Finite Group Databasebeta

← #10778195

Subgroups — #10778195

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3107C211yesmaximal, characteristic, complemented
2#83347C211yesmaximal, characteristic, complemented
3#82525C211yesmaximal, characteristic
4#82519C211yesmaximal, characteristic, complemented
5#71831C211yesmaximal, characteristic, complemented
6#71118C211yesmaximal, characteristic, complemented
7#71116C211yesmaximal, characteristic, complemented
8#71106C211yesmaximal, characteristic, complemented
9#71104C211yesmaximal, characteristic, complemented
10#71055C211yesmaximal, characteristic, complemented
11#71053C211yesmaximal, characteristic, complemented
12#71043C211yesmaximal, characteristic, complemented
13#71041C211yesmaximal, characteristic, complemented
14#70775C211yesmaximal, characteristic, complemented
15#70746C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C2×#950—21no—
20#5477C2211yescharacteristic
21C3×#485—21no—
22C3×#475C2211yescharacteristic
23C3×#474C2211yescharacteristic
24C3×#462C2211yescharacteristic
25C3×#461C2211yescharacteristic
26C3×#452—21no—
27C3×C4×Q16—21no—
28C3×C4×SD16—21no—
29C3×#431C2211yescharacteristic
30#5088C2211yescharacteristic
31#5085C2211yescharacteristic
32#5079C2211yescharacteristic
33#5078C2211yescharacteristic
34#5049—21no—
35#5049—21no—
36#5031C2211yescharacteristic
37#5028—21no—
38#5027C2211yescharacteristic
39#5026—21no—
40#5026—21no—
41Q8×C3⋊C8C2211yescharacteristic
42Q8×C3⋊C8C2211yescharacteristic
43#5019C2211yescharacteristic
44#5016—21no—
45#5015C2211yescharacteristic
46#5012—21no—
47#5012—21no—
48#5012C2211yescharacteristic
49#5012C2211yescharacteristic
50#5002—21no—
51#4926—21no—
52#4926C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C4×#208—21no—
55C4×#208C2211yescharacteristic
56#4547C2211yescharacteristic
57#4547C2211yescharacteristic
58#4547C2211yescharacteristic
59#4547C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3182—31no—
69#3107D311yescharacteristic, complemented
70#3035—31no—
71#2971—31no—
72#1701—31no—
73#1699—31no—
74#1693—31no—
75#1685—31no—
76#1682—31no—
77#1680—31no—
78#1674—31no—
79#1666—31no—
80#1565—31no—
81#1553—31no—
82#1552—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
105#950—44no—
106C2×#216—21no—
107C2×#215—21no—
108C2×#215—21no—
109C2×#215—21no—
110C2×#214—21no—
111C2×#213—21no—
112#930—21no—
113#930—21no—
114C2×#208—21no—
115C2×#208—21no—
116C2×#208D411yescharacteristic
117C2×#208D411yescharacteristic
118C22×C3⋊C8—21no—
119#875—21no—
120#874C2311yescharacteristic
121C4×D12—21no—
122C4×D12—21no—
123#869C2311yescharacteristic
124C3×#107—21no—
125C3×#106—21no—
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×#103—21no—
129C3×#103—21no—
130C3×#103—21no—
131C3×#102—21no—
132C3×C22⋊C8—21no—
133C3×C4×C8—21no—
134#835—21no—
135#835—21no—
136#830—21no—
137#830—21no—
138#830—21no—
139#810—21no—
140#810—21no—
141#809—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#807—21no—
146#807—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)—21no—
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154#803—21no—
155#803D411yescharacteristic
156#803D411yescharacteristic
157C4×C3⋊C8—21no—
158C4×C3⋊C8—21no—
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#533C2×D311yescharacteristic
163#532—31no—
164C4×#41C2×D311yescharacteristic
165#500—31no—
166#492—31no—
167#487—31no—
168#486—31no—
169#485—21no—
170#475C2×D311yescharacteristic
171#474C2×D311yescharacteristic
172#468—61no—
173#468—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#462C2×D311yescharacteristic
178#461—31no—
179#461C2×D311yescharacteristic
180#460—31no—
181#459—31no—
182#452—21no—
183C8×Q8—31no—
184C8×Q8—31no—
185#440—61no—
186#440—61no—
187C4×Q16—21no—
188C4×SD16—21no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—31no—
194#432—61no—
195#431C2×D311yescharacteristic
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332—31no—
209#332—31no—
210#331—31no—
211#331—31no—
212C3×#41—22no—
213C3×#41—41no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8C2×D411yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4#4111yescharacteristic
225C2×#67—41no—
226#235—42no—
227#235—42no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×C4×D3—41no—
232C2×Dic6—21no—
233C3×Q16—42no—
234C3×SD16—42no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—41no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—41no—
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42#4111yescharacteristic
258C3×C42C2411yescharacteristic
259#216—42no—
260#215—42no—
261#215—42no—
262#215—42no—
263#214—42no—
264#213—42no—
265(C2×D3)⋊C4—41no—
266(C2×D3)⋊C4—41no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269#208—22no—
270#208—41no—
271#208—42no—
272#208—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—42no—
278C2×C3⋊C8C2×D411yescharacteristic
279C2×C3⋊C8C2×D411yescharacteristic
280C2×C3⋊C8C2×D411yescharacteristic
281C2×C3⋊C8C2×D411yescharacteristic
282C2×#41—61no—
283C2×#41#6711yescharacteristic
284#135—124no—
285C2×Q16—21no—
286C2×Q16—61no—
