Finite Group Databasebeta

← #10696859

Subgroups — #10696859

Order 768.

434 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 83 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#83021C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic
4C3×#1470C211yesmaximal, characteristic
5#70822C211yesmaximal, characteristic
6#70813C211yesmaximal, characteristic
7#67742C211yesmaximal, characteristic
8#10135—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5069—22no—
15#5069—21no—
16#5069—22no—
17#5064—21no—
18#5063—22no—
19#4971—21no—
20#4970—21no—
21C2×#810—21no—
22C2×#810—21no—
23#4926C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4850—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29C3×#324—22no—
30#4486—22no—
31#4465—22no—
32#3236—31no—
33#3230—31no—
34C2×#491—31no—
35#1556—31no—
36#1531—31no—
37#1470D311yescharacteristic
38#1302—31no—
39C22×Dic6—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C22×C3⋊C8—21no—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D412yes—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×#97—21no—
69C3×C4×C8—21no—
70#830—21no—
71#818—24no—
72#816—24no—
73#810—24no—
74#810—24no—
75#810—22no—
76#810—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82#803D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×#125C2×D311yescharacteristic
86#501—61no—
87#500—61no—
88#491—62no—
89#491—62no—
90#491—61no—
91#487—61no—
92#484—61no—
93#484—62no—
94#483—61no—
95#479—62no—
96#477—62no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100C2×#103—61no—
101C2×#103—61no—
102C2×#97—31no—
103#332—62no—
104#330—62no—
105#324—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×Dic3—41no—
112C2×Dic6—21no—
113C2×Dic6—22no—
114C2×Dic6—21no—
115C2×Dic6—44no—
116C2×Dic6—41no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#217—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—22no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—42no—
146C22×Q8—61no—
147C22×C8—61no—
148C4⋊Q8—32no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154#125—21no—
155#125—21no—
156#125—21no—
157#125—21no—
158#125—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4D1211yescharacteristic
169C2×C4⋊C4D1211yescharacteristic
170C2×C42C22×D311yescharacteristic
171#107—62no—
172#107—64no—
173#106—62no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177#103—64no—
178#103—64no—
179#103—62no—
180#103—62no—
181#103—64no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C22⋊C8#6712yes—
185#97—21no—
186#97—32no—
187#97—61no—
188#97—61no—
189C4×C8—21no—
190C23×C3#12111yescharacteristic
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—81no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12711yescharacteristic
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—42no—
207C2×Dic3—82no—
208Dic6—42no—
209Dic6—44no—
210Dic6—42no—
211Dic6—82no—
212C3×C8—42no—
213C3×C8—81no—
214C3⋊C8—42no—
215C3⋊C8—42no—
216C2×Q8—61no—
217C2×Q8—62no—
218C2×Q8—61no—
219C2×Q8—124no—
220C2×Q8—121no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—121no—
224C22×C4C2×D1211yescharacteristic
225C22×C4#23511yescharacteristic
226C22×C4C2×#6711yescharacteristic
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—122no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—121no—
251C42—121no—
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42D3×D411yescharacteristic
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—81no—
266C3×C4—81no—
267C3×C4#45412yes—
268Dic3—82no—
269Dic3—81no—
270Dic3—81no—
271C23#93011yescharacteristic
272Q8—124no—
273Q8—122no—
274Q8—122no—
275Q8—242no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—122no—
286C2×C4—242no—
287C2×C4#88211yescharacteristic
288C2×C4#88211yescharacteristic
289C2×C4#88411yescharacteristic
290C2×C4#88411yescharacteristic
291C2×C4#93711yescharacteristic
292C2×C4#94011yescharacteristic
293C8—42no—
294C8—81no—
295C8—122no—
296C8—122no—
297C2×C3—22no—
298C2×C3#200311yescharacteristic
299C2×C3#200911yescharacteristic
300C2×C3#202011yescharacteristic
301C22—21no—
302C22—21no—
303C22#496011yescharacteristic
304C22#514312yes—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—81no—
312C4—241no—
313C4—242no—
314C4—241no—
315C4#475312yes—
316C3#1013511yescharacteristic, complemented, Sylow
317C2—22no—
318C2#7000811yescharacteristic
319C2#7005311yescharacteristic
320C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1470C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1470D3
hypocenterC3#10135
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

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
#830212
C2×#50692
#792722
C3×#14702
#708222
#708132
#677422
#101353