Finite Group Databasebeta

← #10950928

Subgroups — #10950928

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83914C211yesmaximal, characteristic, complemented
2D3×#466C211yesmaximal, characteristic, complemented
3C3×#2186C211yesmaximal, characteristic, complemented
4#72082C211yesmaximal, characteristic, complemented
5#71487C211yesmaximal, characteristic, complemented
6#71485C211yesmaximal, characteristic, complemented
7#71469C211yesmaximal, characteristic, complemented
8#71467C211yesmaximal, characteristic, complemented
9#71070C211yesmaximal, characteristic, complemented
10#69232C211yesmaximal, characteristic, complemented
11#69228C211yesmaximal, characteristic, complemented
12#69200C211yesmaximal, characteristic, complemented
13#69195C211yesmaximal, characteristic, complemented
14D3×C22⋊C16C211yesmaximal, characteristic, complemented
15#69097C211yesmaximal, characteristic, complemented
16#33754—31nomaximal, Sylow
17C2×C3×#135C2211yescharacteristic, complemented
18#5828—21no—
19C2×D3×D8C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21D3×#121—21no—
22C2×C3×SD32C2211yescharacteristic, complemented
23C2×C3×D16C2211yescharacteristic, complemented
24C3×#466C2211yescharacteristic
25#5233—21no—
26#5188C2211yescharacteristic
27#5188C2211yescharacteristic, complemented
28C2×#828C2211yescharacteristic, complemented
29#5171—21no—
30#5171—21no—
31#5157C2211yescharacteristic
32#5157C2211yescharacteristic
33#5157C2211yescharacteristic, complemented
34C2×#826C2211yescharacteristic, complemented
35#5115C2211yescharacteristic
36#5106C2211yescharacteristic, complemented
37#5038—21no—
38C2×D3×C16C2211yescharacteristic, complemented
39C2×D3×C16C2211yescharacteristic, complemented
40#4883C2211yescharacteristic
41D3×#107C2211yescharacteristic
42#4775—21no—
43D3×#102—21no—
44C3×#358C2211yescharacteristic, complemented
45C3×#357C2211yescharacteristic, complemented
46C3×C22⋊C16C2211yescharacteristic, complemented
47#4564C2211yescharacteristic, complemented
48#4562C2211yescharacteristic, complemented
49#4562C2211yescharacteristic, complemented
50#4562C2211yescharacteristic, complemented
51#4549C2211yescharacteristic, complemented
52#4509C2211yescharacteristic, complemented
53#4509C2211yescharacteristic, complemented
54(C2×D3)⋊C16C2211yescharacteristic
55(C2×D3)⋊C16C2211yescharacteristic
56#4505C2211yescharacteristic, complemented
57#4504C2211yescharacteristic, complemented
58#4490C2211yescharacteristic, complemented
59#4489C2211yescharacteristic, complemented
60#3050—31no—
61C2×#466—31no—
62#2236—31no—
63#2234—31no—
64#2234—31no—
65#2233—31no—
66#2217—31no—
67#2217—31no—
68#2215—31no—
69#2215—31no—
70#2187—31no—
71#2186—31no—
72#2186D311yescharacteristic, complemented
73#2184—31no—
74C2×C22⋊C16—31no—
75C2×C3×#41—21no—
76C2×D3×D4—21no—
77C2×D3×D4—41no—
78C22×C4×D3D411yescharacteristic
79C3×#135—41no—
80C3×#135—41no—
81C2×C3×Q16C2311yescharacteristic
82C2×C3×SD16—21no—
83C2×C3×D8C2311yescharacteristic
84C2×C3×D8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—21no—
87#946—21no—
88#946—21no—
89#939—41no—
90#938—21no—
91#938—21no—
92#938—21no—
93#937—41no—
94#935—41no—
