Finite Group Databasebeta

← #10782818

Subgroups — #10782818

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×#3238C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#83148C211yesmaximal, characteristic
4#83132C211yesmaximal, characteristic, complemented
5C2×#4546C211yesmaximal, characteristic, complemented
6#71154C211yesmaximal, characteristic, complemented
7#71153C211yesmaximal, characteristic, complemented
8#71150C211yesmaximal, characteristic, complemented
9#71149C211yesmaximal, characteristic, complemented
10#71138C211yesmaximal, characteristic, complemented
11#71137C211yesmaximal, characteristic, complemented
12#71134C211yesmaximal, characteristic, complemented
13#71133C211yesmaximal, characteristic, complemented
14#70816C211yesmaximal, characteristic, complemented
15#70797C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C3×#529C2211yescharacteristic
19#5477C2211yescharacteristic
20C3×#501—21no—
21C3×#489—21no—
22C3×#483—21no—
23C3×#480—21no—
24C3×#477C2211yescharacteristic
25C3×#477C2211yescharacteristic
26C3×#476C2211yescharacteristic
27C3×#476C2211yescharacteristic
28C3×#432C2211yescharacteristic
29#5088C2211yescharacteristic
30#5088C2211yescharacteristic
31#5087—21no—
32#5087—21no—
33#5078C2211yescharacteristic
34#5078C2211yescharacteristic
35#5066—21no—
36#5057—21no—
37#5048—21no—
38#5038—21no—
39#5027C2211yescharacteristic
40#5024C2211yescharacteristic
41Q8×C3⋊C8C2211yescharacteristic
42#5021C2211yescharacteristic
43#5015C2211yescharacteristic
44#5012C2211yescharacteristic
45#5011—21no—
46D4×C3⋊C8—21no—
47D4×C3⋊C8C2211yescharacteristic
48#5008C2211yescharacteristic
49#5000—21no—
50#4999—21no—
51#4967—21no—
52#4967—21no—
53#4925C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56#4546C2211yescharacteristic
57#4546C2211yescharacteristic
58#4546C2211yescharacteristic
59#4546C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3247—31no—
69#3238D311yescharacteristic, complemented
70#3226—31no—
71#2994—31no—
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—
83C22×C3×Q8—21no—
84C2×#235—21no—
85C3×C4⋊Q8—21no—
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×#126—21no—
91C3×#125D411yescharacteristic
92C3×#124D411yescharacteristic
93C3×#123—21no—
94C3×C22⋊Q8D411yescharacteristic
95C3×#121D411yescharacteristic
96C3×C4×Q8C2311yescharacteristic
97C3×C4×D4—21no—
98C3×C4×D4C2311yescharacteristic
99C3×#117—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#215—21no—
104C2×#213—21no—
105#930—21no—
106#930—21no—
107C2×#208—21no—
108C22×C3⋊C8—21no—
109C22×C3⋊C8—21no—
110C22×C3⋊C8—21no—
111#875—21no—
112#874C2311yescharacteristic
113C4×D12—21no—
114C4×D12—21no—
115#869C2311yescharacteristic
116C3×#107—21no—
117C3×#107—21no—
118C3×#106—21no—
119C3×#106—21no—
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×#103—21no—
123C3×#103—21no—
124C3×#102—21no—
125C3×#102—21no—
126C3×C22⋊C8—21no—
127C3×#97—21no—
128#835—21no—
129#832—21no—
130#830—21no—
131#830—21no—
132#830D411yescharacteristic
133#830D411yescharacteristic
134#809—21no—
135#809—21no—
136#809—21no—
137#809—21no—
138#809—21no—
139#809—21no—
140#808—21no—
141#807—21no—
142#807—21no—
143#807—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)—21no—
150C4⋊(C3⋊C8)D411yescharacteristic
151C4⋊(C3⋊C8)D411yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8C2×D311yescharacteristic
162#529C2×D311yescharacteristic
163#501—21no—
164#500—31no—
165#500—31no—
166#489—21no—
167#486—31no—
168#486—31no—
169#483—21no—
170#480—21no—
171#480—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477C2×D311yescharacteristic
176#477C2×D311yescharacteristic
177#476—31no—
178#476C2×D311yescharacteristic
179#476C2×D311yescharacteristic
180#474—61no—
181#474—61no—
182#466—61no—
183#465—61no—
184#463—61no—
185#460—31no—
186#459—31no—
187#446—31no—
188C8×Q8—31no—
189#436—31no—
190#435—61no—
191C8×D4—31no—
192C8×D4—61no—
193#433—61no—
194#432C2×D311yescharacteristic
195#431—31no—
196#424—61no—
197C2×C4⋊C8—31no—
198C2×C4⋊C8—31no—
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×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—41no—
214C2×C3×Q8—82no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—41no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4111yescharacteristic
223C2×#67—41no—
224#235—82no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×C4×D3—41no—
229C2×Dic6—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C42—21no—
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42#4111yescharacteristic
253C3×C42C2411yescharacteristic
254#215—81no—
