Finite Group Databasebeta

← #10708708

Subgroups — #10708708

Order 768.

322 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 67 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#79711C211yesmaximal, characteristic, complemented
2C3×#2207C211yesmaximal, characteristic
3#71291C211yesmaximal, characteristic
4#71286C211yesmaximal, characteristic
5#67765C211yesmaximal, characteristic, complemented
6#67561C211yesmaximal, characteristic, complemented
7#67558C211yesmaximal, characteristic, complemented
8#12330—31nomaximal, Sylow
9#5567—21no—
10C2×C3×SD32—21no—
11C3×#493C2211yescharacteristic
12C3×C4×Q16C2211yescharacteristic
13#5183—21no—
14C2×#827—22no—
15#5075—21no—
16#5031—21no—
17#4873—21no—
18#4795—22no—
19#4699C2211yescharacteristic
20C8×Dic6C2211yescharacteristic
21C3×C4⋊C16C2211yescharacteristic
22C3×#357—21no—
23#4544—21no—
24C4×C3⋊C16C2211yescharacteristic
25#4458C2211yescharacteristic
26#3395—31no—
27#2248—31no—
28#2207D311yescharacteristic
29#1668—31no—
30#1362—31no—
31Q8⋊C16—31no—
32#1332—31no—
33C2×C3×Q16D411yescharacteristic
34C2×C3×D8—21no—
35C2×C3×D8—21no—
36C3×#127—21no—
37C3×C4×Q8—21no—
38#944—22no—
39C2×#216—22no—
40C2×#214—41no—
41C2×Dic12—21no—
42C2×Dic12D412yes—
43#889—42no—
44#887—22no—
45#870—41no—
46#869—21no—
47C4×Dic6—41no—
48C4×Dic6D411yescharacteristic
49C3×SD32—42no—
50C2×C3×C16—21no—
51C3×#107D411yescharacteristic
52C3×#103—21no—
53C3×C4×C8C2311yescharacteristic
54#832—41no—
55#827—44no—
56Dic3⋊C8—21no—
57C8×Dic3—21no—
58C2×C3⋊C16—21no—
59#810—22no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)D411yescharacteristic
62#557—61no—
63C2×SD32—21no—
64C2×SD32—62no—
65#494—31no—
66#494—61no—
67#493C2×D311yescharacteristic
68#462—62no—
69#462—61no—
70#460—61no—
71C8×Q8—31no—
72C4×Q16—61no—
73C4×Q16C2×D311yescharacteristic
74C4⋊C16C2×D311yescharacteristic
75#357—21no—
76C4×C16—31no—
77#335—31no—
78#331—61no—
79C2×C3×Q8—21no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C2×Dic6—22no—
84C2×Dic6D811yescharacteristic
85C3×Q16—21no—
86C3×Q16—21no—
87C3×D8—42no—
88C3×D8—41no—
89C3×D8—41no—
90C2×C3×C8C2×D411yescharacteristic
91C2×C3×C8C2×D411yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4—41no—
94C3×C42—41no—
95C3×C42C2×D411yescharacteristic
96#216—82no—
97#214—82no—
98C4⋊Dic3—41no—
99C4⋊Dic3—41no—
100C4⋊Dic3D811yescharacteristic
101Dic3⋊C4—21no—
102Dic3⋊C4—42no—
103Dic3⋊C4—42no—
104Dic3⋊C4—42no—
105C4×Dic3—21no—
106C4×Dic3—42no—
107C2×C3⋊C8—21no—
108C2×C3⋊C8—41no—
109Dic12—42no—
110Dic12—41no—
111Dic12—41no—
112C3×C16—41no—
113C3⋊C16—22no—
114C2×Q16—32no—
115C2×Q16—62no—
116C2×Q16—61no—
117C2×Q16#6711yescharacteristic
118C2×SD16—121no—
119C2×D8—21no—
120C2×D8—21no—
121C4⋊Q8—62no—
122C4⋊Q8—61no—
123#127—21no—
124#125—121no—
125#125—122no—
126C4×Q8—21no—
127C4×Q8—31no—
128C4×Q8—62no—
129C4×Q8—121no—
130SD32—42no—
131SD32—124no—
132C2×C16—21no—
133C2×C16—61no—
134#107#6711yescharacteristic
135C4⋊C8—31no—
