Finite Group Databasebeta

← #10782866

Subgroups — #10782866

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3131C211yesmaximal, characteristic, complemented
2#83344C211yesmaximal, characteristic, complemented
3#83343C211yesmaximal, characteristic, complemented
4#83291C211yesmaximal, characteristic
5#83290C211yesmaximal, characteristic, complemented
6#78463C211yesmaximal, characteristic, complemented
7#71215C211yesmaximal, characteristic, complemented
8#71214C211yesmaximal, characteristic, complemented
9#71213C211yesmaximal, characteristic, complemented
10#71212C211yesmaximal, characteristic, complemented
11#71183C211yesmaximal, characteristic, complemented
12#71182C211yesmaximal, characteristic, complemented
13#71181C211yesmaximal, characteristic, complemented
14#71180C211yesmaximal, characteristic, complemented
15C2×C8×C3⋊C8C211yesmaximal, characteristic, complemented
16#19600—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C3×#532C2211yescharacteristic
19#5477C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#488—22no—
22C3×#487C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#486C2211yescharacteristic
25C3×#486C2211yescharacteristic
26C3×#416—22no—
27C2×C3×C4×C8C2211yescharacteristic
28#5113—22no—
29#5093C2211yescharacteristic
30#5093C2211yescharacteristic
31#5092C2211yescharacteristic
32#5088C2211yescharacteristic
33#5088C2211yescharacteristic
34#5085C2211yescharacteristic
35#5080C2211yescharacteristic
36#5080C2211yescharacteristic
37#5079C2211yescharacteristic
38#5079C2211yescharacteristic
39#5078C2211yescharacteristic
40#5078C2211yescharacteristic
41#5076C2211yescharacteristic
42#5073C2211yescharacteristic
43#5070C2211yescharacteristic
44#5070C2211yescharacteristic
45#5051—22no—
46#5048—22no—
47#5041—22no—
48#5038—22no—
49C4×#216—22no—
50C4×#215—22no—
51C4×#214—22no—
52C4×#213—22no—
53C2×C4×C3⋊C8C2211yescharacteristic
54C2×C4×C3⋊C8C2211yescharacteristic
55#4697—22no—
56#4694C2211yescharacteristic
57#4694C2211yescharacteristic
58#4681C2211yescharacteristic
59#4681C2211yescharacteristic
60C8×C3⋊C8C2211yescharacteristic
61C8×C3⋊C8C2211yescharacteristic
62C8×C3⋊C8C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3170—31no—
65#3169—31no—
66#3167—31no—
67#3167—31no—
68#3131—31no—
69#3131D311yescharacteristic, complemented
70#1711—31no—
71#1711—31no—
72#1710—31no—
73#1709—31no—
74#1708—31no—
75#1708—31no—
76#1707—31no—
77#1706—31no—
78C2×C82—31no—
79C2×C3×#41—21no—
80C2×#235—21no—
81C22×C3×C8D412yes—
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×#127C2311yescharacteristic
86C3×#125—21no—
87C3×#124—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×#117—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#216—22no—
99C2×#215—22no—
100C2×#214—22no—
101C2×#213—22no—
102#930—21no—
103#930—21no—
104#926—21no—
105#924—21no—
106#924—21no—
107C22×C3⋊C8D412yes—
108C22×C3⋊C8D412yes—
109#875—21no—
110#874C2311yescharacteristic
111C4×D12—21no—
112C4×D12—21no—
113#870—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×C4×C8D412yes—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126#835—22no—
127#835—22no—
128#832—22no—
129#832—22no—
130#821—22no—
131#821—22no—
132#816—22no—
133#816—22no—
134#810—22no—
135#810—22no—
136#809—22no—
137#809—22no—
138#808—22no—
139#808—22no—
140#807—22no—
141#807—22no—
142C4×C3⋊C8D412yes—
143C4×C3⋊C8D412yes—
144C4×C3⋊C8C2311yescharacteristic
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533—31no—
149#533C2×D311yescharacteristic
150#532—31no—
151#532C2×D311yescharacteristic
152#500—31no—
153#500—31no—
154#498—31no—
155#498—31no—
156#494—31no—
157#493—31no—
158#492—31no—
159#492—31no—
160#488—22no—
161#488—62no—
162#487—31no—
163#487—31no—
164#487—31no—
165#487—31no—
166#487—31no—
167#487—31no—
168#487C2×D311yescharacteristic
169#487C2×D311yescharacteristic
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#486C2×D311yescharacteristic
177#486C2×D311yescharacteristic
178#467—62no—
179#466—62no—
180#465—62no—
181#465—62no—
182C4×Q16—62no—
183C4×SD16—62no—
184C4×SD16—62no—
185C4×D8—62no—
186#416—22no—
187#416—62no—
188C2×C4×C8—31no—
189C2×C4×C8—31no—
190C2×C4×C8C2×D311yescharacteristic
191C82—31no—
192C82—31no—
193C82—31no—
194C82—31no—
195C3×#41—44no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—41no—
203C22×C3×C4—41no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C2×#67—41no—
208#235—44no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×C4×D3—41no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×C3×C8—24no—
217C2×C3×C8C2×D412yes—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C22⋊C4—41no—
228C3×C22⋊C4—41no—
229C3×C22⋊C4—41no—
230C3×C42—41no—
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2411yescharacteristic
235#217—41no—
236#216—44no—
237#215—44no—
238#214—44no—
239#213—44no—
240(C2×D3)⋊C4—41no—
241(C2×D3)⋊C4—41no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248Dic3⋊C4—41no—
249Dic3⋊C4—41no—
250C4×Dic3—41no—
251C2×C3⋊C8—24no—
