Finite Group Databasebeta

← #10950917

Subgroups — #10950917

Order 768.

642 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
1D3×#467C211yesmaximal, characteristic, complemented
2D3×#466C211yesmaximal, characteristic
3C3×#2217C211yesmaximal, characteristic, complemented
4#71485C211yesmaximal, characteristic, complemented
5#71483C211yesmaximal, characteristic, complemented
6#71414C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊C16C211yesmaximal, characteristic, complemented
8#71082C211yesmaximal, characteristic, complemented
9#71072C211yesmaximal, characteristic, complemented
10#69352C211yesmaximal, characteristic, complemented
11#69350C211yesmaximal, characteristic, complemented
12#69231C211yesmaximal, characteristic, complemented
13#69223C211yesmaximal, characteristic, complemented
14#69200C211yesmaximal, characteristic, complemented
15#69193C211yesmaximal, characteristic, complemented
16#33753—31nomaximal, Sylow
17C2×D3×Q16C2211yescharacteristic, complemented
18C2×D3×D8C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20D3×C22⋊Q8—21no—
21D3×#121—21no—
22C2×C3×SD32C2211yescharacteristic, complemented
23C22×C3×C16C2211yescharacteristic, complemented
24C3×#467C2211yescharacteristic, complemented
25C3×#466C2211yescharacteristic
26#5188C2211yescharacteristic
27C2×#828C2211yescharacteristic, complemented
28#5157C2211yescharacteristic
29C2×#827C2211yescharacteristic, complemented
30#5116C2211yescharacteristic, complemented
31#5115C2211yescharacteristic
32C22×C3⋊C16C2211yescharacteristic, complemented
33#5051—21no—
34#5038—21no—
35C2×#800C2211yescharacteristic, complemented
36#4887C2211yescharacteristic
37#4883C2211yescharacteristic
38D3×#107C2211yescharacteristic
39D3×#107C2211yescharacteristic, complemented
40#4813—21no—
41D3×#103—21no—
42#4775—21no—
43D3×#102—21no—
44C3×#367C2211yescharacteristic, complemented
45C3×#358C2211yescharacteristic, complemented
46C3×#357C2211yescharacteristic, complemented
47#4564C2211yescharacteristic, complemented
48#4562C2211yescharacteristic, complemented
49#4509C2211yescharacteristic, complemented
50(C2×D3)⋊C16C2211yescharacteristic
51(C2×D3)⋊C16C2211yescharacteristic
52(C2×D3)⋊C16C2211yescharacteristic, complemented
53(C2×D3)⋊C16C2211yescharacteristic, complemented
54#4505C2211yescharacteristic, complemented
55#4503C2211yescharacteristic, complemented
56#4492C2211yescharacteristic, complemented
57#4491C2211yescharacteristic, complemented
58#4490C2211yescharacteristic, complemented
59#4490C2211yescharacteristic, complemented
60C2×#467—31no—
61C2×#466—31no—
62#2235—31no—
63#2235—31no—
64#2233—31no—
65#2233—31no—
66#2217—31no—
67#2217—31no—
68#2217—31no—
69#2217D311yescharacteristic, complemented
70#2187—31no—
71#2187—31no—
72#2185—31no—
73#2185—31no—
74C2×C22⋊C16—31no—
75C2×D3×Q8—21no—
76C2×D3×D4—21no—
77C2×D3×D4—41no—
78C22×C4×D3D411yescharacteristic
79C2×C3×Q16C2311yescharacteristic
80C2×C3×D8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C22⋊Q8—21no—
83C3×#121—21no—
84#946—21no—
85C2×#216—21no—
86#939—41no—
87#938—21no—
88C2×#213—21no—
89#930—21no—
90#924—21no—
91C22×C3⋊C8D411yescharacteristic
92D3×Q16—41no—
93D3×Q16—41no—
94D3×D8—41no—
