Finite Group Databasebeta

← #10696623

Subgroups — #10696623

Order 768.

474 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 85 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#79320C211yesmaximal, characteristic
2#78506C211yesmaximal, characteristic
3#73410C211yesmaximal, characteristic
4#70822C211yesmaximal, characteristic
5#69863C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic
7C3×#1300C211yesmaximal, characteristic
8#10101—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#5121—22no—
16#4936—21no—
17#4935—21no—
18#4926C2211yescharacteristic
19#4829—22no—
20#4722—21no—
21#4721—21no—
22#4703—21no—
23#4702—21no—
24#4702—22no—
25#4682—21no—
26#4680—21no—
27#4661—22no—
28#4656—22no—
29#4486—22no—
30#4465—22no—
31#4452—22no—
32#3236—31no—
33#3165—31no—
34#2660—31no—
35#1556—31no—
36#1528—31no—
37#1506—31no—
38#1300D311yescharacteristic
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—24no—
46C2×C4⋊Dic3D412yes—
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53#870—21no—
54#870—21no—
55#870—21no—
56#870—21no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71C3×#95—21no—
72C3×#95D412yes—
73C3⋊#95—24no—
74#830—21no—
75#818—24no—
76#818—22no—
77#817—24no—
78#817—22no—
79#816—22no—
80#816—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×#125—31no—
86#501—61no—
87#501—62no—
88#500—61no—
89#498—61no—
90#491—61no—
91#484—61no—
92#483—61no—
93#479—62no—
94#432—31no—
95#432C2×D311yescharacteristic
96#428—62no—
97C4×M16C2×D311yescharacteristic
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#328—62no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C3×M16—24no—
119C2×C3×C8—22no—
120C2×C3×C8—22no—
121C3×C42—24no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126#217—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—22no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—84no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—41no—
148C22×Q8—61no—
149C2×M16—21no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
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—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—64no—
173C2×C42—21no—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176#107—62no—
177#106—64no—
178#106—62no—
179C4⋊C8—61no—
180C4⋊C8D1212yes—
181#103—62no—
182#103—62no—
183C22⋊C8—21no—
184C22⋊C8—61no—
185#97—21no—
186#97—61no—
187#97D1212yes—
188C4×C8—21no—
189#95—21no—
190#95—64no—
191#95D1212yes—
192C23×C3#12011yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12711yescharacteristic
205C2×Dic3—42no—
206C2×Dic3—44no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213Dic6—42no—
214Dic6—82no—
215Dic6—82no—
216Dic6—81no—
217C3×C8—24no—
218C3×C8—42no—
219C3⋊C8—81no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—121no—
223C2×Q8—121no—
224C2×Q8—242no—
225C22×C4—22no—
226C22×C4—61no—
227C22×C4—62no—
228C22×C4—61no—
229C22×C4C2×D1211yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232M16—24no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—121no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—244no—
255C22⋊C4—121no—
256C42—24no—
257C42—121no—
258C42C2×D1211yescharacteristic
259C42C2×D1211yescharacteristic
260C42D3×D411yescharacteristic
261C42#23911yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39211yescharacteristic
265C22×C3#45612yes—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—44no—
272C3×C4#45612yes—
273Dic3—81no—
274Dic3—81no—
275Dic3—82no—
276Dic3—81no—
277Dic3—81no—
278C23#88211yescharacteristic
279Q8—122no—
280Q8—241no—
281Q8—242no—
282Q8—242no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—122no—
290C2×C4—124no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4#87411yescharacteristic
298C2×C4#88211yescharacteristic
299C2×C4#88211yescharacteristic
300C2×C4#89511yescharacteristic
301C2×C4#89511yescharacteristic
302C2×C4#89511yescharacteristic
303C8—24no—
304C8—42no—
305C8—241no—
306C2×C3—22no—
307C2×C3#200511yescharacteristic
308C2×C3#200511yescharacteristic
309C2×C3#200811yescharacteristic
310C22—21no—
311C22—21no—
312C22#467111yescharacteristic
313C22#482612yes—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—44no—
320C4—241no—
321C4—241no—
322C4—242no—
323C4—241no—
324C4—241no—
325C4#482612yes—
326C3#1010111yescharacteristic, complemented, Sylow
327C2—22no—
328C2#6876811yescharacteristic
329C2#6876811yescharacteristic
330C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1300D3
hypocenterC3#10101
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
326. C3#10101semidirect

Maximal subgroups

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

subgroupindex
#793202
#785062
#734102
#708222
#698632
#694332
C3×#13002
#101013