Finite Group Databasebeta

← #10697001

Subgroups — #10697001

Order 768.

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

normal subgroups only

This group has 552 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×#4699C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#69879C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7C3×#1278C211yesmaximal, characteristic
8#10155—31nomaximal, Sylow
9C22×Dic12—22no—
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16#5121—22no—
17#4924C2211yescharacteristic
18#4849—22no—
19#4739—22no—
20#4702—22no—
21#4699—22no—
22#4699—21no—
23#4699—22no—
24#4682—21no—
25#4681—21no—
26#4680—21no—
27#4554—22no—
28#4482—22no—
29#4456—22no—
30#3209—31no—
31#3165—31no—
32C2×#494—31no—
33#1997—31no—
34#1549—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—28no—
51C2×Dic12—24no—
52C2×Dic12—48no—
53#870—21no—
54#869—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8D412yes—
73#818—24no—
74#817—24no—
75#816—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C22×Q16—62no—
79#533—31no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—31no—
82#501—62no—
83#500—61no—
84#498—61no—
85#494—62no—
86#494—61no—
87#494—62no—
88#487—61no—
89#462—62no—
90#451—62no—
91#431C2×D311yescharacteristic
92#428—62no—
93#423—31no—
94C4×M16C2×D311yescharacteristic
95C2×C4×C8C2×D311yescharacteristic
96#343—62no—
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—22no—
105C2×Dic6—21no—
106C2×Dic6—22no—
107C2×Dic6—22no—
108C2×Dic6—21no—
109C2×Dic6—44no—
110C2×Dic6—44no—
111C2×Dic6—41no—
112C3×M16—22no—
113C3×M16Q1614yescomplemented
114C2×C3×C8—24no—
115C2×C3×C8—22no—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122#217—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—22no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138#208—42no—
139C2×C3⋊C8—41no—
140Dic12—416no—
141Dic12—84no—
142C22×Q8—32no—
143C22×Q8—61no—
144C2×Q16—68no—
145C2×Q16—64no—
146C2×Q16—128no—
147C2×M16—61no—
148C2×M16D1212yes—
149C22×C8—21no—
150C4⋊Q8—32no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171#107—64no—
172#106—64no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8D1212yes—
176#103—64no—
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×C4C2×Q1612yes—
193C2×C3×C4#13712yes—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—82no—
199C2×Dic3—82no—
200Dic6—44no—
201Dic6—44no—
202Dic6—42no—
203Dic6—44no—
204Dic6—42no—
205Dic6—82no—
206C3×C8—24no—
207C3×C8—24no—
208C3×C8—42no—
209C3⋊C8—81no—
210C2×Q8—62no—
211C2×Q8—61no—
212C2×Q8—62no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—124no—
216C2×Q8—124no—
217C2×Q8—121no—
218C22×C4—62no—
219C22×C4—121no—
220C22×C4C2×D1211yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4#23911yescharacteristic
223Q16—1216no—
224Q16—244no—
225M16—22no—
226M16—122no—
227M16Dic1214yescomplemented
228C2×C8—24no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—121no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—61no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—121no—
248C42—121no—
249C42C2×D1211yescharacteristic
250C42C2×D1211yescharacteristic
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#39211yescharacteristic
256C22×C3#46212yes—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4#45112yes—
260C3×C4#45312yes—
261C3×C4#47312yes—
262C3×C4#49411yescharacteristic
263C3×C4#49711yescharacteristic
264Dic3—82no—
265Dic3—82no—
266Dic3—81no—
267Dic3—81no—
268C23#89511yescharacteristic
269Q8—124no—
270Q8—124no—
271Q8—122no—
272Q8—122no—
273Q8—124no—
274Q8—242no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—242no—
282C2×C4—242no—
283C2×C4#87411yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88211yescharacteristic
286C2×C4#88211yescharacteristic
287C2×C4#89511yescharacteristic
288C2×C4#89511yescharacteristic
289C2×C4C2×Dic1212yes—
290C2×C4#90912yes—
291C8—24no—
292C8—24no—
293C8—42no—
294C8—241no—
295C2×C3—22no—
296C2×C3#200211yescharacteristic
297C2×C3#201611yescharacteristic
298C2×C3#202011yescharacteristic
299C22—21no—
300C22—21no—
301C22#467111yescharacteristic
302C22#484912yes—
303C4—21no—
304C4—21no—
305C4—241no—
306C4—242no—
307C4—242no—
308C4—241no—
309C4#469911yescharacteristic
310C4#471511yescharacteristic
311C4#473912yes—
312C4#482212yes—
313C4#489612yes—
314C3#1015511yescharacteristic, complemented, Sylow
315C2—22no—
316C2#6868811yescharacteristic
317C2#6876111yescharacteristic
318C2#6894411yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1278D3
hypocenterC3#10155
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
113. C3×M16Q16semidirect
227. M16Dic12semidirect
314. C3#10155semidirect

Maximal subgroups

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

subgroupindex
#793132
#785062
C2×#46992
#707802
#698792
#697382
C3×#12782
#101553