Finite Group Databasebeta

← #10950925

Subgroups — #10950925

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#80714C211yesmaximal, characteristic, complemented
2D3×#466C211yesmaximal, characteristic, complemented
3C3×#2215C211yesmaximal, characteristic, complemented
4#71468C211yesmaximal, characteristic, complemented
5#71467C211yesmaximal, characteristic, complemented
6#71414C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊C16C211yesmaximal, characteristic, complemented
8#71072C211yesmaximal, characteristic, complemented
9#71071C211yesmaximal, characteristic, complemented
10#69331C211yesmaximal, characteristic, complemented
11#69323C211yesmaximal, characteristic, complemented
12#69200C211yesmaximal, characteristic, complemented
13#69193C211yesmaximal, characteristic, complemented
14#69192C211yesmaximal, characteristic, complemented
15#69182C211yesmaximal, characteristic, complemented
16#33754—31nomaximal, Sylow
17C2×#912C2211yescharacteristic, complemented
18C2×D3×D8C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20#5606—21no—
21D3×#121—21no—
22C2×C3×D16C2211yescharacteristic, complemented
23C22×C3×C16C2211yescharacteristic, complemented
24C3×#466C2211yescharacteristic
25C3×#466C2211yescharacteristic, complemented
26#5157C2211yescharacteristic
27#5157C2211yescharacteristic
28C2×#827C2211yescharacteristic, complemented
29C2×#826C2211yescharacteristic, complemented
30#5116C2211yescharacteristic, complemented
31#5115C2211yescharacteristic
32C22×C3⋊C16C2211yescharacteristic, complemented
33#5041—21no—
34#5038—21no—
35C2×#800C2211yescharacteristic, complemented
36#4887C2211yescharacteristic
37#4883C2211yescharacteristic
38#4880C2211yescharacteristic, complemented
39D3×#107C2211yescharacteristic
40#4778—21no—
41#4775—21no—
42#4772—21no—
43D3×#102—21no—
44C3×#366C2211yescharacteristic, complemented
45C3×#357C2211yescharacteristic, complemented
46C3×#357C2211yescharacteristic, complemented
47#4562C2211yescharacteristic, 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#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#2215D311yescharacteristic, complemented
70#2187—31no—
71#2186—31no—
72#2186—31no—
73#2184—31no—
74C2×C22⋊C16—31no—
75C2×#237—21no—
76C2×D3×D4—21no—
77C2×D3×D4—41no—
78C22×C4×D3D411yescharacteristic
79C2×C3×D8C2311yescharacteristic
80C2×C3×D8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#121—21no—
83C3×#121—21no—
84#939—41no—
85#938—21no—
86#938—21no—
87#937—41no—
88D4×Dic3—41no—
89C2×#214—21no—
90C2×#213—21no—
91#930—21no—
92#924—21no—
93C22×C3⋊C8D411yescharacteristic
94#912—41no—
95#912—41no—
96D3×D8—41no—
97D3×D8—41no—
98C2×Dic12C2311yescharacteristic
99C2×D24C2311yescharacteristic
100C2×D3×C8—41no—
101C2×D3×C8—41no—
102C2×D3×C8D411yescharacteristic
103C2×D3×C8D411yescharacteristic
104C2×D3×C8C2311yescharacteristic
105C2×D3×C8C2311yescharacteristic
106#897—21no—
107#895—21no—
108#892—21no—
109D3×C4⋊C4—21no—
110#884—41no—
111#883—41no—
112#67⋊C4—41no—
113D3×C22⋊C4—41no—
114C3×D16—41no—
115C2×C3×C16—41no—
116C2×C3×C16C2311yescharacteristic
117C2×C3×C16C2311yescharacteristic
118C3×#107C2311yescharacteristic
119C3×#107C2311yescharacteristic
120C3×#102—21no—
121C3×#102—21no—
122#832—21no—
123#832—21no—
124#827—41no—
125#826—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#808—21no—
136#807—21no—
137#800—41no—
138C2×#135—31no—
139C22×D8—31no—
140C23×C8—31no—
141#525—61no—
142C2×#121—61no—
143C2×SD32—31no—
144C2×SD32—31no—
145C2×D16—31no—
146C2×D16C2×D311yescharacteristic, complemented
147C22×C16—31no—
148C22×C16C2×D311yescharacteristic, complemented
149#467—31no—
150#467—31no—
151#466—31no—
152#466—31no—
153#466—31no—
154#466—31no—
155#466—61no—
156#466—61no—
157#466C2×D311yescharacteristic
158#466C2×D311yescharacteristic, complemented
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#366C2×D311yescharacteristic, complemented
169#358—31no—
170#358—31no—
171#357—31no—
172#357—31no—
173#357—31no—
174#357—31no—
175#357C2×D311yescharacteristic, complemented
176#357C2×D311yescharacteristic, complemented
177C22⋊C16—31no—
178C22⋊C16—31no—
179C22⋊C16—31no—
180C22⋊C16—31no—
181C23×D3—41no—
182C23×D3D811yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C22×C3×C4C2×D411yescharacteristic
188C2×#67—41no—
189C2×#67—41no—
190C2×#67—41no—
191C2×#67—41no—
192C2×#67—41no—
193C22×Dic3—41no—
194C22×Dic3D811yescharacteristic
195#237—42no—
196#237—42no—
197D3×D4—42no—
198D3×D4—42no—
199D3×D4—84no—
200C2×D12—21no—
201C2×D12—41no—
202C2×C4×D3—41no—
203C2×C4×D3—41no—
204C2×C4×D3—41no—
205C2×C4×D3D811yescharacteristic
206C2×C4×D3D811yescharacteristic
207C2×C4×D3C2×D411yescharacteristic
208C2×C4×D3C2×D411yescharacteristic
209C2×Dic6—21no—
210C3×D8—21no—
211C3×D8—21no—
212C3×D8—21no—
213C3×D8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8C2×D411yescharacteristic
217C2×C3×C8C2411yescharacteristic
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—41no—
