Finite Group Databasebeta

← #10697901

Subgroups — #10697901

Order 768.

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

normal subgroups only

This group has 588 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#79272C211yesmaximal, characteristic
3#73311C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#70779C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7C3×#1276C211yesmaximal, characteristic
8#10293—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869—22no—
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C43C2211yescharacteristic
16#4935—22no—
17#4935—22no—
18#4924C2211yescharacteristic
19#4850—22no—
20#4849—22no—
21#4739—22no—
22#4722—22no—
23#4699—21no—
24#4682—21no—
25#4681—21no—
26#4681—21no—
27#4523—22no—
28#4482—22no—
29#4481—22no—
30#3230—31no—
31#3209—31no—
32#2868—31no—
33#1994—31no—
34#1549—31no—
35#1548—31no—
36#1276D311yescharacteristic
37C22×Dic6—21no—
38C22×Dic6D412yes—
39C2×C3×C42—21no—
40C2×C3×C42—21no—
41C2×C3×C42D412yes—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44C2×C4⋊Dic3—28no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D412yes—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×Dic12—24no—
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#869—21no—
62#869—48no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8D412yes—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73#818—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—
81C2×C4⋊Q8—62no—
82#500—61no—
83#494—61no—
84#487—61no—
85#487—61no—
86#484—62no—
87#477—62no—
88#462—62no—
89#451—62no—
90#431C2×D311yescharacteristic
91#431C2×D311yescharacteristic
92#423—31no—
93#384—62no—
94#384—62no—
95C43C2×D311yescharacteristic
96#339—62no—
97#333—62no—
98#332—62no—
99C22×C3×C4—22no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—22no—
104C22×Dic3—22no—
105C2×Dic6—21no—
106C2×Dic6—22no—
107C2×Dic6—21no—
108C2×Dic6—22no—
109C2×Dic6—44no—
110C2×Dic6—44no—
111C2×Dic6—41no—
112C2×Dic6—41no—
113C2×C3×C8—22no—
114C2×C3×C8—22no—
115C3×C42—24no—
116C3×C42—24no—
117C3×C42—22no—
118C3×C42Q1614yescomplemented
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—22no—
126C4⋊Dic3—21no—
127C4⋊Dic3—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—416no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
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—
142Dic12—84no—
143C22×Q8—32no—
144C22×Q8—61no—
145C2×Q16—64no—
146C2×M16—61no—
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—
156C4⋊Q8—61no—
157C4⋊Q8—128no—
158#125—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—68no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42D1212yes—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8D1212yes—
177C4⋊C8D1212yes—
178#103—64no—
179#103—64no—
180C22⋊C8—21no—
181C22⋊C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C23×C3#12011yescharacteristic
185C2×C3×C4—24no—
186C2×C3×C4—24no—
187C2×C3×C4—24no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12711yescharacteristic
196C2×C3×C4C2×Q1612yes—
197C2×C3×C4#13712yes—
198C2×Dic3—44no—
199C2×Dic3—44no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204Dic6—42no—
205Dic6—44no—
206Dic6—42no—
207Dic6—82no—
208Dic6—82no—
209Dic6—82no—
210C3×C8—42no—
211C3×C8—42no—
212C3⋊C8—81no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—124no—
218C2×Q8—124no—
219C2×Q8—121no—
220C2×Q8—121no—
221C22×C4—22no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227Q16—244no—
228M16—122no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—121no—
232C4⋊C4—61no—
233C4⋊C4—62no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—1216no—
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—24no—
249C42—24no—
250C42—22no—
251C42—121no—
252C42Dic1214yescomplemented
253C42C2×D1211yescharacteristic
254C42C2×D1211yescharacteristic
255C42D3×D411yescharacteristic
256C42#23911yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#39211yescharacteristic
260C22×C3#45112yes—
261C3×C4—28no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4#45312yes—
265C3×C4#45612yes—
266C3×C4#46212yes—
267C3×C4#49411yescharacteristic
268C3×C4#49711yescharacteristic
269Dic3—82no—
270Dic3—81no—
271Dic3—82no—
272Dic3—81no—
273C23#88211yescharacteristic
274Q8—122no—
275Q8—124no—
276Q8—122no—
277Q8—242no—
278Q8—242no—
279Q8—242no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—124no—
286C2×C4—124no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4#87411yescharacteristic
292C2×C4#88211yescharacteristic
293C2×C4#88211yescharacteristic
294C2×C4#89511yescharacteristic
295C2×C4#89511yescharacteristic
296C2×C4#89511yescharacteristic
297C2×C4C2×Dic1212yes—
298C2×C4#90912yes—
299C8—42no—
300C8—42no—
301C8—241no—
302C2×C3—22no—
303C2×C3#200211yescharacteristic
304C2×C3#200511yescharacteristic
305C2×C3#201611yescharacteristic
306C22—21no—
307C22—21no—
308C22#467111yescharacteristic
309C22#473912yes—
310C4—28no—
311C4—21no—
312C4—21no—
313C4—242no—
314C4—242no—
315C4—241no—
316C4—241no—
317C4#469911yescharacteristic
318C4#471511yescharacteristic
319C4#482212yes—
320C4#482612yes—
321C4#484912yes—
322C3#1029311yescharacteristic, complemented, Sylow
323C2—22no—
324C2#6868811yescharacteristic
325C2#6876111yescharacteristic
326C2#6876811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1276D3
hypocenterC3#10293
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
118. C3×C42Q16semidirect
252. C42Dic12semidirect
322. C3#10293semidirect

Maximal subgroups

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

subgroupindex
#793132
#792722
#733112
#707802
#707792
#693962
C3×#12762
#102933