Finite Group Databasebeta

← #10696857

Subgroups — #10696857

Order 768.

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

normal subgroups only

This group has 434 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#83224C211yesmaximal, characteristic
2C2×#5082C211yesmaximal, characteristic
3#79320C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic
5#70831C211yesmaximal, characteristic
6#70813C211yesmaximal, characteristic
7#67751C211yesmaximal, characteristic
8#10135—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C2×#835—21no—
15C2×#835—21no—
16#5093—21no—
17#5091—22no—
18#5082—22no—
19#5082—22no—
20#5078—21no—
21#4970—22no—
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#3230—31no—
34C2×#491—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
47#926—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C22×C3⋊C8—21no—
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D412yes—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97—21no—
70#835—24no—
71#835—24no—
72#830D412yes—
73#818—22no—
74#817—22no—
75#816—22no—
76#816—22no—
77#810—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#500—61no—
88#491—61no—
89#491—62no—
90#491—62no—
91#487—61no—
92#484—61no—
93#484—62no—
94#483—61no—
95#479—62no—
96#477—62no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#103—61no—
102C2×#97—31no—
103#332—22no—
104#330—62no—
105#324—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—44no—
108C2×C3×Q8—41no—
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—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—42no—
146C22×Q8—21no—
147C22×C8—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153C4⋊Q8D1212yes—
154#125—61no—
155#125—61no—
156#125—61no—
157#125—61no—
158#125—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171#107—64no—
172#107—62no—
173#106—62no—
174C4⋊C8—21no—
175C4⋊C8—61no—
176C4⋊C8#6712yes—
177#103—64no—
178#103—62no—
179#103—62no—
180#103—64no—
181#103—64no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—32no—
185#97—21no—
186#97—21no—
187#97—32no—
188#97—61no—
189C4×C8—61no—
190C23×C3#12111yescharacteristic
191C3×Q8—44no—
192C3×Q8—82no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—82no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12711yescharacteristic
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—81no—
210Dic6—42no—
211Dic6—42no—
212C3×C8—42no—
213C3×C8—81no—
214C3⋊C8—42no—
215C3⋊C8—42no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—61no—
220C2×Q8—61no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4#23511yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—122no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—121no—
251C42—121no—
252C42C2×D1211yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#39211yescharacteristic
259C22×C3#47312yes—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—82no—
266C3×C4#45412yes—
267Dic3—81no—
268Dic3—81no—
269Dic3—81no—
270Dic3—81no—
271C23#88411yescharacteristic
272Q8—44no—
273Q8—82no—
274Q8—122no—
275Q8—122no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—122no—
286C2×C4—241no—
287C2×C4#88211yescharacteristic
288C2×C4#88211yescharacteristic
289C2×C4#88411yescharacteristic
290C2×C4#93011yescharacteristic
291C2×C4#93711yescharacteristic
292C2×C4#94011yescharacteristic
293C8—42no—
294C8—81no—
295C8—122no—
296C8—122no—
297C2×C3—22no—
298C2×C3#200311yescharacteristic
299C2×C3#200911yescharacteristic
300C2×C3#202011yescharacteristic
301C22—21no—
302C22—21no—
303C22#475512yes—
304C22#496011yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—241no—
312C4—241no—
313C4—241no—
314C4—241no—
315C4#520112yes—
316C3#1013511yescharacteristic, complemented, Sylow
317C2—22no—
318C2#7012511yescharacteristic
319C2#7032611yescharacteristic
320C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1556D3
hypocenterC3#10135
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
316. C3#10135semidirect

Maximal subgroups

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

subgroupindex
#832242
C2×#50822
#793202
C3×#15562
#708312
#708132
#677512
#101353