95D3×D8—41no—
96C2×Dic12C2311yescharacteristic
97C2×D24C2311yescharacteristic
98C2×D3×C8—41no—
99C2×D3×C8—41no—
100C2×D3×C8D411yescharacteristic
101C2×D3×C8D411yescharacteristic
102C2×D3×C8C2311yescharacteristic
103C2×D3×C8C2311yescharacteristic
104#897—21no—
105#896—41no—
106#895—21no—
107D3×C4⋊C4—21no—
108D3×C4⋊C4—21no—
109D3×C4⋊C4—41no—
110#884—41no—
111D3×C22⋊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#828—41no—
125#827—41no—
126#821—21no—
127#818C2311yescharacteristic
128#818C2311yescharacteristic
129#816—21no—
130C2×C3⋊C16—41no—
131C2×C3⋊C16C2311yescharacteristic
132C2×C3⋊C16C2311yescharacteristic
133#810—21no—
134#809—21no—
135#807—21no—
136#807—21no—
137#800—41no—
138C22×Q16—31no—
139C22×D8—31no—
140C23×C8—31no—
141C2×C22⋊Q8—61no—
142C2×#121—61no—
143C2×SD32—31no—
144C2×SD32—31no—
145C2×SD32—31no—
146C2×SD32C2×D311yescharacteristic, complemented
147C22×C16—31no—
148C22×C16C2×D311yescharacteristic, complemented
149#467—31no—
150#467—31no—
151#467—31no—
152#467—61no—
153#467—61no—
154#467C2×D311yescharacteristic, complemented
155#466—31no—
156#466—31no—
157#466—31no—
158#466—61no—
159#466—61no—
160#466C2×D311yescharacteristic
161C2×#107—31no—
162C2×#107—31no—
163C2×#103—61no—
164C2×#102—61no—
165#367—31no—
166#367—31no—
167#367—31no—
168#367C2×D311yescharacteristic, complemented
169#358—31no—
170#358—31no—
171#358—31no—
172#358C2×D311yescharacteristic, complemented
173#357—31no—
174#357—31no—
175#357—31no—
176#357C2×D311yescharacteristic, complemented
177C22⋊C16—31no—
178C22⋊C16—31no—
179C22⋊C16—31no—
180C22⋊C16—31no—
181C23×D3—41no—
182C23×D3D811yescharacteristic
183C2×C3×Q8—21no—
184C2×C3×D4—21no—
185C2×C3×D4—41no—
186C22×C3×C4C2×D411yescharacteristic
187C2×#67—41no—
188C2×#67—41no—
189C2×#67—41no—
190C22×Dic3D811yescharacteristic
191D3×Q8—42no—
192D3×Q8—42no—
193D3×D4—42no—
194D3×D4—42no—
195D3×D4—84no—
196C2×D12—21no—
197C2×D12—41no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C4×D3—41no—
201C2×C4×D3—41no—
202C2×C4×D3—41no—
203C2×C4×D3D811yescharacteristic
204C2×C4×D3D811yescharacteristic
205C2×C4×D3C2×D411yescharacteristic
206C2×C4×D3C2×D411yescharacteristic
207C2×Dic6—21no—
208C2×Dic6—41no—
209C3×Q16—21no—
210C3×Q16—21no—
211C3×D8—21no—
212C3×D8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8C2×D411yescharacteristic
216C2×C3×C8C2411yescharacteristic
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—41no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—41no—
222#217—41no—
223#217—41no—
224#216—42no—
225#213—42no—
226(C2×D3)⋊C4—41no—
227(C2×D3)⋊C4—41no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—41no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—41no—
233Dic3⋊C4—41no—
234Dic3⋊C4—41no—
235Dic3⋊C4—41no—
236Dic3⋊C4—41no—
237C2×C3⋊C8—41no—
238C2×C3⋊C8C2×D411yescharacteristic
239C2×C3⋊C8C2×D411yescharacteristic
240Dic12—21no—
241Dic12—21no—
242D24—21no—
243D24—21no—
244D3×C8—21no—
245D3×C8—21no—
246D3×C8—21no—
247D3×C8—21no—
248D3×C8—41no—
249D3×C8—41no—
250C3×C16—21no—
251C3×C16—21no—
252C3⋊C16—21no—
253C3⋊C16—21no—
254C22×Q8—61no—
255C22×D4—61no—
256C22×D4—121no—
257C23×C4—31no—
258C2×Q16—31no—
259C2×Q16—61no—
260C2×Q16—121no—
