Finite Group Databasebeta

← #10696898

Subgroups — #10696898

Order 768.

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

normal subgroups only

This group has 472 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×#3229C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3C2×#5089C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic, complemented
5#71777C211yesmaximal, characteristic, complemented
6#70784C211yesmaximal, characteristic
7#67742C211yesmaximal, characteristic, complemented
8#10141—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#498—21no—
13C3×#487—21no—
14C3×#483—22no—
15C3×#475—22no—
16C3×#431C2211yescharacteristic
17#5089—21no—
18#5089—22no—
19#5089—22no—
20#5086—21no—
21#5050—21no—
22#5046—21no—
23#5028—22no—
24C2×#810—21no—
25C2×#809—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4546—22no—
30#4543—22no—
31#4468—22no—
32#3229D311yescharacteristic, complemented
33#3215—31no—
34C2×#489—31no—
35#1558—31no—
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D412yes—
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#216—22no—
58C2×#215—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C22×C3⋊C8—21no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66C3×#106—24no—
67C3×C4⋊C8—21no—
68C3×#103—24no—
69C3×C22⋊C8D412yes—
70C3×C4×C8—21no—
71#830—21no—
72#830—21no—
73#810—24no—
74#810—22no—
75#809—24no—
76#809—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#498—21no—
87#497—61no—
88#489—62no—
89#489—62no—
90#489—61no—
91#487—21no—
92#483—22no—
93#475—22no—
94#464—62no—
95#451—61no—
96#450—61no—
97#432—31no—
98#432—31no—
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#102—61no—
102C2×#97—31no—
103#333—62no—
104#330—62no—
105#323—62no—
106C23×D3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—42no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—42no—
123C2×C3×C8—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#216—82no—
141#215—82no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—42no—
149C2×C3⋊C8—41no—
150C22×Q8—21no—
151C22×Q8—31no—
152C22×D4—31no—
153C2×Q16—62no—
154C2×SD16—62no—
155C22×C8—61no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8#6712yes—
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162#125—21no—
163#124—61no—
164#124—61no—
165#124—61no—
166#124—61no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171#117—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C42C22×D311yescharacteristic
177#106—24no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181#103—24no—
182#103—64no—
183#103—62no—
184#102—64no—
185#102—62no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188C22⋊C8#6712yes—
189#97—32no—
190#97—61no—
191#97—61no—
192#97—61no—
193C4×C8—21no—
194C23×C3#12111yescharacteristic
195C22×D3—41no—
196C22×D3—42no—
197C22×D3—81no—
198C22×D3—81no—
199C3×Q8—42no—
200C3×Q8—42no—
201C3×Q8—82no—
202C3×Q8—82no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—82no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—81no—
218D12—42no—
219D12—42no—
220Dic6—42no—
221Dic6—42no—
222C3×C8—42no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225C3⋊C8—81no—
226C24—61no—
227C2×Q8—21no—
228C2×Q8—22no—
229C2×Q8—21no—
230C2×Q8—41no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—82no—
234C2×Q8—122no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—122no—
238C22×C4—41no—
239C22×C4—61no—
240C22×C4D3×D411yescharacteristic
241C22×C4#23711yescharacteristic
242C22×C4C2×#6711yescharacteristic
243Q16—242no—
244SD16—242no—
245C2×C8—22no—
246C2×C8—62no—
247C2×C8—62no—
248C2×C8—122no—
249C2×C8—121no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—41no—
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C42C2×#6711yescharacteristic
269C42C2×#6711yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#39211yescharacteristic
273C22×C3#47112yes—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—162no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4—82no—
286C3×C4#45512yes—
287Dic3—81no—
288C23—121no—
289C23—122no—
290C23—241no—
291C23—241no—
292C23#93811yescharacteristic
293Q8—42no—
294Q8—42no—
295Q8—82no—
296Q8—82no—
297Q8—122no—
298Q8—122no—
299D4—122no—
300D4—122no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—121no—
309C2×C4—241no—
310C2×C4#93711yescharacteristic
311C2×C4#93711yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#94011yescharacteristic
315C2×C4#95311yescharacteristic
316C8—42no—
317C8—122no—
318C8—122no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#200911yescharacteristic
323C2×C3#201911yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—482no—
331C22#517712yes—
332C22#522311yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—81no—
341C4—241no—
342C4#520212yes—
343C3#1014111yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7186911yescharacteristic
347C2#7205911yescharacteristic
348C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3229D3
hypocenterC3#10141
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C3×#3229C2semidirect
4. #71822C2semidirect
5. #71777C2semidirect
7. #67742C2semidirect
32. #3229D3semidirect
343. C3#10141semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833572
C2×#50892
#718222
#717772
#707842
#677422
#101413