Finite Group Databasebeta

← #10777922

Subgroups — #10777922

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3107C211yesmaximal, characteristic, complemented
2#83343C211yesmaximal, characteristic, complemented
3#82529C211yesmaximal, characteristic
4#82517C211yesmaximal, characteristic, complemented
5#71829C211yesmaximal, characteristic, complemented
6#71123C211yesmaximal, characteristic, complemented
7#71120C211yesmaximal, characteristic, complemented
8#71107C211yesmaximal, characteristic, complemented
9#71104C211yesmaximal, characteristic, complemented
10#71059C211yesmaximal, characteristic, complemented
11#71056C211yesmaximal, characteristic, complemented
12#71043C211yesmaximal, characteristic, complemented
13#71040C211yesmaximal, characteristic, complemented
14#70773C211yesmaximal, characteristic, complemented
15#70745C211yesmaximal, characteristic, complemented
16#19159—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5477C2211yescharacteristic
20C3×#485—21no—
21C3×#475C2211yescharacteristic
22C3×#474C2211yescharacteristic
23C3×#462C2211yescharacteristic
24C3×#461C2211yescharacteristic
25C3×#452—21no—
26C3×C4×Q16—21no—
27C3×C4×SD16—21no—
28C3×#431C2211yescharacteristic
29#5238—21no—
30#5088C2211yescharacteristic
31#5085C2211yescharacteristic
32#5080C2211yescharacteristic
33#5079C2211yescharacteristic
34#5048—21no—
35#5048—21no—
36#5027C2211yescharacteristic
37C4×#215—21no—
38C4×#215—21no—
39C4×#215—21no—
40#5024C2211yescharacteristic
41#5024C2211yescharacteristic
42#5021C2211yescharacteristic
43#5015C2211yescharacteristic
44C4×#213—21no—
45#5012C2211yescharacteristic
46#5012C2211yescharacteristic
47#5011—21no—
48#5011—21no—
49#5008C2211yescharacteristic
50#5003—21no—
51#4925—21no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54C4×#208—21no—
55C2×C4×C3⋊C8C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3167—31no—
69#3107D311yescharacteristic, complemented
70#3106—31no—
71#2965—31no—
72#1704—31no—
73#1703—31no—
74#1702—31no—
75#1701—31no—
76#1692—31no—
77#1691—31no—
78#1686—31no—
79#1685—31no—
80#1564—31no—
81#1544—31no—
82#1543—31no—
83C2×C3×#41D411yescharacteristic
84C2×#235—21no—
85C2×C3×Q16—21no—
86C2×C3×SD16—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×#125—21no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4D411yescharacteristic
99C3×C4×D4C2311yescharacteristic
100C3×#117—21no—
101C3×#117—21no—
102C3×#117D411yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#215—21no—
106C2×#215—21no—
107C2×#215—21no—
108C2×#213—21no—
109#930—21no—
110#930—21no—
111C2×#208—21no—
112C2×#208—21no—
113C22×C3⋊C8D411yescharacteristic
114C22×C3⋊C8D411yescharacteristic
115#875—21no—
116#874C2311yescharacteristic
117C4×D12—21no—
118C4×D12—21no—
119#869C2311yescharacteristic
120C3×#107—21no—
121C3×#106—21no—
122C3×C4⋊C8C2311yescharacteristic
123C3×C4⋊C8C2311yescharacteristic
124C3×#103—21no—
125C3×#103—21no—
126C3×#103—21no—
127C3×#102—21no—
128C3×C22⋊C8—21no—
129C3×C4×C8—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#832—21no—
134#830—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#809—21no—
139#809—21no—
140#809—21no—
141#809—21no—
142#808—21no—
143#808—21no—
144#808—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152#803—21no—
153#803—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8D411yescharacteristic
157C4×C3⋊C8D411yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161#532—31no—
162C4×#41C2×D311yescharacteristic
163#498—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#485—21no—
168#482—61no—
169#477—31no—
170#476—31no—
171#475C2×D311yescharacteristic
172#474C2×D311yescharacteristic
173#465—61no—
174#465—61no—
175#462C2×D311yescharacteristic
176#461C2×D311yescharacteristic
177#460—31no—
178#459—31no—
179#452—21no—
180#449—61no—
181#446—31no—
182#446—31no—
183C4×Q16—21no—
184C4×SD16—21no—
185C4×SD16—61no—
186C4×SD16—61no—
187C4×SD16—61no—
188C4×D8—61no—
189#436—31no—
190#436—31no—
191#435—61no—
192#435—61no—
193#431—31no—
194#431—31no—
195#431—61no—
196#431C2×D311yescharacteristic
197C4×M16—61no—
198C2×C4×C8—31no—
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4#4111yescharacteristic
224C2×#67—41no—
225#235—82no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×C4×D3—41no—
230C2×Dic6—21no—
231C3×Q16—42no—
232C3×SD16—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—41no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—41no—
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255C3×C42#4111yescharacteristic
256C3×C42C2411yescharacteristic
257#215—42no—
258#215—42no—
259#215—42no—
260#213—42no—
261(C2×D3)⋊C4—41no—
262(C2×D3)⋊C4—41no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265#208—42no—
266#208—42no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—41no—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277C2×#41—61no—
278C2×#41#6711yescharacteristic
