Finite Group Databasebeta

← #10696885

Subgroups — #10696885

Order 768.

446 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 83 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×#5081C211yesmaximal, characteristic
3#79320C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic
5#70831C211yesmaximal, characteristic
6#70813C211yesmaximal, characteristic
7#67751C211yesmaximal, characteristic
8#10139—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5228—22no—
15#5092—21no—
16#5090—22no—
17#5081—22no—
18#5081—22no—
19#5079—21no—
20C2×#808—21no—
21C2×#807—21no—
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4829—22no—
25#4722—21no—
26#4721—21no—
27#4703—21no—
28#4702—21no—
29C3×#332—22no—
30#4469—22no—
31#4465—22no—
32#3236—31no—
33#3209—31no—
34C2×#501—31no—
35#1556D311yescharacteristic
36#1531—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—24no—
48#926—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51#924—21no—
52#924—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C22×C3⋊C8—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59#870—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97—21no—
71#835—24no—
72#830D412yes—
73#818—22no—
74#817—22no—
75#816—22no—
76#816—22no—
77#808—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—61no—
87#501—62no—
88#501—62no—
89#494—61no—
90#491—61no—
91#487—61no—
92#484—61no—
93#483—61no—
94#479—62no—
95#462—62no—
96#451—62no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#107—61no—
101C2×#106—61no—
102C2×#97—31no—
103#332—22no—
104#330—62no—
105#324—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127#217—41no—
128#216—84no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C22×Q8—21no—
148C2×Q16—64no—
149C22×C8—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8D1212yes—
156#125—61no—
157#125—61no—
158#125—61no—
159#125—61no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174#107—62no—
175#106—64no—
176#106—62no—
177C4⋊C8—21no—
178C4⋊C8—61no—
179C4⋊C8#6712yes—
180#103—62no—
181#103—62no—
182#103—64no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—32no—
186#97—21no—
187#97—21no—
188#97—32no—
189#97—61no—
190C4×C8—61no—
191C23×C3#12111yescharacteristic
192C3×Q8—44no—
193C3×Q8—82no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—82no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12711yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—42no—
210C2×Dic3—81no—
211Dic6—42no—
212Dic6—42no—
213C3×C8—42no—
214C3×C8—81no—
215C3⋊C8—24no—
216C3⋊C8—42no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—82no—
222C22×C4—41no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4#23511yescharacteristic
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228Q16—244no—
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—64no—
232C2×C8—62no—
233C2×C8—62no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C22⋊C4—121no—
253C42—121no—
254C42C2×D1211yescharacteristic
255C42D3×D411yescharacteristic
256C42D3×D411yescharacteristic
257C42C2×#6711yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#39211yescharacteristic
261C22×C3#47312yes—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—82no—
268C3×C4#45612yes—
269Dic3—81no—
270Dic3—81no—
271Dic3—81no—
272Dic3—81no—
273C23#88411yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—122no—
277Q8—122no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4—241no—
289C2×C4#88211yescharacteristic
290C2×C4#88211yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4#93011yescharacteristic
293C2×C4#93711yescharacteristic
294C2×C4#94011yescharacteristic
295C8—42no—
296C8—64no—
297C8—81no—
298C8—122no—
299C2×C3—22no—
300C2×C3#200511yescharacteristic
301C2×C3#200811yescharacteristic
302C2×C3#202011yescharacteristic
303C22—21no—
304C22—21no—
305C22#475512yes—
306C22#496011yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—241no—
314C4—241no—
315C4—241no—
316C4—241no—
317C4#520512yes—
318C3#1013911yescharacteristic, complemented, Sylow
319C2—22no—
320C2#7012511yescharacteristic
321C2#7033111yescharacteristic
322C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1556D3
hypocenterC3#10139
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
318. C3#10139semidirect

Maximal subgroups

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

subgroupindex
#832222
C2×#50812
#793202
C3×#15562
#708312
#708132
#677512
#101393