Finite Group Databasebeta

← #10697111

Subgroups — #10697111

Order 768.

488 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 103 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#83222C211yesmaximal, characteristic
2C2×#5079C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10171—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5228—22no—
15C2×#835—22no—
16#5120—22no—
17#5092—21no—
18#5090—22no—
19#5079—21no—
20#5079—22no—
21#5079—22no—
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4703—22no—
25#4681—21no—
26#4681—21no—
27C3×#343—22no—
28#4484—22no—
29#4482—22no—
30#3209—31no—
31#3139—31no—
32C2×#487—31no—
33#1997D311yescharacteristic
34#1546—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8D412yes—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#216—24no—
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×#97—21no—
67C3×C4×C8—21no—
68#835—28no—
69#835—24no—
70#830—21no—
71#816—24no—
72#816—24no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#494—61no—
81#491—62no—
82#487—61no—
83#487—61no—
84#487—62no—
85#487—62no—
86#487—61no—
87#462—62no—
88#451—62no—
89#431—31no—
90#423C2×D311yescharacteristic
91#419—62no—
92C2×#103—62no—
93C4×M16C2×D311yescharacteristic
94C2×C4×C8—31no—
95#343—22no—
96#332—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—44no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—22no—
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×Dic6—42no—
110C2×Dic6—42no—
111C2×Dic6—41no—
112C3×M16—22no—
113C3×M16—22no—
114C3×M16—42no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#217—41no—
126#216—84no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—41no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—42no—
141C22×Q8—61no—
142C22×Q8D1212yes—
143C2×Q16—64no—
144C2×M16—21no—
145C2×M16#6712yes—
146C22×C8—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—32no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
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—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C42C22×D311yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—32no—
171#103—64no—
172#103—64no—
173#103—68no—
174#103—64no—
175C22⋊C8—61no—
176#97—21no—
177C4×C8—21no—
178C4×C8—32no—
179C4×C8—61no—
180C4×C8—61no—
181C23×C3#12111yescharacteristic
182C3×Q8—44no—
183C3×Q8—44no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—82no—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12711yescharacteristic
194C2×C3×C4#13512yes—
195C2×C3×C4#13712yes—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—82no—
200Dic6—44no—
201Dic6—42no—
202Dic6—42no—
203Dic6—81no—
204Dic6—81no—
205C3×C8—42no—
206C3×C8—81no—
207C3⋊C8—42no—
208C3⋊C8—42no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—44no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—121no—
216C2×Q8—122no—
217C2×Q8—122no—
218C22×C4—22no—
219C22×C4—121no—
220C22×C4C2×D1211yescharacteristic
221C22×C4#23511yescharacteristic
222C22×C4C2×#6711yescharacteristic
223Q16—244no—
224M16—22no—
225M16—22no—
226M16—42no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—122no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—62no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—121no—
247C42—121no—
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39211yescharacteristic
255C22×C3#46412yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4#45212yes—
260C3×C4#45512yes—
261C3×C4#47312yes—
262C3×C4#49511yescharacteristic
263C3×C4#49711yescharacteristic
264Dic3—81no—
265Dic3—82no—
266Dic3—81no—
267C23#93011yescharacteristic
268Q8—44no—
269Q8—44no—
270Q8—122no—
271Q8—122no—
272Q8—124no—
273Q8—241no—
274Q8—241no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—122no—
282C2×C4—242no—
283C2×C4#88211yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4#88411yescharacteristic
287C2×C4#93711yescharacteristic
288C2×C4#94011yescharacteristic
289C2×C4#94212yes—
290C2×C4#95012yes—
291C8—42no—
292C8—81no—
293C8—122no—
294C8—122no—
295C2×C3—22no—
296C2×C3#200411yescharacteristic
297C2×C3#201511yescharacteristic
298C2×C3#202011yescharacteristic
299C22—21no—
300C22—21no—
301C22#496011yescharacteristic
302C22#502812yes—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—242no—
307C4—241no—
308C4—241no—
309C4#474612yes—
310C4#475512yes—
311C4#504712yes—
312C4#505911yescharacteristic
313C4#508611yescharacteristic
314C3#1017111yescharacteristic, complemented, Sylow
315C2—22no—
316C2#6968911yescharacteristic
317C2#7008411yescharacteristic
318C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10171
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
314. C3#10171semidirect

Maximal subgroups

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

subgroupindex
#832222
C2×#50792
#785372
C3×#19972
#705062
#705012
#676672
#101713