Finite Group Databasebeta

← #10782814

Subgroups — #10782814

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2994C211yesmaximal, characteristic, complemented
2#83148C211yesmaximal, characteristic
3#83123C211yesmaximal, characteristic, complemented
4#79268C211yesmaximal, characteristic, complemented
5#70965C211yesmaximal, characteristic, complemented
6#70960C211yesmaximal, characteristic, complemented
7#70945C211yesmaximal, characteristic, complemented
8#70940C211yesmaximal, characteristic, complemented
9#70874C211yesmaximal, characteristic, complemented
10#70869C211yesmaximal, characteristic, complemented
11#70853C211yesmaximal, characteristic, complemented
12#70848C211yesmaximal, characteristic, complemented
13#70752C211yesmaximal, characteristic, complemented
14C2×#4481C211yesmaximal, characteristic, complemented
15#70433C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18#5477C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×#436C2211yescharacteristic
23C3×#435—21no—
24C3×C8×D4—21no—
25C3×C8×D4C2211yescharacteristic
26C3×#433—21no—
27C3×#431C2211yescharacteristic
28C3×#424—21no—
29C2×C3×C4⋊C8C2211yescharacteristic
30#5066—21no—
31#5064—21no—
32#5057—21no—
33#5055—21no—
34#5048—21no—
35#5043—21no—
36#5038—21no—
37#5033—21no—
38#5027C2211yescharacteristic
39#5021C2211yescharacteristic
40#5021C2211yescharacteristic
41#5021C2211yescharacteristic
42#5015C2211yescharacteristic
43#5008C2211yescharacteristic
44#5008C2211yescharacteristic
45#5008C2211yescharacteristic
46#4926C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4846—21no—
49#4846—21no—
50#4736—21no—
51#4736—21no—
52#4694C2211yescharacteristic
53#4694C2211yescharacteristic
54#4682C2211yescharacteristic
55#4682C2211yescharacteristic
56#4485C2211yescharacteristic
57#4485C2211yescharacteristic
58#4481C2211yescharacteristic
59#4481C2211yescharacteristic
60#4481C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3247—31no—
69#3238—31no—
70#3226—31no—
71#2994D311yescharacteristic, complemented
72#1698—31no—
73#1697—31no—
74#1696—31no—
75#1695—31no—
76#1678—31no—
77#1677—31no—
78#1676—31no—
79#1675—31no—
80#1573—31no—
81#1551—31no—
82C2×#333—31no—
83C2×#235—21no—
84C22×Dic6—21no—
85C2×C3×M16—21no—
86C22×C3×C8—21no—
87C22×C3×C8—21no—
88C3×#126—21no—
89C3×#125D411yescharacteristic
90C3×#124D411yescharacteristic
91C3×#123—21no—
92C3×C22⋊Q8D411yescharacteristic
93C3×#121D411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4C2311yescharacteristic
97C3×#117—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#215—21no—
100C2×#213—21no—
101#930—21no—
102#930—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C22×C3⋊C8—21no—
106#875—21no—
107#874C2311yescharacteristic
108C4×D12—21no—
109C4×D12—21no—
110#869—21no—
111#869—21no—
112#869—21no—
113#869C2311yescharacteristic
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8—21no—
119C3×C4⋊C8D411yescharacteristic
120C3×C4⋊C8D411yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C22⋊C8—21no—
124C3×C22⋊C8—21no—
125C3×C22⋊C8D411yescharacteristic
126C3×C22⋊C8D411yescharacteristic
127C3×#97—21no—
128C3×C4×C8—21no—
129C3×C4×C8C2311yescharacteristic
130C3×C4×C8C2311yescharacteristic
131#835—21no—
132#835—21no—
133#835—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#830—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142#818—21no—
143#818—21no—
144#818—21no—
145#818—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159#803—21no—
160#532—31no—
161C2×C4⋊Q8—31no—
162#529C2×D311yescharacteristic
163#501—61no—
164#500—31no—
165#500—31no—
166#489—61no—
167#486—31no—
168#486—31no—
169#483—61no—
170#480—61no—
171#480—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#477—31no—
177#476—31no—
178#476—31no—
179#476—31no—
180#474—61no—
181#474—61no—
182#466—61no—
183#465—61no—
184#463—61no—
185#460—31no—
186#459—31no—
187#446C2×D311yescharacteristic
188C8×Q8C2×D311yescharacteristic
189#436C2×D311yescharacteristic
190#435—21no—
191C8×D4—21no—
192C8×D4C2×D311yescharacteristic
193#433—21no—
194#432—31no—
195#431C2×D311yescharacteristic
196#424—21no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#333—31no—
208#333—31no—
209#331—31no—
210#331—31no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C2×#67—41no—
219C22×Dic3—41no—
220#235—82no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×C4×D3—41no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—41no—
229C2×Dic6—82no—
230C3×M16—81no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—81no—
234C2×C3×C8—81no—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C42—21no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255#215—81no—
256#213—81no—
257(C2×D3)⋊C4—41no—
258(C2×D3)⋊C4—41no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—41no—
266C4⋊Dic3—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—81no—
272C2×#41—61no—
273C22×Q8—61no—
274C2×SD16—61no—
275C2×D8—61no—