95C2×#213—21no—
96#930—21no—
97C4×#67—41no—
98C4×#67—41no—
99#926—21no—
100C22×C3⋊C8D411yescharacteristic
101D3×D8—41no—
102D3×D8—41no—
103C2×D24C2311yescharacteristic
104C2×D3×C8—41no—
105C2×D3×C8—41no—
106C2×D3×C8D411yescharacteristic
107C2×D3×C8D411yescharacteristic
108C2×D3×C8C2311yescharacteristic
109C2×D3×C8C2311yescharacteristic
110#895—21no—
111D3×C4⋊C4—21no—
112#884—41no—
113D3×C22⋊C4—41no—
114C3×SD32—41no—
115C3×D16—41no—
116C2×C3×C16C2311yescharacteristic
117C2×C3×C16C2311yescharacteristic
118C3×#107C2311yescharacteristic
119C3×#102—21no—
120#835—21no—
121#835—21no—
122#832—21no—
123#832—21no—
124#832—21no—
125#828—41no—
126#826—41no—
127#821—21no—
128#818C2311yescharacteristic
129#818C2311yescharacteristic
130#818C2311yescharacteristic
131#817—21no—
132C2×C3⋊C16C2311yescharacteristic
133C2×C3⋊C16C2311yescharacteristic
134#809—21no—
135#807—21no—
136D3×C16—41no—
137D3×C16—41no—
138C2×#135C2×D311yescharacteristic, complemented
139C22×D8—31no—
140C23×C8—31no—
141#525—61no—
142C2×#121—61no—
143C2×SD32—31no—
144C2×SD32C2×D311yescharacteristic, complemented
145C2×D16—31no—
146C2×D16C2×D311yescharacteristic, complemented
147C22×C16—31no—
148C22×C16—31no—
149#467—31no—
150#467—31no—
151#466—31no—
152#466—31no—
153#466—31no—
154#466—31no—
155#466—31no—
156#466—61no—
157#466—61no—
158#466C2×D311yescharacteristic
159#465—61no—
160#465—61no—
161#427—31no—
162C2×#107—31no—
163#416—61no—
164C2×#102—61no—
165#367—31no—
166#367—31no—
167#366—31no—
168#366—31no—
169#358—31no—
170#358C2×D311yescharacteristic, complemented
171#357—31no—
172#357—31no—
173#357—31no—
174#357—31no—
175#357—31no—
176#357C2×D311yescharacteristic, complemented
177C22⋊C16—31no—
178C22⋊C16—31no—
179C22⋊C16—31no—
180C22⋊C16C2×D311yescharacteristic, complemented
181C23×D3—41no—
182C23×D3D811yescharacteristic
183C3×#41—42no—
184C3×#41—42no—
185C2×C3×Q8—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—41no—
189C2×C3×D4—41no—
190C22×C3×C4—41no—
191C22×C3×C4C2×D411yescharacteristic
192C2×#67—41no—
193C2×#67—41no—
194C2×#67—41no—
195C2×#67—41no—
196C2×#67—41no—
197C22×Dic3D811yescharacteristic
198D3×D4—42no—
199D3×D4—42no—
200D3×D4—84no—
201C2×D12—21no—
202C2×D12—41no—
203C2×C4×D3—41no—
204C2×C4×D3—41no—
205C2×C4×D3—41no—
206C2×C4×D3D811yescharacteristic
207C2×C4×D3D811yescharacteristic
208C2×C4×D3C2×D411yescharacteristic
209C2×C4×D3C2×D411yescharacteristic
210C3×Q16—21no—
211C3×Q16—21no—
212C3×SD16—42no—
213C3×D8—21no—
214C3×D8—21no—
215C3×D8—21no—
216C3×D8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8C2×D411yescharacteristic
220C2×C3×C8C2411yescharacteristic
221C3×C4⋊C4—21no—
222C3×C22⋊C4—41no—
223#217—41no—
224#217—41no—
225#217—41no—
226#217—41no—
227#213—42no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—41no—
231(C2×D3)⋊C4—41no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235Dic3⋊C4—41no—
236Dic3⋊C4—41no—
237Dic3⋊C4—41no—
238C4×Dic3—41no—
239C2×C3⋊C8—41no—
240C2×C3⋊C8C2×D411yescharacteristic
241C2×C3⋊C8C2×D411yescharacteristic
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—
254C2×#41—21no—
255C22×D4—61no—
256C22×D4—121no—
257C23×C4—31no—
258#135—41no—
