Finite Group Databasebeta

← #10695097

Subgroups — #10695097

Order 768.

454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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#78475C211yesmaximal, characteristic
2C3×C4×#97C211yesmaximal, characteristic
3#69738C212yesmaximal
4#69730C211yesmaximal, characteristic
5#69396C212yesmaximal
6#9850—31nomaximal, Sylow
7#5470C2212yes—
8C2×C3×#97—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C2×C3×C4×C8C2211yescharacteristic
11C3×C43C2211yescharacteristic
12#5116—22no—
13#4924C2212yes—
14#4699C411yescharacteristic, complemented
15C4×Dic12—22no—
16#4682C411yescharacteristic, complemented
17#4681C412yescomplemented
18#4523—24no—
19#4457—22no—
20#4456—24no—
21#3170—31no—
22#1994—32no—
23C4×#97D311yescharacteristic
24#1566—31no—
25#1493—32no—
26C22×C3×C8—21no—
27C22×C3×C8D412yes—
28C2×C3×C42—21no—
29C2×C3×C42—21no—
30C2×C3×C42D412yes—
31C2×C3×C42C2311yescharacteristic
32#926—22no—
33#924—22no—
34#924—22no—
35C2×#208—22no—
36C2×Dic12—22no—
37#870—22no—
38#869C2×C412yes—
39#869C2×C412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C3×#97—28no—
43C3×C4×C8—21no—
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8C2×C411yescharacteristic
46C3×C4×C8D412yes—
47C3×C4×C8D412yes—
48#818—22no—
49#816—24no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52#533—32no—
53#500—31no—
54#494—31no—
55#487—32no—
56#467—62no—
57C4×Q16—62no—
58#423—32no—
59C2×#97—22no—
60C2×C4×C8C2×D311yescharacteristic
61C2×C4×C8C2×D311yescharacteristic
62C43C2×D311yescharacteristic
63#339—64no—
64#334—62no—
65Q8⋊C8—64no—
66C22×C3×C4—22no—
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4#4111yescharacteristic
70C2×Dic6—22no—
71C2×Dic6—22no—
72C2×C3×C8—24no—
73C2×C3×C8—22no—
74C2×C3×C8—24no—
75C2×C3×C8C22⋊C412yes—
76C2×C3×C8C22⋊C412yes—
77C3×C42—24no—
78C3×C42—24no—
79C3×C42—42no—
80C3×C42C22×C411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84#217—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89Dic3⋊C4—42no—
90Dic3⋊C4—42no—
91C4×Dic3—42no—
92#208—44no—
93C2×C3⋊C8—42no—
94Dic12—44no—
95C2×Q16—62no—
96C2×M16—62no—
97C22×C8—21no—
98C22×C8#6712yes—
99C4⋊Q8—32no—
100C4⋊Q8—32no—
101#125—62no—
102C22⋊Q8—62no—
103C22⋊Q8—62no—
104C4×Q8—62no—
105C4×Q8—62no—
106#117—62no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42D1212yes—
110C2×C42C22×D311yescharacteristic
111#107—62no—
112C4⋊C8—62no—
113C4⋊C8—62no—
114#103—64no—
115#97—28no—
116C4×C8—21no—
117C4×C8C4×D311yescharacteristic
118C4×C8C4×D311yescharacteristic
119C4×C8D1212yes—
120C4×C8D1212yes—
121C23×C3#12111yescharacteristic
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—24no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4C2×C22⋊C411yescharacteristic
128C2×C3×C4C4×D411yescharacteristic
129C2×C3×C4C4×D411yescharacteristic
130C2×C3×C4#12111yescharacteristic
131C2×C3×C4#12411yescharacteristic
132C2×C3×C4#12711yescharacteristic
133C2×C3×C4#13714yes—
134C2×Dic3—42no—
135C2×Dic3—42no—
136Dic6—44no—
137Dic6—44no—
138C3×C8—24no—
139C3×C8—22no—
140C3×C8#10414yes—
141C3⋊C8—82no—
142C2×Q8—62no—
143C2×Q8—62no—
144C22×C4—22no—
145C22×C4C2×D1211yescharacteristic
146C22×C4#23511yescharacteristic
147C22×C4C2×#6711yescharacteristic
148Q16—124no—
149M16—124no—
150C2×C8—24no—
151C2×C8—22no—
152C2×C8—24no—
153C2×C8—122no—
154C2×C8(C2×D3)⋊C412yes—
155C2×C8(C2×D3)⋊C412yes—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—122no—
161C4⋊C4—122no—
162C22⋊C4—122no—
163C42—24no—
164C42—24no—
165C42—42no—
166C42—122no—
167C42C2×C4×D311yescharacteristic
168C42C2×D1211yescharacteristic
169C42C2×D1211yescharacteristic
170C42#23511yescharacteristic
171C22×C3—21no—
172C22×C3—21no—
173C22×C3#39011yescharacteristic
174C22×C3#47012yes—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—44no—
178C3×C4C2×#10412yes—
179C3×C4#44112yes—
180C3×C4#44412yes—
181C3×C4#49712yes—
182Dic3—82no—
183Dic3—82no—
184C23#93011yescharacteristic
185Q8—124no—
186Q8—124no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4C4×D1211yescharacteristic
195C2×C4C4×D1211yescharacteristic
196C2×C4#87411yescharacteristic
197C2×C4#87511yescharacteristic
198C2×C4#90914yes—
199C2×C4C2×(C2×D3)⋊C411yescharacteristic
200C2×C4#93011yescharacteristic
201C8—24no—
202C8—22no—
203C8—242no—
204C8#80514yes—
205C2×C3—22no—
206C2×C3#196211yescharacteristic
207C2×C3#197111yescharacteristic
208C2×C3#197511yescharacteristic
209C22—21no—
210C22—21no—
211C22#493911yescharacteristic
212C22#513712yes—
213C4—21no—
214C4—21no—
215C4—44no—
216C4—242no—
217C4—242no—
218C4#471512yes—
219C4#471612yes—
220C4#471712yes—
221C4C2×#80512yes—
222C3#985011yescharacteristic, complemented, Sylow
223C2—22no—
224C2#6939511yescharacteristic
225C2#6982411yescharacteristic
226C2#6984511yescharacteristic
subgroupG/H
centerC2×C4#874
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#97C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenterC4×#97D3
hypocenterC3#9850
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
14. #4699C4semidirect
16. #4682C4semidirect
17. #4681C4semidirect
222. C3#9850semidirect

Maximal subgroups

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

subgroupindex
#784752
C3×C4×#972
#697382
#697302
#693962
#98503