Finite Group Databasebeta

← #10697688

Subgroups — #10697688

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#82431C212yesmaximal
2#79037C211yesmaximal, characteristic
3C3×#1984C211yesmaximal, characteristic
4#69738C212yesmaximal
5#67705C211yesmaximal, characteristic
6#10259—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8#5470C2212yes—
9C2×C3×C4⋊C8C2211yescharacteristic
10C2×C3×C4×C8C2211yescharacteristic
11#5225—22no—
12#5116—22no—
13#5022—24no—
14#5021—22no—
15#5020—22no—
16#4964—22no—
17#4924C2212yes—
18#4850—22no—
19#4849—22no—
20C3×#336—22no—
21#4456—24no—
22#3249—31no—
23#3078—32no—
24#1984D311yescharacteristic
25#1493—32no—
26#1292—31no—
27C22×C3×Q8D411yescharacteristic
28C22×C3×C8—21no—
29C22×C3×C8D412yes—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C42—21no—
35C2×C3×C42C2311yescharacteristic
36#926—22no—
37#924—22no—
38#924—22no—
39C2×#208—22no—
40C2×Dic12—22no—
41#870—22no—
42#869D412yes—
43#869D412yes—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C4×C8—22no—
48#835—24no—
49#835—24no—
50#818—22no—
51#816—24no—
52#808—24no—
53#807—24no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yes—
56#533—32no—
57C2×C4×Q8C2×D311yescharacteristic
58#479—64no—
59#477—62no—
60#477—62no—
61#475—62no—
62#467—62no—
63#462—62no—
64#428—62no—
65#423—32no—
66C2×C4⋊C8C2×D311yescharacteristic
67#419—62no—
68C2×C4×C8C2×D311yescharacteristic
69#336—22no—
70Q8⋊C8—64no—
71C2×C3×Q8—21no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8Q1612yescomplemented
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4C2×D411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4#4111yescharacteristic
80C2×Dic6—22no—
81C2×Dic6—22no—
82C2×C3×C8—22no—
83C2×C3×C8—22no—
84C2×C3×C8—41no—
85C2×C3×C8—41no—
86C2×C3×C8—42no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—42no—
89C3×C42—41no—
90C3×C42—41no—
91C3×C42C2×D412yes—
92C3×C42C2×D412yes—
93#217—42no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98Dic3⋊C4—42no—
99Dic3⋊C4—42no—
100C4×Dic3—42no—
101#208—42no—
102#208—42no—
103C2×C3⋊C8—24no—
104Dic12—82no—
105C22×Q8D1211yescharacteristic
106C2×Q16—62no—
107C2×M16—62no—
108C22×C8—21no—
109C22×C8#6712yes—
110C4⋊Q8—32no—
111C4⋊Q8—32no—
112#125—62no—
113C22⋊Q8—62no—
114C22⋊Q8—62no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—62no—
118C4×Q8—62no—
119#117—62no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4D1211yescharacteristic
122C2×C42—21no—
123C2×C42C22×D311yescharacteristic
124#107—64no—
125#107—62no—
126#106—64no—
127C4⋊C8—22no—
128C4⋊C8—34no—
129C4⋊C8—62no—
130#103—64no—
131#103—64no—
132#103—64no—
133C4×C8—22no—
134C23×C3#12111yescharacteristic
135C3×Q8—24no—
136C3×Q8—42no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—42no—
144C2×C3×C4—41no—
145C2×C3×C4#12012yes—
146C2×C3×C4#12011yescharacteristic
147C2×C3×C4#12112yes—
148C2×C3×C4#12711yescharacteristic
149C2×C3×C4C2×Q1611yescharacteristic
150C2×C3×C4#13511yescharacteristic
151C2×C3×C4#13712yes—
152C2×Dic3—42no—
153C2×Dic3—42no—
154Dic6—44no—
155Dic6—42no—
156Dic6—42no—
157C3×C8—42no—
158C3×C8—42no—
159C3⋊C8—44no—
160C2×Q8—21no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—62no—
164C2×Q8—62no—
165C2×Q8Dic1212yescomplemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C2×D1211yescharacteristic
169C22×C4#23511yescharacteristic
170C22×C4C2×#6711yescharacteristic
171Q16—242no—
172M16—122no—
173M16—122no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—64no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—122no—
187C4⋊C4—122no—
188C22⋊C4—122no—
189C42—41no—
190C42—41no—
191C42—122no—
192C42D3×D412yes—
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—42no—
202C3×C4—81no—
203C3×C4#45112yes—
204C3×C4#45212yes—
205C3×C4#45511yescharacteristic
206C3×C4#45611yescharacteristic
207Dic3—82no—
208Dic3—82no—
209C23#93011yescharacteristic
210Q8—24no—
211Q8—42no—
212Q8—124no—
213Q8—122no—
214Q8—122no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4#88212yes—
226C2×C4#88412yes—
227C2×C4#90411yescharacteristic
228C2×C4C2×Dic1211yescharacteristic
229C2×C4#90912yes—
230C2×C4#93711yescharacteristic
231C2×C4#94011yescharacteristic
232C8—42no—
233C8—42no—
234C8—124no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#200111yescharacteristic
238C2×C3#200711yescharacteristic
239C2×C3#201711yescharacteristic
240C22—21no—
241C22—21no—
242C22#496011yescharacteristic
243C22#511611yescharacteristic
244C22#513711yescharacteristic
245C4—21no—
246C4—22no—
247C4—42no—
248C4—81no—
249C4—242no—
250C4—242no—
251C4#473312yes—
252C4#473912yes—
253C4#506111yescharacteristic
254C4#509511yescharacteristic
255C3#1025911yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#6969411yescharacteristic
259C2#6996411yescharacteristic
260C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1984C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1984D3
hypocenterC3#10259
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
74. C2×C3×Q8Q16semidirect
165. C2×Q8Dic12semidirect
255. C3#10259semidirect

Maximal subgroups

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

subgroupindex
#824312
#790372
C3×#19842
#697382
#677052
#102593