2×D311yescharacteristic
155#532C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157#529—22no—
158#520C2×D312yes—
159C2×C4×Q8—31no—
160C2×C4×Q8C2×D311yescharacteristic
161#479—62no—
162#477—61no—
163#475—61no—
164#464—62no—
165#462—61no—
166#460—61no—
167#428—32no—
168#423—31no—
169#423—31no—
170C2×#104—32no—
171C2×#104—32no—
172#419—32no—
173#417—32no—
174C2×#101—32no—
175#391—31no—
176#391—31no—
177#387—62no—
178#387—62no—
179#382—31no—
180#376—31no—
181#343—32no—
182#339—32no—
183#339—32no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8C2×D412yes—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C2×D412yes—
201C22×C3×C4—24no—
202C22×C3×C4—22no—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6#4112yes—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4C2×D412yes—
232C3×C4⋊C4C2×D412yes—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C42—24no—
240C3×C42—22no—
241C3×C42—22no—
242C3×C42—22no—
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2×D411yescharacteristic
246C3×C42C2×D411yescharacteristic
247#216—44no—
248#214—44no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251Dic3⋊C4—28no—
252Dic3⋊C4—41no—
253Dic3⋊C4—41no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—41no—
258C4×Dic3—41no—
259C4×Dic3#4112yes—
260#208—22no—
261#208—22no—
262#208—22no—
263#208—22no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—22no—
266C2×#41C22×D312yes—
267C22×Q8—31no—
268C22×Q8C22×D311yescharacteristic
269#137—64no—
270#137—122no—
271C2×Q16—62no—
272C2×SD16—62no—
273C2×M16—32no—
274C2×M16—32no—
275C4⋊Q8—22no—
276C4⋊Q8—62no—
277C4⋊Q8—62no—
278C4⋊Q8#6711yescharacteristic
279C4⋊Q8#6711yescharacteristic
280C4⋊Q8#6711yescharacteristic
281#127#6711yescharacteristic
282#126—24no—
283#126—24no—
284#125—22no—
285#125—24no—
286#125#6712yes—
287#124—22no—
288#124—22no—
289#124#6712yes—
290#123—24no—
291#123—24no—
292C22⋊Q8—24no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—21no—
298#121—21no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—61no—
308C4×Q8—61no—
309C4×Q8—61no—
310C4×Q8—61no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—21no—
315C4×D4—21no—
316#117—24no—
317#117—22no—
318#117C22×D312yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—62no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4#6712yes—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42#6712yes—
329C2×C42C22×D311yescharacteristic
330#107—62no—
331#106—62no—
332C4⋊C8—61no—
333C4⋊C8—61no—
334C4⋊C8—61no—
335C4⋊C8—61no—
336#104—64no—
337#104—122no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—62no—
342#101—122no—
343#95—61no—
344#95—61no—
345#95—62no—
346#95—62no—
347#95—62no—
348C23×C3—22no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—42no—
353C3×Q8—41no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—81no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—81no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—84no—
377C2×C3×C4#12011yescharacteristic
378C2×C3×C4#12011yescharacteristic
379C2×C3×C4#12011yescharacteristic
380C2×C3×C4#12011yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4H(5,𝔽2)11yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—42no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398Dic6—22no—
399Dic6—22no—
400Dic6—41no—
401Dic6—41no—
402C3⋊C8—42no—
403C3⋊C8—42no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8C2×#6712yes—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×#6712yes—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4C2×#6712yes—
434C22×C4C2×#6712yes—
435C22×C4C2×#6711yescharacteristic
436C22×C4C2×#6711yescharacteristic
437C22×C4C23×D311yescharacteristic
438Q16—124no—
439SD16—124no—
440M16—62no—
441M16—62no—
442M16—62no—
443M16—62no—
444C2×C8—62no—
445C2×C8—62no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—68no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—121no—
463C4⋊C4—121no—
464C4⋊C4C2×#6712yes—
465C4⋊C4C2×#6712yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—32no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—121no—
481C42—121no—
482C42C2×#6711yescharacteristic
483C42C2×#6711yescharacteristic
484C42C2×#6711yescharacteristic
485C42C2×#6711yescharacteristic
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#52511yescharacteristic
493C22×C3#53711yescharacteristic
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4#54611yescharacteristic
504C3×C4#54711yescharacteristic
505C3×C4#54811yescharacteristic
506C3×C4#55011yescharacteristic
507Dic3—42no—
508Dic3—42no—
509Dic3—81no—
510Dic3—81no—
511C23—22no—
512C23C22×#6711yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—42no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—62no—
521Q8—62no—
522Q8—81no—
523Q8—121no—
524Q8—121no—
525D4—22no—
526D4—22no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—81no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—84no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4#95311yescharacteristic
556C2×C4#95311yescharacteristic
557C2×C4#95311yescharacteristic
558C2×C4#95311yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#101011yescharacteristic
563C2×C4C22×#6711yescharacteristic
564C2×C4C22×#6711yescharacteristic
565C8—122no—
566C8—122no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—82no—
570C2×C3#240711yescharacteristic
571C2×C3#302011yescharacteristic
572C2×C3#302211yescharacteristic
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22#582811yescharacteristic
579C22#583311yescharacteristic
580C22C2×#95311yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—122no—
591C4—122no—
592C4—241no—
593C4—241no—
594C4#582411yescharacteristic
595C4#582911yescharacteristic
596C4#583111yescharacteristic
597C4#583411yescharacteristic
598C3#2418311yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—82no—
602C2#8409711yescharacteristic
603C2#8411611yescharacteristic
604C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3507C2
socleC22×C3C2×#120
prime radicalC3×#532C22
hypercenter#3507D3
hypocenterC3#24183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #83367C2semidirect
4. #82456C2semidirect
5. #71868C2semidirect
6. #71863C2semidirect
7. #71840C2semidirect
9. #71666C2semidirect
10. #71665C2semidirect
11. #71637C2semidirect
598. C3#24183semidirect

Maximal subgroups

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

subgroupindex
C3×#35072
#833672
#833142
#824562
#718682
#718632
#718402
#718162
#716662
#716652
#716372
#241833