Finite Group Databasebeta

← #10696719

Subgroups — #10696719

Order 768.

466 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#83198C211yesmaximal, characteristic
3#79069C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic
5#69862C211yesmaximal, characteristic
6#69856C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#10115—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5228—22no—
15#5225—22no—
16#5137—22no—
17#5092—21no—
18#5090—22no—
19#5082—22no—
20#5079—21no—
21#5079—21no—
22#5079—21no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4851—21no—
26#4851—21no—
27#4849—21no—
28#4849—21no—
29C3×#343—22no—
30#4465—22no—
31#4456—22no—
32#3255—31no—
33#3209—31no—
34#3139—31no—
35#1997D311yescharacteristic
36#1526—31no—
37#1485—31no—
38#1299—31no—
39C22×C3×Q8D412yes—
40C2×C3×M16—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—24no—
50#926—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57C2×Dic12—22no—
58#870—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#835—24no—
73#835—24no—
74#835—24no—
75#830—21no—
76#817—22no—
77#816—22no—
78#816—22no—
79C4⋊(C3⋊C8)D412yes—
80#803D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#494—61no—
87#491—62no—
88#487—61no—
89#487—61no—
90#487—61no—
91#475—61no—
92#475—61no—
93#470—62no—
94#462—61no—
95#462—61no—
96#462—62no—
97#451—62no—
98#431—31no—
99#423C2×D311yescharacteristic
100#419—62no—
101C4×M16—31no—
102C4×M16C2×D311yescharacteristic
103#343—22no—
104#330—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—22no—
108C2×C3×Q8—44no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C3×M16—22no—
118C3×M16—41no—
119C3×M16—42no—
120C2×C3×C8—22no—
121C2×C3×C8—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#217—41no—
131#217—41no—
132#216—84no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147#208—42no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150Dic12—82no—
151C22×Q8D1212yes—
152C2×Q16—62no—
153C2×Q16—64no—
154C2×M16—21no—
155C2×M16—61no—
156C2×M16#6712yes—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165#125—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C42C22×D311yescharacteristic
180#106—62no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—32no—
184#103—62no—
185#103—62no—
186#103—64no—
187#103—64no—
188#103—64no—
189C22⋊C8—61no—
190#97—21no—
191#97—32no—
192C4×C8—21no—
193C4×C8—61no—
194C4×C8—61no—
195C23×C3#12111yescharacteristic
196C3×Q8—44no—
197C3×Q8—44no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—41no—
215Dic6—41no—
216Dic6—42no—
217Dic6—41no—
218Dic6—41no—
219Dic6—42no—
220C3×C8—42no—
221C3×C8—42no—
222C3⋊C8—42no—
223C3⋊C8—42no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C22×C4—22no—
232C22×C4C2×D1211yescharacteristic
233C22×C4#23511yescharacteristic
234C22×C4C2×#6711yescharacteristic
235Q16—242no—
236Q16—244no—
237M16—22no—
238M16—41no—
239M16—42no—
240M16—122no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—62no—
244C2×C8—62no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C42—121no—
264C42—121no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C42D3×D411yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#39211yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4#45512yes—
278C3×C4#45511yescharacteristic
279C3×C4#45611yescharacteristic
280C3×C4#47312yes—
281Dic3—81no—
282Dic3—81no—
283Dic3—81no—
284Dic3—81no—
285C23#93011yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—122no—
289Q8—122no—
290Q8—121no—
291Q8—121no—
292Q8—121no—
293Q8—121no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4#88211yescharacteristic
305C2×C4#88211yescharacteristic
306C2×C4#88411yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#94011yescharacteristic
310C8—42no—
311C8—42no—
312C8—122no—
313C8—122no—
314C2×C3—41no—
315C2×C3#200411yescharacteristic
316C2×C3#200711yescharacteristic
317C2×C3#202011yescharacteristic
318C22—21no—
319C22—21no—
320C22—21no—
321C22#496011yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—241no—
327C4—241no—
328C4—241no—
329C4—241no—
330C4#474612yes—
331C4#475512yes—
332C4#506111yescharacteristic
333C4#509511yescharacteristic
334C3#1011511yescharacteristic, complemented, Sylow
335C2—41no—
336C2#6969411yescharacteristic
337C2#7008411yescharacteristic
338C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10115
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
334. C3#10115semidirect

Maximal subgroups

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

subgroupindex
#832222
#831982
#790692
C3×#19972
#698622
#698562
#677102
#101153