Finite Group Databasebeta

← #10696856

Subgroups — #10696856

Order 768.

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

normal subgroups only

This group has 434 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×#3230C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic
3C2×#5082C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic
6#70787C211yesmaximal, characteristic
7#67742C211yesmaximal, characteristic
8#10135—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#500—21no—
13C3×#487—21no—
14C3×#484—22no—
15C3×#477—22no—
16C3×#431C2211yescharacteristic
17C2×#835—21no—
18C2×#835—21no—
19#5082—21no—
20#5082—22no—
21#5082—22no—
22#5081—21no—
23#5044—21no—
24#5043—21no—
25#5022—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4547—22no—
30#4543—22no—
31#4469—22no—
32#3236—31no—
33#3230D311yescharacteristic
34C2×#491—31no—
35#1556—31no—
36#1531—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C22×C3⋊C8—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64C3×#107—24no—
65C3×C4⋊C8—21no—
66C3×#103—24no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69#835—24no—
70#835—24no—
71#835—22no—
72#835—22no—
73#830—21no—
74#830—21no—
75#808—22no—
76#807—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×#125—31no—
86#501—61no—
87#500—21no—
88#491—62no—
89#491—62no—
90#491—61no—
91#487—21no—
92#484—22no—
93#484—61no—
94#483—61no—
95#479—62no—
96#477—22no—
97#432—31no—
98#432—31no—
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#103—61no—
102C2×#97—31no—
103#332—62no—
104#330—62no—
105#324—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—44no—
110C2×C3×Q8—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×C3×C8—22no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—42no—
145C2×C3⋊C8—41no—
146C22×Q8—21no—
147C22×C8—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8#6712yes—
152C4⋊Q8#6711yescharacteristic
153C4⋊Q8#6711yescharacteristic
154#125—21no—
155#125—61no—
156#125—61no—
157#125—61no—
158#125—61no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171#107—24no—
172#107—62no—
173#106—62no—
174C4⋊C8—21no—
175C4⋊C8—32no—
176C4⋊C8—61no—
177#103—24no—
178#103—64no—
179#103—64no—
180#103—62no—
181#103—62no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184C22⋊C8#6712yes—
185#97—32no—
186#97—61no—
187#97—61no—
188#97—61no—
189C4×C8—21no—
190C23×C3#12111yescharacteristic
191C3×Q8—42no—
192C3×Q8—44no—
193C3×Q8—42no—
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—82no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211C2×Dic3—81no—
212C3×C8—42no—
213C3⋊C8—42no—
214C3⋊C8—42no—
215C3⋊C8—81no—
216C2×Q8—21no—
217C2×Q8—22no—
218C2×Q8—21no—
219C2×Q8—44no—
220C2×Q8—41no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4D3×D411yescharacteristic
225C22×C4#23711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227C2×C8—22no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—122no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—41no—
251C42—41no—
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42C2×#6711yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#39211yescharacteristic
259C22×C3#47312yes—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—82no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4#45412yes—
269Dic3—81no—
270Dic3—81no—
271C23#93811yescharacteristic
272Q8—44no—
273Q8—42no—
274Q8—42no—
275Q8—82no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—122no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—241no—
287C2×C4#93711yescharacteristic
288C2×C4#93711yescharacteristic
289C2×C4#93911yescharacteristic
290C2×C4#93911yescharacteristic
291C2×C4#94011yescharacteristic
292C2×C4#95311yescharacteristic
293C8—42no—
294C8—122no—
295C8—122no—
296C8—241no—
297C2×C3—22no—
298C2×C3#200311yescharacteristic
299C2×C3#200911yescharacteristic
300C2×C3#202011yescharacteristic
301C22—21no—
302C22—21no—
303C22#519212yes—
304C22#522311yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—81no—
312C4—81no—
313C4—241no—
314C4—241no—
315C4#520112yes—
316C3#1013511yescharacteristic, complemented, Sylow
317C2—22no—
318C2#7186911yescharacteristic
319C2#7207411yescharacteristic
320C2#7209611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3230C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3230D3
hypocenterC3#10135
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
316. C3#10135semidirect

Maximal subgroups

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

subgroupindex
C3×#32302
#833262
C2×#50822
#718222
#717762
#707872
#677422
#101353