Finite Group Databasebeta

← #10695576

Subgroups — #10695576

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×#3007C211yesmaximal, characteristic, complemented
2#84096C211yesmaximal, characteristic
3#82819C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71689C211yesmaximal, characteristic, complemented
6#69967C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9942—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#937—21no—
13C22×(C2×D3)⋊C4C2211yescharacteristic
14C3×#452—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19C2×#837—22no—
20#5231—21no—
21#5231—21no—
22#5223—21no—
23#5220—21no—
24#5213—22no—
25#5211C2211yescharacteristic
26C4×#217C2211yescharacteristic
27#5207C2211yescharacteristic
28#4992—21no—
29#4990—22no—
30#4955—21no—
31#4943—21no—
32#4538—22no—
33#4468—22no—
34#3007D311yescharacteristic, complemented
35#2561—31no—
36#2404—31no—
37#1789—31no—
38#1511—31no—
39#1462—31no—
40#1284—31no—
41C24×D3D411yescharacteristic
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×Q8D411yescharacteristic, complemented
49C22×C3×D4—21no—
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×#67—22no—
54C23×Dic3D411yescharacteristic
55C3×#137—44no—
56C2×C3×Q16—22no—
57C2×C3×SD16—22no—
58C2×C3×M16—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—21no—
62C3×C22⋊Q8—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic, complemented
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#217—22no—
69#937—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4×Dic3D412yescomplemented
80C2×#208—21no—
81C3×#103—22no—
82C3×#102—22no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yescomplemented
85#837—44no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95D412yes—
92#830—21no—
93#830—21no—
94C22×#41C2×D311yescharacteristic
95C2×#137—21no—
96C2×C22⋊Q8C2×D311yescharacteristic
97C2×#120—61no—
98C22×C22⋊C4—31no—
99#452—22no—
100#451—21no—
101#450—21no—
102C2×#104—62no—
103#417—21no—
104#407—31no—
105#407C2×D311yescharacteristic
106#397—61no—
107#393—61no—
108#393—61no—
109#392—61no—
110#388—62no—
111#386—62no—
112#386—61no—
113#381—31no—
114#381—61no—
115#379—61no—
116C4×C22⋊C4—31no—
117#328—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—42no—
125C23×D3—42no—
126C23×D3—42no—
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—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×D4—24no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—42no—
145C2×C3×D4—42no—
146C2×C3×D4—41no—
147C2×C3×D4—41no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4#4111yescharacteristic
162C2×#67—44no—
163C2×#67—44no—
164C2×#67—82no—
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—42no—
171C3×Q16—44no—
172C3×SD16—44no—
173C3×M16—42no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181#217—42no—
182#217—42no—
183#217—82no—
184(C2×D3)⋊C4—44no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—41no—
187(C2×D3)⋊C4—41no—
188(C2×D3)⋊C4—41no—
189(C2×D3)⋊C4—41no—
190(C2×D3)⋊C4—82no—
191C4×Dic3—42no—
192#208—42no—
193C2×C3⋊C8—41no—
194C25—31no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—44no—
200C22×Q8#6711yescharacteristic
201C22×Q8#6711yescharacteristic, complemented
202C22×D4—21no—
203C22×D4—62no—
204C22×D4#6711yescharacteristic
205C23×C4—21no—
206C23×C4—31no—
207C23×C4C22×D311yescharacteristic
208#137—44no—
209C2×Q16—22no—
210C2×SD16—22no—
211C2×M16—21no—
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—21no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C42—32no—
234#104—124no—
235#103—22no—
236#102—22no—
237C22⋊C8—21no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240C22⋊C8#6712yescomplemented
241#95—32no—
242#95—62no—
243#95—62no—
244#95—62no—
245#95—61no—
246#95—61no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—42no—
255C23×C3#12011yescharacteristic
256C23×C3#12011yescharacteristic
257C23×C3#12111yescharacteristic
258C22×D3—22no—
259C22×D3—22no—
260C22×D3—22no—
261C22×D3—42no—
262C22×D3—42no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—88no—
270C22×D3—81no—
271C22×D3—81no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—41no—
275C3×Q8—41no—
276C3×Q8—42no—
277C3×Q8—81no—
278C3×D4—22no—
279C3×D4—22no—
280C3×D4—41no—
281C3×D4—41no—
282C3×D4—44no—
283C3×D4—44no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4#12011yescharacteristic
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12711yescharacteristic
303#67—88no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—81no—
314C3×C8—42no—
315C3⋊C8—81no—
316C24—21no—
317C24—61no—
318C24—61no—
319C24—61no—
320C24—122no—
321C24—122no—
322C24—122no—
323C24D3×D411yescharacteristic
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—21no—
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×D4—24no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—124no—
346C2×D4—124no—
347C2×D4—242no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—122no—
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D411yescharacteristic
364C22×C4#23711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—44no—
369SD16—44no—
370M16—42no—
371M16—122no—
372C2×C8—22no—
373C2×C8—121no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—124no—
381C22⋊C4—124no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—121no—
385C22⋊C4—121no—
386C22⋊C4—121no—
387C22⋊C4—121no—
388C22⋊C4—242no—
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—42no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3#39211yescharacteristic
405C22×C3#45512yes—
406C22×C3B(4,𝔽2)12yes—
407C2×D3—44no—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—81no—
414C2×D3—81no—
415C2×D3—82no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—42no—
421C3×C4—81no—
422C3×C4—81no—
423Dic3—82no—
424Dic3—82no—
425Dic3—81no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—62no—
435C23—62no—
436C23—62no—
437C23—122no—
438C23—122no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23—122no—
443C23—121no—
444C23—122no—
445C23—241no—
446C23—241no—
447C23—248no—
448C23#93711yescharacteristic
449C23#93711yescharacteristic
450C23#93811yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—41no—
454Q8—41no—
455Q8—42no—
456Q8—81no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—44no—
462D4—44no—
463D4—248no—
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—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—241no—
489C2×C4#93911yescharacteristic
490C2×C4#93911yescharacteristic
491C2×C4#94011yescharacteristic
492C2×C4#95311yescharacteristic
493C8—42no—
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—42no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—124no—
517C22—124no—
518C22—122no—
519C22—242no—
520C22—242no—
521C22—242no—
522C22—242no—
523C22—241no—
524C22—241no—
525C22#503212yes—
526C22#520212yes—
527C22#522311yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—242no—
536C4—242no—
537C4—241no—
538C3#994211yescharacteristic, complemented, Sylow
539C2—22no—
540C2—22no—
541C2—41no—
542C2—82no—
543C2—242no—
544C2#7167111yescharacteristic
545C2#7210611yescharacteristic
546C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#392
prime radical#84096C2
hypercenter#3007D3
hypocenterC3#9942
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
1. C3×#3007C2semidirect
3. #82819C2semidirect
4. #72138C2semidirect
5. #71689C2semidirect
6. #69967C2semidirect
7. #67682C2semidirect
34. #3007D3semidirect
48. C22×C3×Q8D4semidirect
64. C2×C3×C4⋊C4D4semidirect
79. C2×C4×Dic3D4semidirect
84. C3×C22⋊C8D4semidirect
201. C22×Q8#67semidirect
219. C2×C4⋊C4#67semidirect
240. C22⋊C8#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×#30072
#840962
#828192
#721382
#716892
#699672
#676822
#99423