Finite Group Databasebeta

← #10695577

Subgroups — #10695577

Order 768.

894 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2561C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic, complemented
3#82125C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic, complemented
6#69454C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9942—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#953—21no—
13C2×#942—21no—
14C3×#397—21no—
15C3×#393—21no—
16C3×#388—22no—
17C3×#381—21no—
18C3×#379—21no—
19C3×C4×C22⋊C4C2211yescharacteristic
20#5213—21no—
21#5211C2211yescharacteristic
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5050—21no—
25#5047—22no—
26#5046—21no—
27#4966—21no—
28#4960—21no—
29#4956—22no—
30#4947—21no—
31C2×#805—22no—
32#4538—22no—
33#4525—22no—
34#3007—31no—
35#2561D311yescharacteristic, complemented
36#2404—31no—
37#1789—31no—
38#1511—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic, complemented
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C22×#67—22no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—44no—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic, complemented
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42D412yescomplemented
72#953—44no—
73C2×#217—22no—
74C2×#217—21no—
75C2×#216—22no—
76#942—44no—
77C2×#215—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×#208—21no—
80C2×#208—21no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—21no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95D412yes—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yescomplemented
91#810—22no—
92#809—22no—
93#805—44no—
94C22×#41—31no—
95C2×#137—61no—
96C2×C22⋊Q8C2×D311yescharacteristic
97C2×#120—61no—
98C22×C22⋊C4C2×D311yescharacteristic
99#452—62no—
100#451—61no—
101#450—61no—
102C2×#104—62no—
103#417—61no—
104#407—31no—
105#407—31no—
106#397—21no—
107#393—21no—
108#393—61no—
109#392—61no—
110#388—22no—
111#386—62no—
112#386—61no—
113#381—21no—
114#381—31no—
115#379—21no—
116C4×C22⋊C4C2×D311yescharacteristic
117#328—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—42no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3C2×D411yescharacteristic
126C23×D3—21no—
127C2×C3×Q8—21no—
128C2×C3×Q8—21no—
129C2×C3×Q8—42no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C2×#67—24no—
145C2×#67—42no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—44no—
149C2×#67—44no—
150C2×#67—82no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153#235—24no—
154#235—24no—
155#235—42no—
156#235—42no—
157#235—42no—
158#235—42no—
159#235—42no—
160#235—42no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—42no—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
166C2×C4×D3—41no—
167C2×C4×D3—41no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—42no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—82no—
184C3×C42—42no—
185#217—44no—
186#217—42no—
187#216—44no—
188#215—44no—
189(C2×D3)⋊C4—82no—
190#208—42no—
191#208—42no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—41no—
194C25#6711yescharacteristic
195C2×#41—61no—
196C2×#41—61no—
197C2×#41—61no—
198C2×#41—61no—
199C2×#41—124no—
200C22×Q8—31no—
201C22×Q8#6711yescharacteristic, complemented
202C22×D4—31no—
203C22×D4—61no—
204C22×D4—62no—
205C23×C4—61no—
206C23×C4#6711yescharacteristic
207C23×C4C22×D311yescharacteristic
208#137—124no—
209C2×Q16—62no—
210C2×SD16—62no—
211C2×M16—61no—
212C2×M16—61no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217#120—124no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4#6711yescharacteristic, complemented
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—62no—
233C2×C42#6712yescomplemented
234#104—124no—
235#103—62no—
236#102—62no—
237C22⋊C8—32no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240C22⋊C8—61no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—32no—
245#95—62no—
246#95—61no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—81no—
262C23×C3—81no—
263C23×C3—88no—
264C23×C3#12011yescharacteristic
265C23×C3#12011yescharacteristic
266C23×C3#12111yescharacteristic
