Finite Group Databasebeta

← #10697057

Subgroups — #10697057

Order 768.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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#79313C211yesmaximal, characteristic
2#78506C211yesmaximal, characteristic
3C2×#4681C211yesmaximal, characteristic
4#70779C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7C3×#1278C211yesmaximal, characteristic
8#10163—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5121—22no—
16C2×#816—22no—
17#4924C2211yescharacteristic
18#4849—22no—
19#4739—22no—
20#4702—22no—
21#4699—21no—
22#4682—21no—
23#4681—22no—
24#4681—21no—
25#4681—22no—
26#4680—21no—
27#4554—22no—
28#4481—22no—
29#4456—22no—
30#3209—31no—
31#3165—31no—
32C2×#487—31no—
33#1997—31no—
34#1548—31no—
35#1493—31no—
36#1278D311yescharacteristic
37C22×Dic6—21no—
38C22×Dic6D412yes—
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⋊Dic3—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C2×Dic12—24no—
51#870—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D412yes—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8D412yes—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70C3×C4×C8D412yes—
71#818—24no—
72#817—24no—
73#816—28no—
74#816—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8—31no—
80#501—62no—
81#500—61no—
82#498—61no—
83#494—61no—
84#487—62no—
85#487—61no—
86#487—62no—
87#462—62no—
88#451—62no—
89#431C2×D311yescharacteristic
90#428—62no—
91#423—31no—
92C2×#103—62no—
93C4×M16C2×D311yescharacteristic
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#333—62no—
97Q8⋊C8—62no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—22no—
102C22×Dic3—41no—
103C2×Dic6—22no—
104C2×Dic6—22no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—22no—
108C2×Dic6—44no—
109C2×Dic6—41no—
110C2×Dic6—82no—
111C3×M16—22no—
112C3×M16Q1614yescomplemented
113C2×C3×C8—22no—
114C2×C3×C8—22no—
115C2×C3×C8—22no—
116C2×C3×C8—42no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121#217—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—22no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—41no—
137#208—42no—
138C2×C3⋊C8—41no—
139Dic12—84no—
140C22×Q8—32no—
141C22×Q8—61no—
142C2×Q16—64no—
143C2×M16—61no—
144C2×M16D1212yes—
145C22×C8—21no—
146C4⋊Q8—32no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156#125—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161#117—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#106—64no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8D1212yes—
172#103—68no—
173#103—64no—
174C22⋊C8—21no—
175#97—21no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8D1212yes—
180C23×C3#12111yescharacteristic
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12711yescharacteristic
189C2×C3×C4C2×Q1612yes—
190C2×C3×C4#13712yes—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—82no—
196C2×Dic3—82no—
197Dic6—44no—
198Dic6—42no—
199Dic6—44no—
200Dic6—42no—
201Dic6—82no—
202Dic6—82no—
203C3×C8—24no—
204C3×C8—42no—
205C3×C8—42no—
206C3⋊C8—81no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—62no—
212C2×Q8—124no—
213C2×Q8—121no—
214C2×Q8—242no—
215C22×C4—62no—
216C22×C4—121no—
217C22×C4C2×D1211yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4#23911yescharacteristic
220Q16—244no—
221M16—22no—
222M16—122no—
223M16Dic1214yescomplemented
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—42no—
228C2×C8—121no—
229C4⋊C4—62no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—62no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—121no—
244C42—121no—
245C42C2×D1211yescharacteristic
246C42C2×D1211yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#39211yescharacteristic
252C22×C3#46212yes—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4#45112yes—
256C3×C4#45512yes—
257C3×C4#47112yes—
258C3×C4#49411yescharacteristic
259C3×C4#49711yescharacteristic
260Dic3—82no—
261Dic3—81no—
262Dic3—81no—
263Dic3—82no—
264C23#89511yescharacteristic
265Q8—124no—
266Q8—122no—
267Q8—122no—
268Q8—124no—
269Q8—242no—
270Q8—242no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—242no—
278C2×C4—242no—
279C2×C4#87411yescharacteristic
280C2×C4#88211yescharacteristic
281C2×C4#88211yescharacteristic
282C2×C4#88211yescharacteristic
283C2×C4#89511yescharacteristic
284C2×C4#89511yescharacteristic
285C2×C4C2×Dic1212yes—
286C2×C4#90912yes—
287C8—24no—
288C8—42no—
289C8—42no—
290C8—241no—
291C2×C3—22no—
292C2×C3#200411yescharacteristic
293C2×C3#201611yescharacteristic
294C2×C3#201911yescharacteristic
295C22—21no—
296C22—21no—
297C22#467111yescharacteristic
298C22#484912yes—
299C4—21no—
300C4—21no—
301C4—242no—
302C4—241no—
303C4—242no—
304C4—241no—
305C4#469911yescharacteristic
306C4#471511yescharacteristic
307C4#473912yes—
308C4#482512yes—
309C4#489812yes—
310C3#1016311yescharacteristic, complemented, Sylow
311C2—22no—
312C2#6868811yescharacteristic
313C2#6876211yescharacteristic
314C2#6894311yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1278D3
hypocenterC3#10163
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
112. C3×M16Q16semidirect
223. M16Dic12semidirect
310. C3#10163semidirect

Maximal subgroups

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

subgroupindex
#793132
#785062
C2×#46812
#707792
#698792
#697382
C3×#12782
#101633