Finite Group Databasebeta

← #10695068

Subgroups — #10695068

Order 768.

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

normal subgroups only

This group has 478 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×#1567C211yesmaximal, characteristic
3#70514C212yesmaximal
4#70501C212yesmaximal
5C4×C8×Dic3C211yesmaximal, characteristic
6#9845—31nomaximal, Sylow
7#5470C2212yes—
8C3×#423C2212yes—
9C2×C3×C4×C8C2211yescharacteristic
10#5116—22no—
11C2×C8×Dic3—22no—
12C42×Dic3C2211yescharacteristic
13C2×C4×C3⋊C8C2211yescharacteristic
14#4699C411yescharacteristic, complemented
15C4×Dic12—22no—
16#4682C411yescharacteristic, complemented
17#4681C412yescomplemented
18C3×#335—22no—
19#4555—24no—
20#4484—24no—
21#3170—31no—
22#1994—32no—
23C42×C8—31no—
24#1567D311yescharacteristic
25#1493—32no—
26C2×C3×M16—22no—
27C22×C3×C8D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—22no—
30#924—22no—
31#924—22no—
32C2×C4×Dic3—21no—
33C2×C4×Dic3—21no—
34C2×C4×Dic3D412yes—
35C22×C3⋊C8—21no—
36C2×Dic12—22no—
37#870—22no—
38#869C2×C412yes—
39#869C2×C412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C3×C4⋊C8—22no—
43C3×C4⋊C8—22no—
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8C2×C411yescharacteristic
46C3×C4×C8D412yes—
47#818—22no—
48#816—24no—
49C8×Dic3—28no—
50C4×C3⋊C8—21no—
51C4×C3⋊C8D412yes—
52#533—32no—
53#500—31no—
54#494—31no—
55#487—32no—
56#467—62no—
57C4×Q16—62no—
58#423C2×D312yes—
59C2×C4×C8—31no—
60C2×C4×C8—62no—
61C2×C4×C8C2×D311yescharacteristic
62C43—31no—
63#339—64no—
64#335—22no—
65Q8⋊C8—64no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4#4111yescharacteristic
69C22×Dic3—22no—
70C2×Dic6—22no—
71C2×Dic6—22no—
72C3×M16—44no—
73C2×C3×C8—24no—
74C2×C3×C8—42no—
75C2×C3×C8C22⋊C412yes—
76C2×C3×C8C22⋊C412yes—
77C3×C42C22×C411yescharacteristic
78C3×C42C2×D411yescharacteristic
79C3×C42C2×D411yescharacteristic
80C3×C42#4111yescharacteristic
81#217—42no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87Dic3⋊C4—42no—
88C4×Dic3—24no—
89C4×Dic3—24no—
90C4×Dic3—22no—
91C4×Dic3—42no—
92C4×Dic3Q1614yescomplemented
93C2×C3⋊C8—24no—
94C2×C3⋊C8—22no—
95Dic12—44no—
96C2×Q16—62no—
97C2×M16—22no—
98C22×C8—61no—
99C22×C8D1212yes—
100C4⋊Q8—32no—
101C4⋊Q8—32no—
102#125—62no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—62no—
106C4×Q8—62no—
107#117—62no—
108C2×C42—32no—
109C2×C42—61no—
110C2×C42—61no—
111C2×C42C22×D311yescharacteristic
112#107—62no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115#103—64no—
116C4×C8—32no—
117C4×C8—68no—
118C4×C8—61no—
119C4×C8C4×D311yescharacteristic
120C4×C8C4×D311yescharacteristic
121C4×C8#6712yes—
122C23×C3#12111yescharacteristic
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×C22⋊C411yescharacteristic
126C2×C3×C4C4×D411yescharacteristic
127C2×C3×C4C4×D411yescharacteristic
128C2×C3×C4#12111yescharacteristic
129C2×C3×C4#12411yescharacteristic
130C2×C3×C4#12711yescharacteristic
131C2×C3×C4C2×Q1612yes—
132C2×C3×C4#13512yes—
133C2×Dic3—24no—
134C2×Dic3—24no—
135C2×Dic3—24no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138Dic6—44no—
139Dic6—44no—
140C3×C8—22no—
141C3×C8—82no—
142C3×C8#10414yes—
143C3⋊C8—24no—
144C2×Q8—62no—
145C2×Q8—62no—
146C22×C4—62no—
147C22×C4C2×D1211yescharacteristic
148C22×C4D3×D411yescharacteristic
149C22×C4#23911yescharacteristic
150Q16—124no—
151M16—44no—
152C2×C8—24no—
153C2×C8—42no—
154C2×C8—64no—
155C2×C8—62no—
156C2×C8(C2×D3)⋊C412yes—
157C2×C8(C2×D3)⋊C412yes—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C22⋊C4—122no—
165C42—34no—
166C42—64no—
167C42—64no—
168C42—62no—
169C42—122no—
170C42C2×C4×D311yescharacteristic
171C42D3×D411yescharacteristic
172C42#23911yescharacteristic
173C42C2×#6711yescharacteristic
174C22×C3—21no—
175C22×C3—21no—
176C22×C3#39011yescharacteristic
177C22×C3#46712yes—
178C3×C4—21no—
179C3×C4—21no—
180C3×C4C2×#10412yes—
181C3×C4C4×Q1612yes—
182C3×C4#44312yes—
183C3×C4#49411yescharacteristic
184C3×C4#49511yescharacteristic
185Dic3—28no—
186Dic3—82no—
187Dic3—82no—
188C23#89511yescharacteristic
189Q8—124no—
190Q8—124no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—64no—
194C2×C4—64no—
195C2×C4—64no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4#89311yescharacteristic
199C2×C4#89311yescharacteristic
200C2×C4#89511yescharacteristic
201C2×C4#91212yes—
202C2×C4D3×Q1612yes—
203C2×C4C2×(C2×D3)⋊C411yescharacteristic
204C2×C4#94011yescharacteristic
205C2×C4#94711yescharacteristic
206C8—22no—
207C8—64no—
208C8—82no—
209C8#82514yes—
210C2×C3—22no—
211C2×C3#196211yescharacteristic
212C2×C3#196911yescharacteristic
213C2×C3#197411yescharacteristic
214C22—21no—
215C22—21no—
216C22#488712yes—
217C22#498811yescharacteristic
218C4—21no—
219C4—21no—
220C4—68no—
221C4—242no—
222C4—242no—
223C4#487312yes—
224C4#489512yes—
225C4C2×#82512yes—
226C4#515511yescharacteristic
227C4#518311yescharacteristic
228C3#984511yescharacteristic, complemented, Sylow
229C2—22no—
230C2#7051211yescharacteristic
231C2#7062711yescharacteristic
232C2#7069611yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1567C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1567D3
hypocenterC3#9845
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
14. #4699C4semidirect
16. #4682C4semidirect
17. #4681C4semidirect
92. C4×Dic3Q16semidirect
228. C3#9845semidirect

Maximal subgroups

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

subgroupindex
#784752
C3×#15672
#705142
#705012
C4×C8×Dic32
#98453