287C2×SD16—21no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16—61no—
296C2×M16—61no—
297C22×C8—61no—
298C4⋊Q8—31no—
299C4⋊Q8C22×D311yescharacteristic
300C4⋊Q8C22×D311yescharacteristic
301#127—31no—
302#125—21no—
303#124—61no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306#121—61no—
307#121—61no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8#6711yescharacteristic
311C4×Q8C22×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—61no—
315C4×D4—61no—
316C4×D4#6711yescharacteristic
317C4×D4C22×D311yescharacteristic
318#117—21no—
319#117—21no—
320#117#6711yescharacteristic
321C2×C42—21no—
322C2×C42C22×D311yescharacteristic
323#107—21no—
324#107—61no—
325#107—61no—
326#106—21no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—31no—
330C4⋊C8—31no—
331C4⋊C8—61no—
332C4⋊C8—61no—
333C4⋊C8—61no—
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—21no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—21no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—31no—
353#97—31no—
354#97—61no—
355C4×C8—21no—
356C4×C8—31no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C23×C3—21no—
361C23×C3#12111yescharacteristic
362C22×D3—41no—
363C3×Q8—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8AGL(1,ℤ/8)12yes—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—42no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—41no—
380C2×C3×C4—42no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—81no—
386C2×C3×C4#12111yescharacteristic
387C2×C3×C4#12111yescharacteristic
388C2×C3×C4#12111yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392#67—82no—
393C2×Dic3—41no—
394D12—41no—
395D12—41no—
396D12—41no—
397D12—41no—
398D12—81no—
399C4×D3—82no—
400Dic6—41no—
401Dic6—41no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8#13512yescomplemented
412#41—22no—
413#41—41no—
414#41—122no—
415#41—122no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—61no—
419C2×Q8C2×#6711yescharacteristic
420C2×D4—21no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—121no—
425C2×D4C2×#6711yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—121no—
430C22×C4D3×D411yescharacteristic
431C22×C4#23711yescharacteristic
432C22×C4C2×#6711yescharacteristic
433Q16—42no—
434Q16—122no—
435SD16—42no—
436SD16—122no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—62no—
442M16—122no—
443M16—121no—
444M16—122no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—31no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—122no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4C2×#6711yescharacteristic
467C4⋊C4C2×#6711yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—121no—
473C22⋊C4—121no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—41no—
478C42D3×D411yescharacteristic
479C42#23711yescharacteristic
480C42C2×#6711yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3C2×#12111yescharacteristic
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4C2×#13511yescharacteristic
502C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C3×C4#57511yescharacteristic
504C3×C4#57511yescharacteristic
505Dic3—81no—
506C23—21no—
507C23—121no—
508C23#93911yescharacteristic
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—121no—
515Q8—121no—
516Q8#94912yes—
517D4—21no—
518D4—21no—
519D4—42no—
520D4—121no—
521D4—121no—
522D4—121no—
523D4—121no—
524D4—241no—
525D4—242no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—81no—
540C2×C4—121no—
541C2×C4—242no—
542C2×C4#93911yescharacteristic
543C2×C4#93911yescharacteristic
544C2×C4#93911yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×#23711yescharacteristic
547C2×C4C22×#6711yescharacteristic
548C8—32no—
549C8—41no—
550C8—41no—
551C8—61no—
552C8—61no—
553C8—61no—
554C8—121no—
555C8—121no—
556C8—121no—
557C8—121no—
558C2×C3—41no—
559C2×C3—81no—
560C2×C3#303511yescharacteristic
561C2×C3#305511yescharacteristic
562C2×C3#306911yescharacteristic
563D3—161no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—241no—
571C22—241no—
572C22—241no—
573C22C2×#93911yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—241no—
584C4D3×#13511yescharacteristic
585C4#576811yescharacteristic
586C4C2×#94911yescharacteristic
587C4#582211yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—81no—
591C2—481no—
592C2#8292411yescharacteristic
593C2#8390011yescharacteristic
594C2#8397811yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3107C2
socleC22×C3C2×#121
prime radical#82525C2
hypercenter#3107D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. C3×#3107C2semidirect
2. #83347C2semidirect
4. #82519C2semidirect
5. #71831C2semidirect
6. #71118C2semidirect
7. #71116C2semidirect
8. #71106C2semidirect
9. #71104C2semidirect
10. #71055C2semidirect
11. #71053C2semidirect
12. #71043C2semidirect
13. #71041C2semidirect
14. #70775C2semidirect
15. #70746C2semidirect
69. #3107D3semidirect
411. C3⋊C8#135semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
C3×#31072
#833472
#825252
#825192
#718312
#711182
#711162
#711062
#711042
#710552
#710532
#710432
#710412
#707752
#707462
#191823