Finite Group Databasebeta

← #10950941

Subgroups — #10950941

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#467C211yesmaximal, characteristic, complemented
2#80714C211yesmaximal, characteristic
3C3×#2217C211yesmaximal, characteristic, complemented
4#71486C211yesmaximal, characteristic, complemented
5#71483C211yesmaximal, characteristic, complemented
6#71416C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊C16C211yesmaximal, characteristic, complemented
8#71081C211yesmaximal, characteristic, complemented
9#71072C211yesmaximal, characteristic, complemented
10#69352C211yesmaximal, characteristic, complemented
11#69342C211yesmaximal, characteristic, complemented
12#69225C211yesmaximal, characteristic, complemented
13#69214C211yesmaximal, characteristic, complemented
14#69199C211yesmaximal, characteristic, complemented
15#69194C211yesmaximal, characteristic, complemented
16#33756—31nomaximal, Sylow
17C2×D3×Q16C2211yescharacteristic
18C2×#912C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20D3×C22⋊Q8—21no—
21#5606—21no—
22C2×C3×SD32C2211yescharacteristic, complemented
23C22×C3×C16C2211yescharacteristic
24C3×#467C2211yescharacteristic
25C3×#466C2211yescharacteristic
26#5188C2211yescharacteristic
27C2×#829C2211yescharacteristic, complemented
28#5157C2211yescharacteristic
29C2×#827C2211yescharacteristic, complemented
30#5116C2211yescharacteristic
31#5116C2211yescharacteristic
32C22×C3⋊C16C2211yescharacteristic
33#5051—21no—
34#5041—21no—
35C2×Dic24C2211yescharacteristic, complemented
36#4887C2211yescharacteristic
37#4887C2211yescharacteristic
38#4880C2211yescharacteristic
39D3×#107C2211yescharacteristic
40#4813—21no—
41D3×#103—21no—
42#4778—21no—
43#4772—21no—
44C3×#367C2211yescharacteristic, complemented
45C3×#358C2211yescharacteristic, complemented
46C3×#357C2211yescharacteristic, complemented
47#4564C2211yescharacteristic, complemented
48#4562C2211yescharacteristic, complemented
49(C2×D3)⋊C16C2211yescharacteristic
50(C2×D3)⋊C16C2211yescharacteristic
51(C2×D3)⋊C16C2211yescharacteristic
52(C2×D3)⋊C16C2211yescharacteristic
53#4504C2211yescharacteristic, complemented
54#4503C2211yescharacteristic, complemented
55#4503C2211yescharacteristic, complemented
56#4492C2211yescharacteristic, complemented
57#4492C2211yescharacteristic, complemented
58#4490C2211yescharacteristic, complemented
59#4489C2211yescharacteristic, complemented
60#3050—31no—
61C2×#467—31no—
62#2237—31no—
63#2237—31no—
64#2235—31no—
65#2232—31no—
66#2217—31no—
67#2217D311yescharacteristic, complemented
68#2216—31no—
69#2216—31no—
70#2189—31no—
71#2189—31no—
72#2188—31no—
73#2185—31no—
74C2×C22⋊C16—31no—
75C2×D3×Q8—21no—
76C2×#237—21no—
77C22×C4×D3D411yescharacteristic
78C2×C3×Q16C2311yescharacteristic
79C2×C3×D8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C22⋊Q8—21no—
82C3×#121—21no—
83#946—21no—
84C2×#216—21no—
85#938—21no—
86#937—41no—
87D4×Dic3—41no—
88C2×#214—21no—
89#924—21no—
90#924—21no—
91C22×C3⋊C8D411yescharacteristic
92D3×Q16—41no—
93D3×Q16—41no—
94#912—41no—
95#912—41no—
96C2×Dic12C2311yescharacteristic
