Finite Group Databasebeta

← #10696621

Subgroups — #10696621

Order 768.

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

normal subgroups only

This group has 474 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#83246C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic
3#79320C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic
5#70836C211yesmaximal, characteristic
6#70812C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic
8#10101—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5230—22no—
15#5081—22no—
16#5080—21no—
17#5078—21no—
18#4983—22no—
19#4977—21no—
20#4975—21no—
21#4964—22no—
22#4932C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4829—22no—
25#4722—21no—
26#4721—21no—
27#4703—21no—
28#4702—21no—
29C3×#332—22no—
30#4467—22no—
31#4465—22no—
32#3236—31no—
33#3165—31no—
34#2660—31no—
35#1556D311yescharacteristic
36#1528—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—24no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57#870—21no—
58#870—21no—
59#870—21no—
60#870—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97—21no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#818—22no—
76#817—22no—
77#816—22no—
78#816—22no—
79#808—24no—
80#807—24no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
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#432C2×D311yescharacteristic
95#432C2×D311yescharacteristic
96#428—62no—
97C4×M16—31no—
98#398—62no—
99#395—62no—
100#384—61no—
101#382—61no—
102#376—31no—
103#332—22no—
104#330—62no—
105#328—62no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128#217—41no—
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—
143Dic3⋊C4—84no—
144C4×Dic3—24no—
145C4×Dic3—41no—
146#208—24no—
147C2×C3⋊C8—22no—
148C22×Q8—21no—
149C2×M16—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
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—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—64no—
172C2×C4⋊C4D1212yes—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176#107—62no—
177#106—64no—
178#106—62no—
179C4⋊C8—21no—
180C4⋊C8#6712yes—
181#103—62no—
182#103—62no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185#97—21no—
186#97—21no—
187#97—32no—
188C4×C8—61no—
189#95—32no—
190#95—64no—
191#95—61no—
192C23×C3#12011yescharacteristic
193C3×Q8—82no—
194C3×Q8—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—44no—
199C2×C3×C4—42no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12711yescharacteristic
206C2×Dic3—24no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215Dic6—42no—
216Dic6—81no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—24no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—62no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4C2×D1211yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232M16—64no—
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—62no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
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—64no—
257C42—121no—
258C42#23511yescharacteristic
259C42D3×D411yescharacteristic
260C42D3×D411yescharacteristic
261C42C2×#6711yescharacteristic
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—82no—
272C3×C4#45612yes—
273Dic3—44no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277Dic3—81no—
278C23#88211yescharacteristic
279Q8—82no—
280Q8—82no—
281Q8—122no—
282Q8—241no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—64no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—122no—
297C2×C4#88211yescharacteristic
298C2×C4#88411yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4#93011yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#94011yescharacteristic
303C8—42no—
304C8—64no—
305C8—81no—
306C2×C3—22no—
307C2×C3#200511yescharacteristic
308C2×C3#200511yescharacteristic
309C2×C3#200811yescharacteristic
310C22—21no—
311C22—21no—
312C22#475212yes—
313C22#496011yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—82no—
320C4—124no—
321C4—241no—
322C4—241no—
323C4—241no—
324C4—241no—
325C4#520512yes—
326C3#1010111yescharacteristic, complemented, Sylow
327C2—22no—
328C2#7011911yescharacteristic
329C2#7033111yescharacteristic
330C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1556D3
hypocenterC3#10101
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
326. C3#10101semidirect

Maximal subgroups

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

subgroupindex
#832462
#832072
#793202
C3×#15562
#708362
#708122
#677482
#101013