261C2×Q16—121no—
262C2×Q16C22×D311yescharacteristic
263C2×D8—31no—
264C2×D8—61no—
265C2×D8—121no—
266C2×D8—121no—
267C2×D8C22×D311yescharacteristic
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—31no—
273C22×C8—121no—
274C22×C8—121no—
275C22×C8C22×D311yescharacteristic
276C22⋊Q8—21no—
277C22⋊Q8—61no—
278C22⋊Q8—61no—
279C22⋊Q8—61no—
280C22⋊Q8—121no—
281C22⋊Q8—121no—
282#121—21no—
283#121—61no—
284#121—61no—
285#121—61no—
286#121—121no—
287#121—121no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—121no—
291C2×C22⋊C4—121no—
292C2×C22⋊C4—121no—
293SD32—41no—
294SD32—121no—
295SD32—121no—
296SD32—121no—
297C2×C16—31no—
298C2×C16—31no—
299C2×C16—41no—
300C2×C16—121no—
301C2×C16C22×D311yescharacteristic
302C2×C16C22×D311yescharacteristic
303#107—31no—
304#107—31no—
305#107—61no—
306#107—61no—
307#107C22×D311yescharacteristic
308#107C22×D311yescharacteristic
309#103—21no—
310#103—61no—
311#103—61no—
312#103—61no—
313#102—21no—
314#102—61no—
315#102—61no—
316#102—61no—
317C23×C3—41no—
318C23×C3C2×D811yescharacteristic
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—82no—
323C22×D3—82no—
324C22×D3—82no—
325C22×D3SD3214yescomplemented
326C22×D3C2×D811yescharacteristic
327C22×D3C2×D811yescharacteristic
328C3×Q8—41no—
329C3×Q8—41no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—82no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4C2×D811yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340#67—82no—
341#67—82no—
342#67—82no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3C2×D811yescharacteristic
348C2×Dic3C2×D811yescharacteristic
349D12—41no—
350D12—41no—
351D12—82no—
352C4×D3—21no—
353C4×D3—21no—
354C4×D3—21no—
355C4×D3—21no—
356C4×D3—41no—
357C4×D3—41no—
358C4×D3—82no—
359C4×D3—82no—
360C4×D3—82no—
361Dic6—41no—
362Dic6—41no—
363Dic6—82no—
364C3×C8—21no—
365C3×C8H(5,𝔽2)11yescharacteristic
366C3×C8H(5,𝔽2)11yescharacteristic
367C3⋊C8—21no—
368C3⋊C8—21no—
369C24—31no—
370C24—121no—
371C2×Q8—21no—
372C2×Q8—61no—
373C2×Q8—121no—
374C2×Q8—122no—
375C2×Q8—122no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—61no—
379C2×D4—121no—
380C2×D4—121no—
381C2×D4—122no—
382C2×D4—122no—
383C2×D4—121no—
384C2×D4—121no—
385C2×D4—244no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—121no—
392C22×C4—121no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—121no—
396C22×C4D3×D411yescharacteristic
397Q16—21no—
398Q16—21no—
399Q16—61no—
400Q16—61no—
401Q16—122no—
402D8—21no—
403D8—21no—
404D8—61no—
405D8—61no—
406D8—122no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—121no—
416C2×C8—121no—
417C2×C8—121no—
418C2×C8D3×D411yescharacteristic
419C2×C8C23×D311yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—121no—
426C4⋊C4—121no—
427C4⋊C4—121no—
428C4⋊C4—121no—
429C4⋊C4—121no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C22⋊C4—121no—
437C22⋊C4—121no—
438C16—21no—
439C16—21no—
440C16—61no—
441C16—61no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3C2×SD3212yes—
448C22×C3C22×D811yescharacteristic
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—21no—
453C2×D3—44no—
454C2×D3—81no—
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—164no—
458C2×D3C2×SD3212yes—
