Finite Group Databasebeta

← #10697689

Subgroups — #10697689

Order 768.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 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#79068C212yesmaximal
2#79037C211yesmaximal, characteristic
3#70501C212yesmaximal
4#70472C211yesmaximal, characteristic
5C3×#1292C211yesmaximal, characteristic
6#10259—31nomaximal, Sylow
7#5470C2212yes—
8C2×C4×Dic6C2211yescharacteristic
9C3×#423C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11#5121—22no—
12#5120—22no—
13#5116—22no—
14C2×C4×C3⋊C8C2211yescharacteristic
15#4851—22no—
16#4850—22no—
17#4850—22no—
18#4849—22no—
19#4829—24no—
20#4550—22no—
21#4484—24no—
22#3249—31no—
23#3078—32no—
24#1984—31no—
25#1493—32no—
26#1292D311yescharacteristic
27C22×Dic6D411yescharacteristic
28C2×C3×M16—22no—
29C22×C3×C8D412yes—
30C2×C3×C42C2311yescharacteristic
31#926—22no—
32C2×C4⋊Dic3D411yescharacteristic
33#924—22no—
34#924—22no—
35C2×Dic3⋊C4—21no—
36C2×C4×Dic3—21no—
37C22×C3⋊C8—21no—
38C2×Dic12—22no—
39#870—22no—
40#869D412yes—
41#869D412yes—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8D414yes—
49#818—24no—
50#818—22no—
51#817—24no—
52#816—24no—
53#816—24no—
54#816—24no—
55C4×C3⋊C8—22no—
56#533—32no—
57C2×C4×Q8—31no—
58#479—64no—
59#477—62no—
60#477—62no—
61#475—62no—
62#467—62no—
63#462—62no—
64#428—62no—
65#423C2×D312yes—
66C2×C4⋊C8C2×D311yescharacteristic
67#419—62no—
68C2×C4×C8—31no—
69#336—62no—
70Q8⋊C8—64no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4#4111yescharacteristic
74C22×Dic3—21no—
75C22×Dic3—21no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—21no—
79C2×Dic6—22no—
80C2×Dic6—22no—
81C2×Dic6Q1612yescomplemented
82C3×M16—42no—
83C3×M16—42no—
84C2×C3×C8—24no—
85C2×C3×C8—22no—
86C2×C3×C8—42no—
87C3×C42C2×D412yes—
88C3×C42C2×D412yes—
89#217—42no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95Dic3⋊C4—42no—
96Dic3⋊C4—42no—
97Dic3⋊C4—42no—
98C4×Dic3—41no—
99C4×Dic3—42no—
100C4×Dic3—41no—
101C2×C3⋊C8—22no—
102C2×C3⋊C8—41no—
103C2×C3⋊C8—41no—
104Dic12—82no—
105C22×Q8—31no—
106C2×Q16—62no—
107C2×M16—22no—
108C22×C8—61no—
109C22×C8D1212yes—
110C4⋊Q8—32no—
111C4⋊Q8—32no—
112#125—62no—
113C22⋊Q8—62no—
114C22⋊Q8—62no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×Q8—62no—
118C4×Q8—62no—
119#117—62no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—61no—
122C2×C42—61no—
123C2×C42C22×D311yescharacteristic
124#107—64no—
125#107—62no—
126#106—64no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8D1214yes—
130#103—64no—
131#103—64no—
132#103—64no—
133C4×C8—62no—
134C23×C3#12111yescharacteristic
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—41no—
138C2×C3×C4#12012yes—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4#12112yes—
141C2×C3×C4#12711yescharacteristic
142C2×C3×C4C2×Q1611yescharacteristic
143C2×C3×C4#13511yescharacteristic
144C2×C3×C4#13712yes—
145C2×Dic3—22no—
146C2×Dic3—41no—
147C2×Dic3—41no—
148C2×Dic3—42no—
149C2×Dic3—41no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152Dic6—24no—
153Dic6—42no—
154Dic6—44no—
155Dic6—42no—
156Dic6—42no—
157C3×C8—44no—
158C3×C8—42no—
159C3⋊C8—42no—
160C2×Q8—32no—
161C2×Q8—61no—
162C2×Q8—62no—
163C2×Q8—62no—
164C2×Q8—62no—
165C2×Q8—62no—
166C22×C4—61no—
167C22×C4—61no—
168C22×C4C2×D1211yescharacteristic
169C22×C4D3×D411yescharacteristic
170C22×C4#23911yescharacteristic
171Q16—242no—
172M16—42no—
173M16—42no—
174C2×C8—22no—
175C2×C8—24no—
176C2×C8—42no—
177C2×C8—62no—
178C2×C8—121no—
179C2×C8—121no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—122no—
186C4⋊C4—122no—
187C4⋊C4—122no—
188C22⋊C4—122no—
189C42—121no—
190C42—121no—
191C42—122no—
192C42C2×D1212yes—
193C42D3×D412yes—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#39211yescharacteristic
197C22×C3#46711yescharacteristic
198C22×C3#47011yescharacteristic
199C3×C4—21no—
200C3×C4—22no—
201C3×C4#45112yes—
202C3×C4#45212yes—
203C3×C4#45511yescharacteristic
204C3×C4#45611yescharacteristic
205Dic3—42no—
206Dic3—82no—
207Dic3—82no—
208Dic3—81no—
209C23#89511yescharacteristic
210Q8—64no—
211Q8—124no—
212Q8—122no—
213Q8—122no—
214Q8—122no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—62no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4—122no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4#87411yescharacteristic
226C2×C4#88211yescharacteristic
227C2×C4#88212yes—
228C2×C4#89512yes—
229C2×C4#91211yescharacteristic
230C2×C4#91512yes—
231C2×C4D3×Q1611yescharacteristic
232C8—42no—
233C8—44no—
234C8—122no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#200111yescharacteristic
238C2×C3#200711yescharacteristic
239C2×C3#201711yescharacteristic
240C22—21no—
241C22—21no—
242C22#467111yescharacteristic
243C22#486711yescharacteristic
244C22#488711yescharacteristic
245C4—22no—
246C4—21no—
247C4—122no—
248C4—241no—
249C4—242no—
250C4—242no—
251C4#474611yescharacteristic
252C4#475211yescharacteristic
253C4#478312yes—
254C4#480912yes—
255C3#1025911yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#6872711yescharacteristic
259C2#6888811yescharacteristic
260C2#6900911yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1292D3
hypocenterC3#10259
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
81. C2×Dic6Q16semidirect
255. C3#10259semidirect

Maximal subgroups

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

subgroupindex
#790682
#790372
#705012
#704722
C3×#12922
#102593