97C2×Dic12C2311yescharacteristic
98C2×D3×C8—41no—
99C2×D3×C8—41no—
100C2×D3×C8D411yescharacteristic
101C2×D3×C8D411yescharacteristic
102C2×D3×C8C2311yescharacteristic
103C2×D3×C8C2311yescharacteristic
104#897—21no—
105#897—21no—
106#896—41no—
107#892—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—41no—
110#883—41no—
111#67⋊C4—41no—
112D3×C22⋊C4—41no—
113#878—41no—
114C3×SD32—41no—
115C2×C3×C16—41no—
116C2×C3×C16C2311yescharacteristic
117C2×C3×C16C2311yescharacteristic
118C3×#107C2311yescharacteristic
119C3×#107C2311yescharacteristic
120C3×#103—21no—
121C3×#102—21no—
122#835—21no—
123#832—21no—
124#829—41no—
125#827—41no—
126#818C2311yescharacteristic
127#818C2311yescharacteristic
128#816—21no—
129#816—21no—
130C2×C3⋊C16—41no—
131C2×C3⋊C16C2311yescharacteristic
132C2×C3⋊C16C2311yescharacteristic
133#810—21no—
134#810—21no—
135#808—21no—
136#807—21no—
137Dic24—41no—
138C2×#135—31no—
139C22×Q16—31no—
140C23×C8—31no—
141#525—61no—
142C2×C22⋊Q8—61no—
143C2×Q32—31no—
144C2×Q32—31no—
145C2×SD32—31no—
146C2×SD32C2×D311yescharacteristic, complemented
147C22×C16—31no—
148C22×C16C2×D311yescharacteristic
149#467—31no—
150#467—31no—
151#467—31no—
152#467—31no—
153#467—31no—
154#467—61no—
155#467—61no—
156#467C2×D311yescharacteristic
157#466—31no—
158#466C2×D311yescharacteristic
159#465—61no—
160#465—61no—
161#427—31no—
162C2×#107—31no—
163#416—61no—
164C2×#103—61no—
165#367—31no—
166#367C2×D311yescharacteristic, complemented
167#366—31no—
168#366—31no—
169#358—31no—
170#358—31no—
171#358—31no—
172#358—31no—
173#358—31no—
174#358C2×D311yescharacteristic, complemented
175#357—31no—
176#357C2×D311yescharacteristic, complemented
177C22⋊C16—31no—
178C22⋊C16—31no—
179C22⋊C16—31no—
180C22⋊C16—31no—
181C23×D3D811yescharacteristic
182C2×C3×Q8—21no—
183C2×C3×D4—21no—
184C2×C3×D4—41no—
185C22×C3×C4C2×D411yescharacteristic
186C2×#67—41no—
187C2×#67—41no—
188C22×Dic3—41no—
189C22×Dic3D811yescharacteristic
190D3×Q8—42no—
191D3×Q8—42no—
192#237—42no—
193#237—42no—
194C2×C4×D3—41no—
195C2×C4×D3—41no—
196C2×C4×D3—41no—
197C2×C4×D3—41no—
198C2×C4×D3D811yescharacteristic
199C2×C4×D3D811yescharacteristic
200C2×C4×D3C2×D411yescharacteristic
201C2×C4×D3C2×D411yescharacteristic
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—41no—
205C3×Q16—21no—
206C3×Q16—21no—
207C3×D8—21no—
208C3×D8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8C2×D411yescharacteristic
212C2×C3×C8C2411yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—41no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—41no—
218#217—41no—
219#217—41no—
220#217—41no—
221#216—42no—
222#214—42no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—41no—
225(C2×D3)⋊C4—41no—
226(C2×D3)⋊C4—41no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—41no—
230C4⋊Dic3—41no—
231Dic3⋊C4—41no—
232Dic3⋊C4—41no—
233Dic3⋊C4—41no—
234Dic3⋊C4—41no—
235C4×Dic3—41no—
236C2×C3⋊C8—41no—
237C2×C3⋊C8C2×D411yescharacteristic
238C2×C3⋊C8C2×D411yescharacteristic
239Dic12—21no—
240Dic12—21no—
241Dic12—21no—
242Dic12—21no—
243D3×C8—21no—
244D3×C8—21no—