276C2×M16—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—61no—
280C4⋊Q8—31no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—61no—
284C4⋊Q8—61no—
285C4⋊Q8—61no—
286#127—31no—
287#126—21no—
288#125D1211yescharacteristic
289#124—61no—
290#124D1211yescharacteristic
291#123—21no—
292C22⋊Q8D1211yescharacteristic
293#121—61no—
294#121—61no—
295#121D1211yescharacteristic
296C4×Q8C22×D311yescharacteristic
297C4×D4—21no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4C22×D311yescharacteristic
301#117—21no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C42C22×D311yescharacteristic
305#107—61no—
306#107—61no—
307#107—61no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#106—61no—
313#106—61no—
314#106—61no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8#6711yescharacteristic
324C4⋊C8#6711yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—61no—
342C22⋊C8#6711yescharacteristic
343C22⋊C8#6711yescharacteristic
344#97—21no—
345#97—61no—
346C4×C8—21no—
347C4×C8C22×D311yescharacteristic
348C4×C8C22×D311yescharacteristic
349C23×C3—21no—
350C23×C3#12111yescharacteristic
351C22×D3—41no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×D4—21no—
355C3×D4—21no—
356C3×D4—42no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4#12111yescharacteristic
368C2×C3×C4#12111yescharacteristic
369C2×C3×C4#12111yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373#67—82no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—82no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—81no—
383C4×D3—82no—
384Dic6—82no—
385Dic6—81no—
386Dic6—81no—
387Dic6—81no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—242no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—121no—
403C2×Q8—242no—
404C2×Q8C2×D1211yescharacteristic
405C2×D4—21no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—121no—
410C2×D4C2×D1211yescharacteristic
411C22×C4—21no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4C2×D1211yescharacteristic
415C22×C4#23511yescharacteristic
416C22×C4C2×#6711yescharacteristic
417SD16—241no—
418D8—241no—
419M16—81no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—81no—
427C2×C8—81no—
428C2×C8—241no—
429C2×C8C2×#6711yescharacteristic
430C2×C8C2×#6711yescharacteristic
431C2×C8C2×#6711yescharacteristic
432C2×C8C2×#6711yescharacteristic
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4C2×D1211yescharacteristic
446C4⋊C4C2×D1211yescharacteristic
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42C2×D1211yescharacteristic
456C42#23511yescharacteristic
457C42C2×#6711yescharacteristic
458C42C23×D311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3#47212yes—
465C22×C3C2×#12111yescharacteristic
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4#57711yescharacteristic
477C3×C4#57711yescharacteristic
478C3×C4#57811yescharacteristic
479C3×C4#57811yescharacteristic
480Dic3—81no—
481Dic3—81no—
482Dic3—81no—
483C23—21no—
484C23—121no—
485C23#93011yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—241no—
489Q8—241no—
490Q8—241no—
491Q8—242no—
492D4—21no—
493D4—21no—
494D4—42no—
495D4—121no—
496D4—121no—
497D4—121no—
498D4—121no—
499D4—241no—
500D4—242no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—242no—
515C2×C4—242no—
516C2×C4#93011yescharacteristic
517C2×C4#93011yescharacteristic
518C2×C4#93011yescharacteristic
519C2×C4C22×D1211yescharacteristic
520C2×C4C2×#23511yescharacteristic
521C2×C4C22×#6711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—121no—
529C8—121no—
530C8—121no—
531C8—121no—
532C2×C3—22no—
533C2×C3—81no—
534C2×C3#304611yescharacteristic
535C2×C3#306311yescharacteristic
536C2×C3C2×#47211yescharacteristic
537D3—161no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22#514212yes—
547C22C2×#93011yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—241no—
556C4—241no—
557C4—241no—
558C4#575111yescharacteristic
559C4#575311yescharacteristic
560C4#583611yescharacteristic
561C4#583811yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—22no—
564C2—81no—
565C2—481no—
566C2#8257211yescharacteristic
567C2C2×#514211yescharacteristic
568C2#8380811yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2994C2
socleC22×C3C2×#121
prime radical#83148C2
hypercenter#2994D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2994

Complements

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

NH ≅ G/Ntype
1. C3×#2994C2semidirect
3. #83123C2semidirect
4. #79268C2semidirect
5. #70965C2semidirect
6. #70960C2semidirect
7. #70945C2semidirect
8. #70940C2semidirect
9. #70874C2semidirect
10. #70869C2semidirect
11. #70853C2semidirect
12. #70848C2semidirect
13. #70752C2semidirect
14. C2×#4481C2semidirect
15. #70433C2semidirect
71. #2994D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
C3×#29942
#831482
#831232
#792682
#709652
#709602
#709452
#709402
#708742
#708692
#708532
#708482
#707522
C2×#44812
#704332
#195913