255#213—81no—
256(C2×D3)⋊C4—41no—
257(C2×D3)⋊C4—41no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260#208—81no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—81no—
266C2×C3⋊C8—81no—
267C2×C3⋊C8—81no—
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×#41—61no—
273C22×Q8—21no—
274C2×SD16—61no—
275C2×D8—61no—
276C2×M16—61no—
277C22×C8—61no—
278C22×C8—61no—
279C22×C8—61no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—31no—
284C4⋊Q8C22×D311yescharacteristic
285C4⋊Q8C22×D311yescharacteristic
286#127—31no—
287#126—21no—
288#125#6711yescharacteristic
289#124—61no—
290#124#6711yescharacteristic
291#123—21no—
292C22⋊Q8#6711yescharacteristic
293#121—61no—
294#121—61no—
295#121#6711yescharacteristic
296C4×Q8C22×D311yescharacteristic
297C4×D4—21no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4C22×D311yescharacteristic
301#117—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C42C22×D311yescharacteristic
305#107—21no—
306#107—21no—
307#107—61no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#106—21no—
313#106—21no—
314#106—61no—
315C4⋊C8—31no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—21no—
329#103—61no—
330#102—21no—
331#102—21no—
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—31no—
341C22⋊C8—31no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—21no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C23×C3—21no—
350C23×C3#12111yescharacteristic
351C22×D3—41no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—81no—
355C3×Q8—81no—
356C3×Q8—82no—
357C3×D4—21no—
358C3×D4—21no—
359C3×D4—42no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—82no—
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—82no—
380C2×Dic3—41no—
381D12—41no—
382D12—41no—
383D12—41no—
384D12—41no—
385D12—81no—
386C4×D3—82no—
387Dic6—81no—
388C3×C8—41no—
389C3×C8—41no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—242no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—41no—
402C2×Q8—61no—
403C2×Q8—82no—
404C2×Q8C2×#6711yescharacteristic
405C2×D4—21no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—121no—
410C2×D4C2×#6711yescharacteristic
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—121no—
414C22×C4D3×D411yescharacteristic
415C22×C4#23711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417SD16—241no—
418D8—241no—
419M16—241no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—241no—
431C2×C8—241no—
432C2×C8—241no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4C2×#6711yescharacteristic
446C4⋊C4C2×#6711yescharacteristic
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42D3×D411yescharacteristic
456C42#23711yescharacteristic
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—81no—
477C3×C4—81no—
478C3×C4#57711yescharacteristic
479C3×C4#57711yescharacteristic
480C3×C4#57811yescharacteristic
481C3×C4#57811yescharacteristic
482Dic3—81no—
483C23—21no—
484C23—121no—
485C23#93911yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—81no—
489Q8—81no—
490Q8—82no—
491Q8—241no—
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—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—121no—
515C2×C4—242no—
516C2×C4#93911yescharacteristic
517C2×C4#93911yescharacteristic
518C2×C4#93911yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4C22×#6711yescharacteristic
522C8—41no—
523C8—41no—
524C8—61no—
525C8—61no—
526C8—61no—
527C8—61no—
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#519912yes—
547C22C2×#93911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4—241no—
558C4#577811yescharacteristic
559C4#577911yescharacteristic
560C4#583711yescharacteristic
561C4#583811yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—22no—
564C2—81no—
565C2—481no—
566C2#8340011yescharacteristic
567C2C2×#519911yescharacteristic
568C2#8397111yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3238C2
socleC22×C3C2×#121
prime radical#83148C2
hypercenter#3238D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3238

Complements

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

NH ≅ G/Ntype
1. C3×#3238C2semidirect
2. #83215C2semidirect
4. #83132C2semidirect
5. C2×#4546C2semidirect
6. #71154C2semidirect
7. #71153C2semidirect
8. #71150C2semidirect
9. #71149C2semidirect
10. #71138C2semidirect
11. #71137C2semidirect
12. #71134C2semidirect
13. #71133C2semidirect
14. #70816C2semidirect
15. #70797C2semidirect
69. #3238D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
C3×#32382
#832152
#831482
#831322
C2×#45462
#711542
#711532
#711502
#711492
#711382
#711372
#711342
#711332
#708162
#707972
#195913