279C2×Q16—21no—
280C2×SD16—21no—
281C2×SD16—61no—
282C2×SD16—61no—
283C2×SD16—61no—
284C2×D8—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—31no—
288C22×C8—31no—
289C4⋊Q8—31no—
290C4⋊Q8C22×D311yescharacteristic
291C4⋊Q8C22×D311yescharacteristic
292#127—31no—
293#125—21no—
294#124—61no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297#121—61no—
298#121—61no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8#6711yescharacteristic
302C4×Q8C22×D311yescharacteristic
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—61no—
306C4×D4—61no—
307C4×D4#6711yescharacteristic
308C4×D4C22×D311yescharacteristic
309#117—21no—
310#117—21no—
311#117#6711yescharacteristic
312C2×C42—21no—
313C2×C42C22×D311yescharacteristic
314#107—21no—
315#107—61no—
316#106—21no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8C22×D311yescharacteristic
327C4⋊C8C22×D311yescharacteristic
328#103—21no—
329#103—21no—
330#103—21no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—61no—
346#97—61no—
347C4×C8—21no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—61no—
353C4×C8—61no—
354C23×C3—21no—
355C23×C3#12111yescharacteristic
356C22×D3—41no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—21no—
364C3×D4—21no—
365C3×D4—42no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386#67—82no—
387C2×Dic3—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391D12—41no—
392D12—81no—
393C4×D3—82no—
394Dic6—81no—
395C3×C8—41no—
396C3×C8—41no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404C3⋊C8#13512yes—
405#41—41no—
406#41—41no—
407#41—242no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—61no—
411C2×Q8C2×#6711yescharacteristic
412C2×D4—21no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C2×D4—121no—
417C2×D4C2×#6711yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—121no—
422C22×C4D3×D411yescharacteristic
423C22×C4#23711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425Q16—42no—
426SD16—42no—
427SD16—122no—
428SD16—122no—
429SD16—122no—
430D8—122no—
431M16—122no—
432M16—122no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—31no—
439C2×C8—62no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—121no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4C2×#6711yescharacteristic
456C4⋊C4C2×#6711yescharacteristic
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—41no—
461C22⋊C4—121no—
462C22⋊C4—121no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—41no—
467C42D3×D411yescharacteristic
468C42#23711yescharacteristic
469C42C2×#6711yescharacteristic
470C42C23×D311yescharacteristic
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×#12111yescharacteristic
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4#47112yes—
490C3×C4C2×#13511yescharacteristic
491C3×C4#57511yescharacteristic
492C3×C4#57511yescharacteristic
493C3×C4#57511yescharacteristic
494Dic3—81no—
495C23—21no—
496C23—121no—
497C23#93911yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—241no—
505D4—21no—
506D4—21no—
507D4—42no—
508D4—121no—
509D4—121no—
510D4—121no—
511D4—121no—
512D4—241no—
513D4—242no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—121no—
529C2×C4—242no—
530C2×C4#93911yescharacteristic
531C2×C4#93911yescharacteristic
532C2×C4#93911yescharacteristic
533C2×C4C2×D3×D411yescharacteristic
534C2×C4C2×#23711yescharacteristic
535C2×C4C22×#6711yescharacteristic
536C8—32no—
537C8—41no—
538C8—41no—
539C8—61no—
540C8—61no—
541C8—61no—
542C8—121no—
543C8—121no—
544C8—121no—
545C8—121no—
546C2×C3—41no—
547C2×C3—81no—
548C2×C3#303511yescharacteristic
549C2×C3#305411yescharacteristic
550C2×C3C2×#47111yescharacteristic
551D3—161no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22C2×#93911yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—241no—
571C4#520312yes—
572C4D3×#13511yescharacteristic
573C4#576811yescharacteristic
574C4#582211yescharacteristic
575C4#582211yescharacteristic
576C3#1915911yescharacteristic, complemented, Sylow
577C2—41no—
578C2—81no—
579C2—481no—
580C2#8292311yescharacteristic
581C2#8390011yescharacteristic
582C2C2×#520311yescharacteristic
subgroupG/H
centerC2×C4#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3107C2
socleC22×C3C2×#121
prime radical#82529C2
hypercenter#3107D3
hypocenterC3#19159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. C3×#3107C2semidirect
2. #83343C2semidirect
4. #82517C2semidirect
5. #71829C2semidirect
6. #71123C2semidirect
7. #71120C2semidirect
8. #71107C2semidirect
9. #71104C2semidirect
10. #71059C2semidirect
11. #71056C2semidirect
12. #71043C2semidirect
13. #71040C2semidirect
14. #70773C2semidirect
15. #70745C2semidirect
69. #3107D3semidirect
576. C3#19159semidirect

Maximal subgroups

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

subgroupindex
C3×#31072
#833432
#825292
#825172
#718292
#711232
#711202
#711072
#711042
#710592
#710562
#710432
#710402
#707732
#707452
#191593