222#217—41no—
223#217—41no—
224#217—41no—
225#214—42no—
226#213—42no—
227(C2×D3)⋊C4—41no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—41no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—41no—
234Dic3⋊C4—41no—
235Dic3⋊C4—41no—
236C4×Dic3—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—
254C2×#41—61no—
255C22×D4—61no—
256C22×D4—121no—
257C23×C4—31no—
258#135—121no—
259#135—121no—
260C2×Q16—31no—
261C2×SD16—61no—
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—21no—
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—121no—
294SD32—121no—
295D16—41no—
296D16—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#107C22×D311yescharacteristic
307#107C22×D311yescharacteristic
308#106—61no—
309#103—61no—
310#103—61no—
311#102—21no—
312#102—21no—
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×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—41no—
333C3×D4—82no—
334C3×D4—82no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4C2×D811yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341#67—82no—
342#67—82no—
343#67—82no—
344#67—82no—
345#67—82no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—82no—
351C2×Dic3C2×D811yescharacteristic
352C2×Dic3C2×D811yescharacteristic
353D12—41no—
354D12—41no—
355D12—82no—
356C4×D3—21no—
357C4×D3—21no—
358C4×D3—21no—
359C4×D3—21no—
360C4×D3—41no—
361C4×D3—41no—
362C4×D3—82no—
363Dic6—41no—
364Dic6—41no—
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—122no—
373#41—122no—
374C2×Q8—61no—
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—121no—
395C22×C4—121no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4D3×D411yescharacteristic
399Q16—61no—
400Q16—61no—
401SD16—122no—
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—21no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—121no—
427C4⋊C4—121no—
428C4⋊C4—121no—
429C22⋊C4—41no—
430C22⋊C4—41no—
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—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3#50912yes—
452C22×C3C22×D811yescharacteristic
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—21no—
458C2×D3—44no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—164no—
463C2×D3#50812yes—
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—121no—
487Q8—121no—
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—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—242no—
519C2×C4—242no—
520C2×C4D3×D811yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C8—21no—
523C8—61no—
524C8—61no—
525C8#100411yescharacteristic
526C8#100911yescharacteristic
527C2×C3—22no—
528C2×C3—161no—
529C2×C3—161no—
530C2×C3#318611yescharacteristic
531C2×C3C2×#50811yescharacteristic
532C2×C3C2×#50911yescharacteristic
533D3—22no—
534D3—41no—
535D3—161no—
536C22—21no—
537C22—21no—
538C22—32no—
539C22—61no—
540C22—61no—
541C22—61no—
542C22—61no—
543C22—61no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—124no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22—484no—
555C22#491112yes—
556C22C2×D3×D811yescharacteristic
557C4—21no—
558C4—61no—
559C4—61no—
560C4—81no—
561C4—81no—
562C4—241no—
563C4—241no—
564C4—241no—
565C4#560511yescharacteristic
566C4#576811yescharacteristic
567C3#3375411yescharacteristic, complemented, Sylow
568C2—22no—
569C2—62no—
570C2—121no—
571C2—161no—
572C2—161no—
573C2—481no—
574C2#8071911yescharacteristic
575C2C2×#491111yescharacteristic
576C2D3×#50811yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2215C2
socleC22×C3C22×D8
hypercenter#2215D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. #80714C2semidirect
2. D3×#466C2semidirect
3. C3×#2215C2semidirect
4. #71468C2semidirect
5. #71467C2semidirect
6. #71414C2semidirect
7. C2×(C2×D3)⋊C16C2semidirect
8. #71072C2semidirect
9. #71071C2semidirect
10. #69331C2semidirect
11. #69323C2semidirect
12. #69200C2semidirect
13. #69193C2semidirect
14. #69192C2semidirect
15. #69182C2semidirect
17. C2×#912C22semidirect
22. C2×C3×D16C22semidirect
23. C22×C3×C16C22semidirect
25. C3×#466C22semidirect
28. C2×#827C22semidirect
29. C2×#826C22semidirect
30. #5116C22semidirect
32. C22×C3⋊C16C22semidirect
35. C2×#800C22semidirect
38. #4880C22semidirect
44. C3×#366C22semidirect
45. C3×#357C22semidirect
46. C3×#357C22semidirect
47. #4562C22semidirect
48. #4562C22semidirect
49. #4509C22semidirect
52. (C2×D3)⋊C16C22semidirect
53. (C2×D3)⋊C16C22semidirect
54. #4505C22semidirect
55. #4503C22semidirect
56. #4492C22semidirect
57. #4491C22semidirect
58. #4490C22semidirect
59. #4489C22semidirect
69. #2215D3semidirect
146. C2×D16C2×D3semidirect
148. C22×C16C2×D3semidirect
158. #466C2×D3semidirect
168. #366C2×D3semidirect
175. #357C2×D3semidirect
176. #357C2×D3semidirect
567. C3#33754semidirect

Maximal subgroups

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

subgroupindex
#807142
D3×#4662
C3×#22152
#714682
#714672
#714142
C2×(C2×D3)⋊C162
#710722
#710712
#693312
#693232
#692002
#691932
#691922
#691822
#337543