Finite Group Databasebeta

← #10697141

Subgroups — #10697141

Order 768.

492 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 105 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#79271C211yesmaximal, characteristic
2#78537C211yesmaximal, characteristic
3C2×#4682C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic
5#69877C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7C3×#1278C211yesmaximal, characteristic
8#10175—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15C2×#824—22no—
16#5120—22no—
17C2×#818—22no—
18#4924C2211yescharacteristic
19#4851—22no—
20#4721—22no—
21#4703—22no—
22#4682—22no—
23#4682—22no—
24#4681—21no—
25#4681—21no—
26#4681—21no—
27#4680—21no—
28#4486—22no—
29#4456—22no—
30#3229—31no—
31C2×#500—31no—
32#3139—31no—
33#1966—31no—
34#1546—31no—
35#1493—31no—
36#1278D311yescharacteristic
37C22×Dic6—21no—
38C22×Dic6—21no—
39C2×C3×M16D412yes—
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3D412yes—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50#870—21no—
51#869—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869D411yescharacteristic
59#869D411yescharacteristic
60#869D412yes—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8D412yes—
70#824—44no—
71#818—28no—
72#817—24no—
73#816—24no—
74#816—24no—
75#816—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#533—31no—
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8—31no—
81#500—62no—
82#500—62no—
83#498—61no—
84#491—62no—
85#487—61no—
86#487—61no—
87#487—61no—
88#483—62no—
89#475—62no—
90#431C2×D311yescharacteristic
91C2×#107—62no—
92#423—31no—
93#419—62no—
94C2×#101—62no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8C2×D311yescharacteristic
97#332—62no—
98Q8⋊C8—62no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—22no—
103C22×Dic3—41no—
104C2×Dic6—21no—
105C2×Dic6—22no—
106C2×Dic6—21no—
107C2×Dic6—42no—
108C2×Dic6—42no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C2×Dic6—42no—
113C2×Dic6—42no—
114C3×M16—22no—
115C3×M16—22no—
116C2×C3×C8—24no—
117C2×C3×C8—22no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139C4×Dic3—41no—
140#208—42no—
141C2×C3⋊C8—41no—
142C22×Q8—61no—
143C22×Q8—61no—
144C2×M16—61no—
145C2×M16D1212yes—
146C22×C8—21no—
147C4⋊Q8—32no—
148C4⋊Q8—31no—
149C4⋊Q8—31no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157#125—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162#117—61no—
163C2×C4⋊C4—32no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C42C22×D311yescharacteristic
168#107—68no—
169#106—64no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8D1212yes—
173#103—64no—
174#103—64no—
175#103—64no—
176#101—124no—
177C22⋊C8—21no—
178#97—21no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8D1212yes—
183C23×C3#12111yescharacteristic
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12711yescharacteristic
192C2×C3×C4#13512yes—
193C2×C3×C4AGL(1,ℤ/8)12yes—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—44no—
199C2×Dic3—82no—
200Dic6—44no—
201Dic6—42no—
202Dic6—42no—
203Dic6—82no—
204Dic6—81no—
205Dic6—81no—
206Dic6—81no—
207Dic6—81no—
208C3×C8—24no—
209C3×C8—42no—
210C3×C8—42no—
211C3⋊C8—81no—
212C2×Q8—61no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—121no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—122no—
219C2×Q8—122no—
220C2×Q8—122no—
221C2×Q8—122no—
222C22×C4—62no—
223C22×C4—121no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4#23911yescharacteristic
227M16—22no—
228M16—22no—
229M16—122no—
230C2×C8—24no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—121no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—61no—
239C4⋊C4—62no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C22⋊C4—121no—
250C42—121no—
251C42C2×D1211yescharacteristic
252C42C2×D1211yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#39211yescharacteristic
258C22×C3#46312yes—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4#44912yes—
262C3×C4#45612yes—
263C3×C4#47312yes—
264C3×C4#49511yescharacteristic
265C3×C4#49611yescharacteristic
266Dic3—81no—
267Dic3—82no—
268Dic3—82no—
269Dic3—81no—
270C23#89511yescharacteristic
271Q8—122no—
272Q8—124no—
273Q8—122no—
274Q8—241no—
275Q8—241no—
276Q8—241no—
277Q8—241no—
278Q8—242no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—124no—
286C2×C4—242no—
287C2×C4#87411yescharacteristic
288C2×C4#88211yescharacteristic
289C2×C4#88211yescharacteristic
290C2×C4#88211yescharacteristic
291C2×C4#89511yescharacteristic
292C2×C4#89511yescharacteristic
293C2×C4#90412yes—
294C2×C4#90812yes—
295C8—24no—
296C8—42no—
297C8—42no—
298C8—241no—
299C2×C3—22no—
300C2×C3#200511yescharacteristic
301C2×C3#201211yescharacteristic
302C2×C3#202011yescharacteristic
303C22—21no—
304C22—21no—
305C22#467111yescharacteristic
306C22#484412yes—
307C4—21no—
308C4—21no—
309C4—242no—
310C4—242no—
311C4—241no—
312C4—241no—
313C4#469611yescharacteristic
314C4#471211yescharacteristic
315C4#473412yes—
316C4#482612yes—
317C4#489612yes—
318C3#1017511yescharacteristic, complemented, Sylow
319C2—22no—
320C2#6867511yescharacteristic
321C2#6876811yescharacteristic
322C2#6894411yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1278D3
hypocenterC3#10175
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
318. C3#10175semidirect

Maximal subgroups

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

subgroupindex
#792712
#785372
C2×#46822
#707782
#698772
#697382
C3×#12782
#101753