Finite Group Databasebeta

← #10692246

Subgroups — #10692246

Order 768.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#82727C211yesmaximal, characteristic
2#78621C211yesmaximal, characteristic
3C3×#1472C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5#69926C211yesmaximal, characteristic
6#67685C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic
8#9346—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5225C411yescharacteristic, complemented
14#5225C411yescharacteristic, complemented
15#5212C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5122—21no—
19#5119—21no—
20#5045—21no—
21#5045—21no—
22#5044—21no—
23#5043—21no—
24#4964C411yescharacteristic, complemented
25#4964C411yescharacteristic, complemented
26#4720—22no—
27#4719—21no—
28#4718—21no—
29C3×#324—22no—
30#4551—21no—
31#4551—21no—
32#4538—21no—
33#4538—21no—
34#3095—31no—
35#2924—31no—
36#1825—31no—
37#1520—31no—
38#1472D311yescharacteristic
39#1305—31no—
40#1298—31no—
41C22×C3×Q8C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×#217—21no—
54#926—21no—
55#926—21no—
56#926C2×C411yescharacteristic
57#926C2×C411yescharacteristic
58#926D411yescharacteristic
59#926D411yescharacteristic
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4×Dic3—21no—
64C2×#208C2×C411yescharacteristic
65C2×#208C2×C411yescharacteristic
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D412yes—
71#835—21no—
72#835—21no—
73#835—21no—
74#835—21no—
75C3⋊#95—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830D411yescharacteristic
80#830D411yescharacteristic
81#815—22no—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—22no—
85#808—21no—
86#808—21no—
87#807—21no—
88#807—21no—
89C2×C22⋊Q8C2×D311yescharacteristic
90C2×#117—31no—
91#485—61no—
92#485—61no—
93#484—61no—
94#483—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98#428—31no—
99#428—31no—
100#424—61no—
101#419—31no—
102#419—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108#380—31no—
109#340—61no—
110#340—61no—
111#328—61no—
112#328—61no—
113#324—22no—
114C24×C3#4111yescharacteristic
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144#217—21no—
145#217—21no—
146#217—21no—
147#217—21no—
148#217—81no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3C22⋊C411yescharacteristic
154C4⋊Dic3C22⋊C411yescharacteristic
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C22×Q8C4×D311yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×M16—21no—
172C2×M16—31no—
173C2×M16—31no—
174C22×C8—21no—
175C22×C8—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180#117—31no—
181#117—31no—
182#117—31no—
183#117—31no—
184#117—61no—
185#117—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4C4×D311yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—61no—
195#107—61no—
196#107—61no—
197#106—61no—
198#106—61no—
199C4⋊C8—62no—
200C4⋊C8—62no—
201#103—61no—
202#103—61no—
203#103—61no—
204#103—61no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—31no—
209C22⋊C8—31no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8#6712yes—
214#97—62no—
215C4×C8—62no—
216#95—61no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3C4×D411yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12311yescharacteristic
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—81no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C24#23511yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—121no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4#23711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278M16—41no—
279M16—41no—
280M16—121no—
281M16—121no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—241no—
311C42—61no—
312C42—61no—
313C42—61no—
314C42—61no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#47111yescharacteristic
327C22×C3#47311yescharacteristic
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4#41011yescharacteristic
334C3×C4#41011yescharacteristic
335Dic3—41no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—161no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23#88311yescharacteristic
345C23#88411yescharacteristic
346C23C4×#6711yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—81no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4D3×C22⋊C411yescharacteristic
374C2×C4#67⋊C411yescharacteristic
375C2×C4#93711yescharacteristic
376C2×C4#93811yescharacteristic
377C8—42no—
378C8—81no—
379C8—121no—
380C8—121no—
381C8—121no—
382C8—121no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3#188111yescharacteristic
388C2×C3#189911yescharacteristic
389C2×C3#190811yescharacteristic
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22#475411yescharacteristic
401C22#475511yescharacteristic
402C22#495611yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—81no—
407C4—81no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4—121no—
412C4—481no—
413C4#474011yescharacteristic
414C4#474011yescharacteristic
415C3#934611yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#7001211yescharacteristic
421C2#7002911yescharacteristic
422C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1472C2
socleC22×C3#386
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenter#1472D3
hypocenterC3#9346
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
13. #5225C4semidirect
14. #5225C4semidirect
24. #4964C4semidirect
25. #4964C4semidirect
415. C3#9346semidirect

Maximal subgroups

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

subgroupindex
#827272
#786212
C3×#14722
#716902
#699262
#676852
#676802
#93463