245D3×C8—21no—
246D3×C8—21no—
247D3×C8—41no—
248D3×C8—41no—
249C3×C16—21no—
250C3×C16—21no—
251C3⋊C16—21no—
252C3⋊C16—21no—
253C2×#41—61no—
254C22×Q8—61no—
255C23×C4—31no—
256#135—121no—
257#135—121no—
258C2×Q16—31no—
259C2×Q16—31no—
260C2×Q16—61no—
261C2×Q16—121no—
262C2×Q16—121no—
263C2×Q16C22×D311yescharacteristic
264C2×SD16—61no—
265C2×D8C22×D311yescharacteristic
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—121no—
272C22×C8—121no—
273C22×C8C22×D311yescharacteristic
274#123—121no—
275C22⋊Q8—21no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278C22⋊Q8—61no—
279C22⋊Q8—61no—
280C22⋊Q8—61no—
281C22⋊Q8—121no—
282C22⋊Q8—121no—
283#121—21no—
284#121—61no—
285#120—121no—
286C4×D4—121no—
287C4×D4—121no—
288#117—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—121no—
291C2×C22⋊C4—121no—
292Q32—121no—
293Q32—121no—
294SD32—41no—
295SD32—121no—
296C2×C16—31no—
297C2×C16—31no—
298C2×C16—41no—
299C2×C16—121no—
300C2×C16C22×D311yescharacteristic
301C2×C16C22×D311yescharacteristic
302#107—31no—
303#107—31no—
304#107—61no—
305#107C22×D311yescharacteristic
306#107C22×D311yescharacteristic
307#106—61no—
308#103—21no—
309#103—61no—
310#103—61no—
311#103—61no—
312#103—61no—
313#103—61no—
314#102—21no—
315#102—61no—
316C23×C3—41no—
317C23×C3C2×D811yescharacteristic
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—41no—
321C22×D3C2×D811yescharacteristic
322C22×D3C2×D811yescharacteristic
323C3×Q8—41no—
324C3×Q8—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—82no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4C2×D811yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335#67—82no—
336#67—82no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—82no—
343C2×Dic3C2×D811yescharacteristic
344C2×Dic3C2×D811yescharacteristic
345C4×D3—21no—
346C4×D3—21no—
347C4×D3—21no—
348C4×D3—21no—
349C4×D3—41no—
350C4×D3—41no—
351C4×D3—82no—
352C4×D3—82no—
353Dic6—41no—
354Dic6—41no—
355Dic6—41no—
356Dic6—41no—
357Dic6—82no—
358C3×C8—21no—
359C3×C8H(5,𝔽2)11yescharacteristic
360C3×C8H(5,𝔽2)11yescharacteristic
361C3⋊C8—21no—
362C3⋊C8—21no—
363C24—31no—
364#41—122no—
365#41—122no—
366C2×Q8—21no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—121no—
370C2×Q8—122no—
371C2×Q8—122no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—121no—
375C2×D4—121no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4D3×D411yescharacteristic
387Q16—21no—
388Q16—21no—
389Q16—61no—
390Q16—61no—
391Q16—61no—
392Q16—61no—
393Q16—122no—
394SD16—122no—
395D8—21no—
396D8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—31no—
400C2×C8—31no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—121no—
406C2×C8—121no—
407C2×C8—121no—
408C2×C8D3×D411yescharacteristic
409C2×C8C23×D311yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4—121no—
416C4⋊C4—121no—
417C4⋊C4—121no—
418C4⋊C4—121no—
419C4⋊C4—121no—
420C4⋊C4—121no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C22⋊C4—121no—
426C22⋊C4—121no—
427C22⋊C4—121no—
428C22⋊C4—121no—
429C22⋊C4—121no—
430C42—121no—
431C16—21no—
432C16—21no—
