Finite Group Databasebeta

← #10696878

Subgroups — #10696878

Order 768.

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

normal subgroups only

This group has 468 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#83213C211yesmaximal, characteristic
2C2×#5081C211yesmaximal, characteristic, complemented
3#79318C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic, complemented
5#70831C211yesmaximal, characteristic, complemented
6#70806C211yesmaximal, characteristic
7#67751C211yesmaximal, characteristic, complemented
8#10138—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5227—22no—
15#5085—21no—
16#5083—22no—
17#5081—22no—
18#5081—22no—
19#5079—21no—
20C2×#808—21no—
21C2×#807—21no—
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4847—22no—
25#4736—21no—
26#4735—21no—
27#4713—21no—
28#4702—21no—
29C3×#332—22no—
30#4469—22no—
31#4464—22no—
32#3234—31no—
33#3206—31no—
34C2×#501—31no—
35#1556D311yescharacteristic, complemented
36#1530—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#215—24no—
49#930—21no—
50#930—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C22×C3⋊C8—21no—
56#875—21no—
57#874D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#869D411yescharacteristic
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#821—22no—
74#821—22no—
75#818—22no—
76#817—22no—
77#808—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—61no—
87#501—62no—
88#501—62no—
89#492—61no—
90#490—61no—
91#487—61no—
92#481—61no—
93#480—61no—
94#478—62no—
95#461—62no—
96#448—62no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#107—61no—
101C2×#106—61no—
102C2×#97—31no—
103#332—22no—
104#329—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
113C2×#67—41no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116#235—82no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132#215—84no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C2×C3⋊C8—24no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×#41—61no—
149C22×Q8—21no—
150C2×SD16—64no—
151C22×C8—61no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8D1212yes—
157#127—31no—
158#125—61no—
159#125—61no—
160#125—61no—
161#125—61no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C42C22×D311yescharacteristic
174#107—64no—
175#107—62no—
176#106—64no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8#6712yes—
181#103—64no—
182#102—62no—
183#102—62no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C22⋊C8—32no—
187#97—21no—
188#97—21no—
189#97—32no—
190#97—61no—
191C4×C8—61no—
192C23×C3#12111yescharacteristic
193C22×D3—41no—
194C3×Q8—44no—
195C3×Q8—82no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—82no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12711yescharacteristic
207#67—82no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—81no—
213D12—42no—
214D12—42no—
215D12—81no—
216C4×D3—82no—
217Dic6—81no—
218C3×C8—42no—
219C3×C8—81no—
220C3⋊C8—24no—
221C3⋊C8—42no—
222#41—242no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—61no—
226C2×Q8—82no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—121no—
231C22×C4—41no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—121no—
235C22×C4#23511yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238SD16—244no—
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—64no—
242C2×C8—62no—
243C2×C8—62no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42C2×D1211yescharacteristic
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C42C2×#6711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3#39211yescharacteristic
267C22×C3#47112yes—
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4#45612yes—
278Dic3—81no—
279Dic3—81no—
280Dic3—81no—
281C23—121no—
282C23#88411yescharacteristic
283Q8—44no—
284Q8—82no—
285Q8—241no—
286D4—122no—
287D4—122no—
288D4—241no—
289D4—242no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4—242no—
300C2×C4—241no—
301C2×C4#88211yescharacteristic
302C2×C4#88211yescharacteristic
303C2×C4#88411yescharacteristic
304C2×C4#93011yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4#94011yescharacteristic
307C8—42no—
308C8—64no—
309C8—81no—
310C8—122no—
311C2×C3—22no—
312C2×C3#200511yescharacteristic
313C2×C3#200811yescharacteristic
314C2×C3#201911yescharacteristic
315D3—161no—
316C22—21no—
317C22—21no—
318C22—241no—
319C22—241no—
320C22—241no—
321C22#475412yes—
322C22#496011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—241no—
330C4—241no—
331C4—241no—
332C4#520512yes—
333C3#1013811yescharacteristic, complemented, Sylow
334C2—22no—
335C2—481no—
336C2#7011711yescharacteristic
337C2#7033111yescharacteristic
338C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1556D3
hypocenterC3#10138
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
2. C2×#5081C2semidirect
4. C3×#1556C2semidirect
5. #70831C2semidirect
7. #67751C2semidirect
35. #1556D3semidirect
333. C3#10138semidirect

Maximal subgroups

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

subgroupindex
#832132
C2×#50812
#793182
C3×#15562
#708312
#708062
#677512
#101383