Finite Group Databasebeta

← #10697463

Subgroups — #10697463

Order 768.

462 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 99 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#83340C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic
5#69555C211yesmaximal, characteristic
6#69546C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic
8#10224—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5081—21no—
15#5081—21no—
16#5079—21no—
17#5078—21no—
18#5064—21no—
19#5064—21no—
20#5044—22no—
21#5021—22no—
22#4998—22no—
23#4925C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4849—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29C3×#328—22no—
30#4529—22no—
31#4452—22no—
32#3230—31no—
33#3209—31no—
34#3162—31no—
35#1993—31no—
36#1518—31no—
37#1518D311yescharacteristic
38#1281—31no—
39C22×Dic6—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×C4⋊Q8D411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53#924—21no—
54#924—21no—
55C22×C3⋊C8D412yes—
56C2×Dic12—24no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71#835—24no—
72#830D412yes—
73#816—24no—
74#808—22no—
75#808—22no—
76#807—22no—
77#807—22no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#501—61no—
87#501—61no—
88#501—61no—
89#501—61no—
90#500—61no—
91#494—61no—
92#487—61no—
93#487—61no—
94#484—62no—
95#477—62no—
96#462—62no—
97#451—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#427—62no—
101C2×#97—31no—
102#382C2×D311yescharacteristic
103#340—62no—
104#328—22no—
105#328—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—41no—
113C2×Dic6—22no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C2×C3×C8—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8Q1614yescomplemented
148Dic12—84no—
149C22×Q8—61no—
150C2×Q16—64no—
151C22×C8—32no—
152C4⋊Q8—32no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8D1211yescharacteristic
159C4⋊Q8D1211yescharacteristic
160#125—21no—
161#125—61no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—21no—
171#117—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#107—62no—
178#107—64no—
179#106—62no—
180#106—62no—
181C4⋊C8—21no—
182C4⋊C8—61no—
183#103—64no—
184#103—64no—
185C22⋊C8—32no—
186C22⋊C8#6712yes—
187#97—61no—
188#97—61no—
189C4×C8—21no—
190C4×C8—61no—
191#95—21no—
192#95—21no—
193C23×C3#12711yescharacteristic
194C3×Q8—42no—
195C3×Q8—81no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4C2×Q1612yes—
210C2×C3×C4#13712yes—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214C2×Dic3—82no—
215Dic6—42no—
216Dic6—44no—
217Dic6—82no—
218Dic6—81no—
219C3×C8—42no—
220C3⋊C8—24no—
221C3⋊C8—42no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—61no—
225C2×Q8—62no—
226C2×Q8—61no—
227C2×Q8—121no—
228C2×Q8—242no—
229C22×C4—41no—
230C22×C4—121no—
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C22×C4C2×#6711yescharacteristic
234Q16—244no—
235C2×C8—22no—
236C2×C8—34no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—62no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—62no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—41no—
258C22⋊C4—121no—
259C42—41no—
260C42—121no—
261C42C2×D1211yescharacteristic
262C42#23511yescharacteristic
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39211yescharacteristic
268C22×C3#49411yescharacteristic
269C22×C3#49711yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—161no—
277C3×C4#45112yes—
278C3×C4#46212yes—
279Dic3—82no—
280Dic3—81no—
281Dic3—81no—
282C23#94011yescharacteristic
283Q8—42no—
284Q8—81no—
285Q8—122no—
286Q8—124no—
287Q8—241no—
288Q8—242no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4—242no—
300C2×C4#88211yescharacteristic
301C2×C4#88211yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4#88411yescharacteristic
304C2×C4#91512yes—
305C2×C4D3×Q1612yes—
306C2×C4#93011yescharacteristic
307C2×C4#93711yescharacteristic
308C8—42no—
309C8—64no—
310C8—122no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#200811yescharacteristic
314C2×C3#200911yescharacteristic
315C2×C3#201611yescharacteristic
316C22—21no—
317C22—21no—
318C22#496011yescharacteristic
319C22#516711yescharacteristic
320C22#518311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—81no—
326C4—81no—
327C4—161no—
328C4—241no—
329C4—242no—
330C4—241no—
331C4#479512yes—
332C4#480912yes—
333C3#1022411yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7005011yescharacteristic
337C2#7028311yescharacteristic
338C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1518D3
hypocenterC3#10224
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
147. C2×C3⋊C8Q16semidirect
333. C3#10224semidirect

Maximal subgroups

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

subgroupindex
#833402
#833242
#793132
C3×#15182
#695552
#695462
#677412
#102243