259#135—41no—
260C2×Q16C22×D311yescharacteristic
261C2×SD16—21no—
262C2×D8—31no—
263C2×D8—61no—
264C2×D8—121no—
265C2×D8—121no—
266C2×D8C22×D311yescharacteristic
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
276#123—121no—
277C22⋊Q8—61no—
278C22⋊Q8—61no—
279#121—21no—
280#121—61no—
281#121—61no—
282#121—61no—
283#121—61no—
284#121—61no—
285#121—121no—
286#121—121no—
287#120—121no—
288C4×D4—121no—
289C4×D4—121no—
290#117—61no—
291C2×C4⋊C4—61no—
292C2×C22⋊C4—121no—
293SD32—41no—
294SD32—121no—
295D16—41no—
296D16—121no—
297C2×C16—31no—
298C2×C16—31no—
299C2×C16—121no—
300C2×C16—121no—
301C2×C16C22×D311yescharacteristic
302C2×C16C22×D311yescharacteristic
303#107—31no—
304#107—31no—
305#107—31no—
306#107—61no—
307#107C22×D311yescharacteristic
308#106—61no—
309#103—61no—
310#103—61no—
311#102—21no—
312#102—61no—
313#102—61no—
314#102—61no—
315#102—61no—
316#102—61no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3C2×D811yescharacteristic
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—82no—
325C22×D3—82no—
326C22×D3—82no—
327C22×D3C2×D811yescharacteristic
328C22×D3C2×D811yescharacteristic
329C3×Q8—41no—
330C3×Q8—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—82no—
336C3×D4—82no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—82no—
342C2×C3×C4C2×D811yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344#67—82no—
345#67—82no—
346#67—82no—
347#67—82no—
348#67—82no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3C2×D811yescharacteristic
354C2×Dic3C2×D811yescharacteristic
355D12—41no—
356D12—41no—
357D12—82no—
358C4×D3—21no—
359C4×D3—21no—
360C4×D3—21no—
361C4×D3—21no—
362C4×D3—41no—
363C4×D3—41no—
364C4×D3—82no—
365C3×C8—21no—
366C3×C8H(5,𝔽2)11yescharacteristic
367C3×C8H(5,𝔽2)11yescharacteristic
368C3⋊C8—21no—
369C3⋊C8—21no—
370C24—31no—
371C24—121no—
372#41—42no—
373#41—42no—
374C2×Q8—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—61no—
380C2×D4—121no—
381C2×D4—121no—
382C2×D4—122no—
383C2×D4—122no—
384C2×D4—121no—
385C2×D4—121no—
386C2×D4—121no—
387C2×D4—121no—
388C2×D4—244no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—41no—
395C22×C4—121no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4D3×D411yescharacteristic
399Q16—21no—
400Q16—21no—
401SD16—42no—
402D8—21no—
403D8—21no—
404D8—21no—
405D8—21no—
406D8—61no—
407D8—61no—
408D8—122no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—121no—
418C2×C8—121no—
419C2×C8—121no—
420C2×C8D3×D411yescharacteristic
421C2×C8C23×D311yescharacteristic
422C4⋊C4—21no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—121no—
427C4⋊C4—121no—
428C4⋊C4—121no—
429C22⋊C4—41no—
430C22⋊C4—121no—
431C22⋊C4—121no—
432C22⋊C4—121no—
433C22⋊C4—121no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C22⋊C4—121no—
437C22⋊C4—121no—
438C42—121no—
439C16—21no—
440C16—21no—
441C16—61no—
442C16—61no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3C22×D811yescharacteristic
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—44no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—164no—
462C2×D3#50812yes—
463C2×D3#50912yes—
464C3×C4—21no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4#53411yescharacteristic