267C22×D3—22no—
268C22×D3—41no—
269C22×D3—42no—
270C22×D3—41no—
271C22×D3—42no—
272C3×Q8—42no—
273C3×Q8—81no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—81no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12111yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12711yescharacteristic
295#67—44no—
296#67—44no—
297#67—88no—
298C2×Dic3—22no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304D12—22no—
305D12—22no—
306D12—41no—
307D12—41no—
308C4×D3—44no—
309C4×D3—44no—
310Dic6—22no—
311Dic6—22no—
312Dic6—41no—
313Dic6—41no—
314C3⋊C8—42no—
315C3⋊C8—81no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—61no—
323C24C2×#6711yescharacteristic
324#41—64no—
325#41—64no—
326#41—122no—
327#41—122no—
328#41—122no—
329#41—122no—
330#41—122no—
331#41—122no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—42no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—122no—
338C2×D4—64no—
339C2×D4—61no—
340C2×D4—61no—
341C2×D4—122no—
342C2×D4—122no—
343C2×D4—121no—
344C2×D4—121no—
345C2×D4—124no—
346C2×D4—124no—
347C2×D4—242no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—122no—
359C22×C4—122no—
360C22×C4—121no—
361C22×C4—121no—
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4#23711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—124no—
369SD16—124no—
370M16—122no—
371M16—122no—
372C2×C8—62no—
373C2×C8—121no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—82no—
387C22⋊C4—124no—
388C22⋊C4—122no—
389C22⋊C4—242no—
390C42—42no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—82no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3#39211yescharacteristic
409C22×C3#45512yes—
410C22×C3B(4,𝔽2)12yes—
411C2×D3—44no—
412C2×D3—42no—
413C2×D3—82no—
414C2×D3—81no—
415C2×D3—81no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—82no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4—81no—
424Dic3—42no—
425Dic3—82no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—41no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—62no—
441C23—81no—
442C23—81no—
443C23—88no—
444C23—121no—
445C23—122no—
446C23—121no—
447C23—122no—
448C23#93711yescharacteristic
449C23#93911yescharacteristic
450C23#95311yescharacteristic
451Q8—42no—
452Q8—62no—
453Q8—62no—
454Q8—81no—
455Q8—121no—
456Q8—121no—
457D4—62no—
458D4—62no—
459D4—121no—
460D4—121no—
461D4—124no—
462D4—124no—
463D4—248no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—81no—
482C2×C4—124no—
483C2×C4—124no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4#93711yescharacteristic
490C2×C4#93811yescharacteristic
491C2×C4#93911yescharacteristic
492C2×C4#94011yescharacteristic
493C8—122no—
494C8—241no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—41no—
498C2×C3—82no—
499C2×C3#200411yescharacteristic
500C2×C3#201011yescharacteristic
501C2×C3#202111yescharacteristic
502D3—82no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22—124no—
521C22—122no—
522C22—242no—
523C22—241no—
524C22—241no—
525C22#506112yes—
526C22#522311yescharacteristic
527C22#524412yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—82no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—122no—
537C4—242no—
538C3#994211yescharacteristic, complemented, Sylow
539C2—22no—
540C2—22no—
541C2—41no—
542C2—82no—
543C2—242no—
544C2#7171511yescharacteristic
545C2#7173111yescharacteristic
546C2#7185111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2561C2
socleC22×C3#392
prime radical#82125C2
hypercenter#2561D3
hypocenterC3#9942
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
1. C3×#2561C2semidirect
2. #82735C2semidirect
4. #71689C2semidirect
5. #71539C2semidirect
6. #69454C2semidirect
7. #67616C2semidirect
35. #2561D3semidirect
42. C22×C3×Q8D4semidirect
60. C2×C3×C4⋊C4D4semidirect
71. C2×C3×C42D4semidirect
90. #830D4semidirect
201. C22×Q8#67semidirect
219. C2×C4⋊C4#67semidirect
233. C2×C42#67semidirect
538. C3#9942semidirect

Maximal subgroups

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

subgroupindex
C3×#25612
#827352
#821252
#716892
#715392
#694542
#676162
#99423