433C16—61no—
434C16—61no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3#51012yes—
441C22×C3C22×D811yescharacteristic
442C2×D3—21no—
443C2×D3—21no—
444C2×D3—21no—
445C2×D3—21no—
446C2×D3—21no—
447C2×D3—44no—
448C2×D3#50812yes—
449C3×C4—21no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4#53411yescharacteristic
454C3×C4#57511yescharacteristic
455Dic3—21no—
456Dic3—21no—
457Dic3—81no—
458Dic3—81no—
459Dic3—81no—
460Dic3—81no—
461C23—31no—
462C23—31no—
463C23—41no—
464C23—62no—
465C23—62no—
466C23—121no—
467C23D3×D811yescharacteristic
468Q8—41no—
469Q8—41no—
470Q8—121no—
471Q8—121no—
472Q8—121no—
473Q8—121no—
474Q8—242no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—242no—
479D4—242no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—31no—
483C2×C4—31no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—242no—
499C2×C4—242no—
500C2×C4—242no—
501C2×C4D3×D811yescharacteristic
502C2×C4C2×D3×D411yescharacteristic
503C8—21no—
504C8—61no—
505C8—61no—
506C8#100411yescharacteristic
507C8#100911yescharacteristic
508C2×C3—22no—
509C2×C3—161no—
510C2×C3#318611yescharacteristic
511C2×C3C2×#50811yescharacteristic
512C2×C3C2×#51011yescharacteristic
513D3—22no—
514D3—41no—
515C22—21no—
516C22—21no—
517C22—32no—
518C22—61no—
519C22—61no—
520C22—61no—
521C22—61no—
522C22—61no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—124no—
527C22#491512yes—
528C22C2×D3×D811yescharacteristic
529C4—21no—
530C4—61no—
531C4—61no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—241no—
536C4—241no—
537C4—241no—
538C4—241no—
539C4#560511yescharacteristic
540C4#576811yescharacteristic
541C3#3375611yescharacteristic, complemented, Sylow
542C2—22no—
543C2—62no—
544C2—121no—
545C2—161no—
546C2#8071911yescharacteristic
547C2C2×#491511yescharacteristic
548C2D3×#50811yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
prime radical#80714C2
hypercenter#2217D3
hypocenterC3#33756
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. D3×#467C2semidirect
3. C3×#2217C2semidirect
4. #71486C2semidirect
5. #71483C2semidirect
6. #71416C2semidirect
7. C2×(C2×D3)⋊C16C2semidirect
8. #71081C2semidirect
9. #71072C2semidirect
10. #69352C2semidirect
11. #69342C2semidirect
12. #69225C2semidirect
13. #69214C2semidirect
14. #69199C2semidirect
15. #69194C2semidirect
22. C2×C3×SD32C22semidirect
27. C2×#829C22semidirect
29. C2×#827C22semidirect
35. C2×Dic24C22semidirect
44. C3×#367C22semidirect
45. C3×#358C22semidirect
46. C3×#357C22semidirect
47. #4564C22semidirect
48. #4562C22semidirect
53. #4504C22semidirect
54. #4503C22semidirect
55. #4503C22semidirect
56. #4492C22semidirect
57. #4492C22semidirect
58. #4490C22semidirect
59. #4489C22semidirect
67. #2217D3semidirect
146. C2×SD32C2×D3semidirect
166. #367C2×D3semidirect
174. #358C2×D3semidirect
176. #357C2×D3semidirect
541. C3#33756semidirect

Maximal subgroups

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

subgroupindex
D3×#4672
#807142
C3×#22172
#714862
#714832
#714162
C2×(C2×D3)⋊C162
#710812
#710722
#693522
#693422
#692252
#692142
#691992
#691942
#337563