Finite Group Databasebeta

← #10695579

Subgroups — #10695579

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×#2404C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#82116C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71539C211yesmaximal, characteristic, complemented
6#69533C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9942—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#120—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#393—21no—
15C3×#392—21no—
16C3×#386—21no—
17C3×#386—22no—
18C3×#381C2211yescharacteristic
19#5247—21no—
20#5239—22no—
21C2×#837—22no—
22#5232—21no—
23#5228—21no—
24#5218—21no—
25#5211—21no—
26C4×#217C2211yescharacteristic
27#5207C2211yescharacteristic
28#5207C2211yescharacteristic
29#4953—21no—
30#4947—21no—
31#4945—22no—
32#4525—22no—
33#4452—22no—
34#3007—31no—
35#2561—31no—
36#2404D311yescharacteristic
37#1789—31no—
38#1511—31no—
39#1462—31no—
40#1284—31no—
41C25×C3D411yescharacteristic
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—44no—
47C22×C3×Q8D411yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×Dic6D411yescharacteristic, complemented
55C3×#120—44no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×#217—22no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#217—22no—
69#951—44no—
70C2×#216—22no—
71C2×#214—22no—
72C2×C4⋊Dic3D411yescharacteristic, complemented
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3D412yescomplemented
79C2×#208—21no—
80C2×#208—21no—
81C3×#95—22no—
82C3×#95—21no—
83C3×#95D412yes—
84#837—44no—
85#835—22no—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90#830—21no—
91#830—21no—
92#830—21no—
93#830D412yescomplemented
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×C22⋊Q8—31no—
97C2×#120—21no—
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—61no—
107#393—21no—
108#393—61no—
109#392—21no—
110#388—62no—
111#386—21no—
112#386—22no—
113#381—61no—
114#381C2×D311yescharacteristic
115#379—61no—
116C4×C22⋊C4—31no—
117#328—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3—42no—
126C24×C3C2×D411yescharacteristic
127C3×#41—24no—
128C3×#41—24no—
129C3×#41—42no—
130C3×#41—42no—
131C3×#41—42no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—42no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—42no—
138C2×C3×D4—24no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—44no—
146C2×C3×D4—44no—
147C2×C3×D4—82no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—42no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4111yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—42no—
171C3×C22⋊C4—44no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—82no—
178#217—42no—
179#217—44no—
180#217—41no—
181#217—41no—
182#217—82no—
183#216—44no—
184#214—44no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187Dic3⋊C4—41no—
188Dic3⋊C4—41no—
189C4×Dic3—42no—
190#208—42no—
191#208—42no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—41no—
194C25#6711yescharacteristic
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—44no—
200C22×Q8—31no—
201C22×Q8#6711yescharacteristic
202C22×D4—21no—
203C22×D4—22no—
204C22×D4#6711yescharacteristic
205C23×C4—21no—
206C23×C4#6711yescharacteristic
207C23×C4C22×D311yescharacteristic
208#137—124no—
209C2×Q16—62no—
210C2×SD16—62no—
211C2×M16—61no—
212C2×M16—61no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217#120—44no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—61no—
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—22no—
229C2×C22⋊C4—62no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—62no—
233C2×C42—32no—
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—21no—
243#95—62no—
244#95—62no—
245#95—61no—
246#95#6712yes—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—41no—
254C23×C3—42no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—81no—
267C23×C3—81no—
268C23×C3—88no—
269C23×C3#12011yescharacteristic
270C23×C3#12011yescharacteristic
271C23×C3#12111yescharacteristic
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—41no—
275C3×Q8—41no—
276C3×D4—22no—
277C3×D4—22no—
278C3×D4—41no—
279C3×D4—41no—
280C3×D4—44no—
281C3×D4—44no—
282C3×D4—88no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—81no—
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12711yescharacteristic
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312Dic6—42no—
313Dic6—81no—
314C3⋊C8—42no—
315C3⋊C8—81no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24C2×#6711yescharacteristic
324#41—24no—
325#41—24no—
326#41—42no—
327#41—42no—
328#41—42no—
329#41—42no—
330#41—42no—
331#41—42no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—42no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—122no—
338C2×D4—24no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—82no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4—61no—
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—61no—
375C4⋊C4—61no—
376C4⋊C4—121no—
377C4⋊C4—121no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C22⋊C4—122no—
386C22⋊C4—124no—
387C22⋊C4—121no—
388C22⋊C4—121no—
389C22⋊C4—242no—
390C42—122no—
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—44no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3#39211yescharacteristic
414C22×C3#45512yes—
415C22×C3B(4,𝔽2)12yes—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—42no—
421C3×C4—82no—
422C3×C4—81no—
423Dic3—82no—
424Dic3—81no—
425Dic3—81no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—81no—
447C23—88no—
448C23#93711yescharacteristic
449C23#93911yescharacteristic
450C23#95311yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—41no—
454Q8—41no—
455Q8—122no—
456Q8—241no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—44no—
462D4—44no—
463D4—88no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—121no—
486C2×C4—121no—
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—82no—
500C2×C3#200411yescharacteristic
501C2×C3#201011yescharacteristic
502C2×C3#202111yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22#503212yes—
526C22#522311yescharacteristic
527C22#524212yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—82no—
534C4—81no—
535C4—242no—
536C4—241no—
537C4—241no—
538C3#994211yescharacteristic, complemented, Sylow
539C2—22no—
540C2—22no—
541C2—41no—
542C2—82no—
543C2—82no—
544C2#7167111yescharacteristic
545C2#7171511yescharacteristic
546C2#7185711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2404C2
socleC22×C3#392
hypercenter#2404D3
hypocenterC3#9942
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #82116C2semidirect
4. #72138C2semidirect
5. #71539C2semidirect
6. #69533C2semidirect
7. #67628C2semidirect
54. C22×Dic6D4semidirect
72. C2×C4⋊Dic3D4semidirect
78. C2×C4×Dic3D4semidirect
93. #830D4semidirect
538. C3#9942semidirect

Maximal subgroups

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

subgroupindex
C3×#24042
#840842
#821162
#721382
#715392
#695332
#676282
#99423