136C4⋊C8—61no—
137C4⋊C8—61no—
138#103—21no—
139#103—62no—
140#102—121no—
141C4×C8—61no—
142C4×C8C22×D311yescharacteristic
143C23×C3—81no—
144C3×Q8—41no—
145C3×Q8—41no—
146C3×D4—81no—
147C3×D4—81no—
148C3×D4—81no—
149C3×D4—81no—
150C3×D4—161no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4#12011yescharacteristic
154C2×C3×C4C2×D811yescharacteristic
155C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—42no—
159Dic6—21no—
160Dic6—42no—
161Dic6—42no—
162Dic6SD3212yescomplemented
163C3×C8—21no—
164C3×C8#13511yescharacteristic
165C3×C8AGL(1,ℤ/8)11yescharacteristic
166C3⋊C8—41no—
167C3⋊C8—81no—
168C2×Q8—21no—
169C2×Q8—31no—
170C2×Q8—62no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174Q16—21no—
175Q16—21no—
176Q16—122no—
177Q16—121no—
178Q16—121no—
179Q16—242no—
180SD16—242no—
181D8—42no—
182D8—41no—
183D8—41no—
184C2×C8—61no—
185C2×C8—121no—
186C2×C8D3×D411yescharacteristic
187C2×C8C2×#6711yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—31no—
190C4⋊C4—41no—
191C4⋊C4—61no—
192C4⋊C4—122no—
193C4⋊C4—122no—
194C4⋊C4—122no—
195C4⋊C4—121no—
196C4⋊C4—121no—
197C42—41no—
198C42—61no—
199C42—122no—
200C42D3×D411yescharacteristic
201C16—41no—
202C16—62no—
203C22×C3—161no—
204C22×C3—161no—
205C22×C3—161no—
206C22×C3#44711yescharacteristic
207C3×C4—21no—
208C3×C4—81no—
209C3×C4—81no—
210C3×C4#44911yescharacteristic
211C3×C4#45311yescharacteristic
212C3×C4C2×SD3211yescharacteristic
213C3×C4#51011yescharacteristic
214Dic3—22no—
215Dic3—41no—
216Dic3—82no—
217C23—81no—
218Q8—32no—
219Q8—41no—
220Q8—41no—
221Q8—61no—
222Q8—122no—
223Q8—122no—
224D4—81no—
225D4—81no—
226D4—81no—
227D4—81no—
228D4—161no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—122no—
234C2×C4D3×D811yescharacteristic
235C2×C4#91111yescharacteristic
236C2×C4#93711yescharacteristic
237C8—21no—
238C8—121no—
239C8—241no—
240C8#93211yescharacteristic
241C8#95011yescharacteristic
242C2×C3—321no—
243C2×C3#162111yescharacteristic
244C2×C3#218511yescharacteristic
245C2×C3#219311yescharacteristic
246C22—161no—
247C22—161no—
248C22—161no—
249C22#515611yescharacteristic
250C4—21no—
251C4—62no—
252C4—81no—
253C4—81no—
254C4—121no—
255C4—242no—
256C4D3×SD3211yescharacteristic
257C4#491511yescharacteristic
258C4#503711yescharacteristic
259C4#519811yescharacteristic
260C3#1233011yescharacteristic, complemented, Sylow
261C2—321no—
262C2#7101311yescharacteristic
263C2#7148311yescharacteristic
264C2#7151811yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#2207C2
socleC22×C3#447
prime radicalC3×#493C22
hypercenter#2207D3
hypocenterC3#12330
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #79711C2semidirect
5. #67765C2semidirect
6. #67561C2semidirect
7. #67558C2semidirect
162. Dic6SD32semidirect
260. C3#12330semidirect

Maximal subgroups

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

subgroupindex
#797112
C3×#22072
#712912
#712862
#677652
#675612
#675582
#123303