252C2×C3⋊C8—24no—
253C2×C3⋊C8C2×D412yes—
254C2×C3⋊C8C2×D412yes—
255C2×C3⋊C8C2×D412yes—
256C2×C3⋊C8C2×D412yes—
257C2×#41—21no—
258C2×#41—61no—
259C2×Q16—62no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×D8—62no—
263C22×C8—32no—
264C22×C8—32no—
265C22×C8#6712yes—
266C4⋊Q8—31no—
267C4⋊Q8—31no—
268C4⋊Q8—31no—
269C4⋊Q8C22×D311yescharacteristic
270C4⋊Q8C22×D311yescharacteristic
271C4⋊Q8C22×D311yescharacteristic
272#127—31no—
273#127C22×D311yescharacteristic
274#125—21no—
275#125—61no—
276#124—21no—
277#124—61no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—61no—
281C22⋊Q8—61no—
282#121—21no—
283#121—21no—
284#121—61no—
285#121—61no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—61no—
289C4×Q8—61no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—61no—
293C4×D4—61no—
294#117—21no—
295#117—61no—
296C2×C42C22×D311yescharacteristic
297#107—62no—
298#107—62no—
299#106—62no—
300#106—62no—
301#103—22no—
302#103—22no—
303#103—62no—
304#103—62no—
305#103—62no—
306#103—62no—
307#103—62no—
308#103—62no—
309#102—22no—
310#102—22no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317C4×C8—32no—
318C4×C8—32no—
319C4×C8—31no—
320C4×C8—31no—
321C4×C8—31no—
322C4×C8—31no—
323C4×C8#6712yes—
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C23×C3—41no—
327C23×C3#12711yescharacteristic
328C22×D3—41no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—82no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—82no—
342C2×C3×C4#12711yescharacteristic
343C2×C3×C4#12711yescharacteristic
344C2×C3×C4#12711yescharacteristic
345C2×C3×C4#13514yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349#67—82no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353D12—42no—
354D12—42no—
355D12—42no—
356C4×D3—82no—
357Dic6—42no—
358Dic6—42no—
359Dic6—42no—
360C3×C8—22no—
361C3×C8—22no—
362C3⋊C8—22no—
363C3⋊C8—22no—
364C3⋊C8#13514yes—
365C3⋊C8#13514yes—
366#41—44no—
367#41—124no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×Q8—61no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—121no—
382C22×C4—41no—
383C22×C4—121no—
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4C2×#6711yescharacteristic
387Q16—124no—
388SD16—124no—
389SD16—124no—
390D8—124no—
391C2×C8—24no—
392C2×C8—32no—
393C2×C8—32no—
394C2×C8—32no—
395C2×C8—32no—
396C2×C8—64no—
397C2×C8—64no—
398C2×C8C2×#6712yes—
399C2×C8C2×#6712yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—121no—
415C4⋊C4—121no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—121no—
420C22⋊C4—121no—
421C22⋊C4—121no—
422C42—41no—
423C42—121no—
424C42D3×D411yescharacteristic
425C42D3×D411yescharacteristic
426C42C2×#6711yescharacteristic
427C42C23×D311yescharacteristic
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3#49512yes—
434C22×C3C2×#12711yescharacteristic
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—81no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4#49512yes—
444C3×C4C2×#13512yes—
445C3×C4C2×#13512yes—
446C3×C4C2×#13512yes—
447Dic3—81no—
448Dic3—81no—
449Dic3—81no—
450C23—41no—
451C23—121no—
452C23#94011yescharacteristic
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—122no—
457Q8—122no—
458Q8—122no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—82no—
463D4—122no—
464D4—122no—
465D4—122no—
466D4—242no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—242no—
477C2×C4#94011yescharacteristic
478C2×C4#94011yescharacteristic
479C2×C4#94011yescharacteristic
480C2×C4#95014yes—
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C8—22no—
485C8—22no—
486C8—34no—
487C8—34no—
488C8—62no—
489C8—62no—
490C2×C3—22no—
491C2×C3—161no—
492C2×C3C2×#49511yescharacteristic
493C2×C3#314711yescharacteristic
494C2×C3#314711yescharacteristic
495D3—161no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22#505912yes—
505C22C2×#94011yescharacteristic
506C4—21no—
507C4—21no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—241no—
512C4—241no—
513C4—241no—
514C4#505912yes—
515C4D3×#13512yes—
516C4D3×#13512yes—
517C4C2×#95012yes—
518C3#1960011yescharacteristic, complemented, Sylow
519C2—22no—
520C2—161no—
521C2—481no—
522C2C2×#505911yescharacteristic
523C2#8390311yescharacteristic
524C2#8390311yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3131C2
socleC22×C3C2×#127
prime radical#83291C2
hypercenter#3131D3
hypocenterC3#19600
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. C3×#3131C2semidirect
2. #83344C2semidirect
3. #83343C2semidirect
5. #83290C2semidirect
6. #78463C2semidirect
7. #71215C2semidirect
8. #71214C2semidirect
9. #71213C2semidirect
10. #71212C2semidirect
11. #71183C2semidirect
12. #71182C2semidirect
13. #71181C2semidirect
14. #71180C2semidirect
15. C2×C8×C3⋊C8C2semidirect
69. #3131D3semidirect
518. C3#19600semidirect

Maximal subgroups

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

subgroupindex
C3×#31312
#833442
#833432
#832912
#832902
#784632
#712152
#712142
#712132
#712122
#711832
#711822
#711812
#711802
C2×C8×C3⋊C82
#196003