468C3×C4#57511yescharacteristic
469Dic3—21no—
470Dic3—21no—
471Dic3—81no—
472Dic3—81no—
473Dic3—81no—
474C23—31no—
475C23—31no—
476C23—41no—
477C23—41no—
478C23—62no—
479C23—62no—
480C23—121no—
481C23—121no—
482C23—242no—
483C23—242no—
484C23—242no—
485C23D3×D811yescharacteristic
486Q8—41no—
487Q8—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—82no—
493D4—82no—
494D4—121no—
495D4—121no—
496D4—242no—
497D4—242no—
498D4—242no—
499D4—242no—
500D4—242no—
501D4—242no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—82no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—242no—
520C2×C4D3×D811yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C8—21no—
523C8—61no—
524C8—61no—
525C8#100411yescharacteristic
526C8#100411yescharacteristic
527C2×C3—41no—
528C2×C3—161no—
529C2×C3—161no—
530C2×C3#318611yescharacteristic
531C2×C3C2×#50811yescharacteristic
532C2×C3C2×#50911yescharacteristic
533D3—22no—
534D3—22no—
535D3—161no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—32no—
540C22—32no—
541C22—61no—
542C22—61no—
543C22—61no—
544C22—61no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—124no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—484no—
556C22C2×D3×D811yescharacteristic
557C4—21no—
558C4—61no—
559C4—61no—
560C4—81no—
561C4—81no—
562C4—241no—
563C4—241no—
564C4—241no—
565C4#558911yescharacteristic
566C4#577311yescharacteristic
567C3#3375411yescharacteristic, complemented, Sylow
568C2—41no—
569C2—62no—
570C2—62no—
571C2—161no—
572C2—161no—
573C2—481no—
574C2#8000611yescharacteristic
575C2D3×#50811yescharacteristic
576C2D3×#50911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2186C2
socleC22×C3C22×D8
hypercenter#2186D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #83914C2semidirect
2. D3×#466C2semidirect
3. C3×#2186C2semidirect
4. #72082C2semidirect
5. #71487C2semidirect
6. #71485C2semidirect
7. #71469C2semidirect
8. #71467C2semidirect
9. #71070C2semidirect
10. #69232C2semidirect
11. #69228C2semidirect
12. #69200C2semidirect
13. #69195C2semidirect
14. D3×C22⋊C16C2semidirect
15. #69097C2semidirect
17. C2×C3×#135C22semidirect
22. C2×C3×SD32C22semidirect
23. C2×C3×D16C22semidirect
27. #5188C22semidirect
28. C2×#828C22semidirect
33. #5157C22semidirect
34. C2×#826C22semidirect
36. #5106C22semidirect
38. C2×D3×C16C22semidirect
39. C2×D3×C16C22semidirect
44. C3×#358C22semidirect
45. C3×#357C22semidirect
46. C3×C22⋊C16C22semidirect
47. #4564C22semidirect
48. #4562C22semidirect
49. #4562C22semidirect
50. #4562C22semidirect
51. #4549C22semidirect
52. #4509C22semidirect
53. #4509C22semidirect
56. #4505C22semidirect
57. #4504C22semidirect
58. #4490C22semidirect
59. #4489C22semidirect
72. #2186D3semidirect
138. C2×#135C2×D3semidirect
144. C2×SD32C2×D3semidirect
146. C2×D16C2×D3semidirect
170. #358C2×D3semidirect
176. #357C2×D3semidirect
180. C22⋊C16C2×D3semidirect
567. C3#33754semidirect

Maximal subgroups

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

subgroupindex
#839142
D3×#4662
C3×#21862
#720822
#714872
#714852
#714692
#714672
#710702
#692322
#692282
#692002
#691952
D3×C22⋊C162
#690972
#337543