459C2×D3C2×SD3212yes—
460C3×C4—21no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4#53411yescharacteristic
465C3×C4#57511yescharacteristic
466Dic3—21no—
467Dic3—21no—
468Dic3—81no—
469Dic3—81no—
470Dic3—81no—
471C23—31no—
472C23—31no—
473C23—34no—
474C23—41no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—242no—
479C23—242no—
480C23—242no—
481C23D3×D811yescharacteristic
482Q8—41no—
483Q8—41no—
484Q8—121no—
485Q8—121no—
486Q8—242no—
487D4—41no—
488D4—41no—
489D4—82no—
490D4—121no—
491D4—121no—
492D4—242no—
493D4—242no—
494D4—242no—
495D4—242no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—242no—
514C2×C4—242no—
515C2×C4—242no—
516C2×C4D3×D811yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C8—21no—
519C8—61no—
520C8—61no—
521C8#100411yescharacteristic
522C8#100911yescharacteristic
523C2×C3—22no—
524C2×C3—161no—
525C2×C3#318611yescharacteristic
526C2×C3C22×SD3211yescharacteristic
527C2×C3#341411yescharacteristic
528D3—22no—
529D3—22no—
530D3—161no—
531C22—21no—
532C22—21no—
533C22—32no—
534C22—32no—
535C22—61no—
536C22—61no—
537C22—61no—
538C22—61no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—124no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22—484no—
547C22D3×SD3212yes—
548C22C2×D3×D811yescharacteristic
549C4—21no—
550C4—61no—
551C4—61no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—241no—
556C4—241no—
557C4—241no—
558C4#560511yescharacteristic
559C4#576811yescharacteristic
560C3#3375311yescharacteristic, complemented, Sylow
561C2—22no—
562C2—62no—
563C2—62no—
564C2—161no—
565C2—481no—
566C2#8071911yescharacteristic
567C2C2×D3×SD3211yescharacteristic
568C2#8203011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
prime radicalD3×#466C2
hypercenter#2217D3
hypocenterC3#33753
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. #71485C2semidirect
5. #71483C2semidirect
6. #71414C2semidirect
7. C2×(C2×D3)⋊C16C2semidirect
8. #71082C2semidirect
9. #71072C2semidirect
10. #69352C2semidirect
11. #69350C2semidirect
12. #69231C2semidirect
13. #69223C2semidirect
14. #69200C2semidirect
15. #69193C2semidirect
17. C2×D3×Q16C22semidirect
22. C2×C3×SD32C22semidirect
23. C22×C3×C16C22semidirect
24. C3×#467C22semidirect
27. C2×#828C22semidirect
29. C2×#827C22semidirect
30. #5116C22semidirect
32. C22×C3⋊C16C22semidirect
35. C2×#800C22semidirect
39. D3×#107C22semidirect
44. C3×#367C22semidirect
45. C3×#358C22semidirect
46. C3×#357C22semidirect
47. #4564C22semidirect
48. #4562C22semidirect
49. #4509C22semidirect
52. (C2×D3)⋊C16C22semidirect
53. (C2×D3)⋊C16C22semidirect
54. #4505C22semidirect
55. #4503C22semidirect
56. #4492C22semidirect
57. #4491C22semidirect
58. #4490C22semidirect
59. #4490C22semidirect
69. #2217D3semidirect
146. C2×SD32C2×D3semidirect
148. C22×C16C2×D3semidirect
154. #467C2×D3semidirect
168. #367C2×D3semidirect
172. #358C2×D3semidirect
176. #357C2×D3semidirect
325. C22×D3SD32semidirect
560. C3#33753semidirect

Maximal subgroups

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

subgroupindex
D3×#4672
D3×#4662
C3×#22172
#714852
#714832
#714142
C2×(C2×D3)⋊C162
#710822
#710722
#693522
#693502
#692312
#692232
#692002
#691932
#337533