Finite Group Databasebeta

← #10695114

Subgroups — #10695114

Order 768.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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×#1744C211yesmaximal, characteristic
3#69738C212yesmaximal
4#69730C211yesmaximal, characteristic
5#69396C212yesmaximal
6#9853—31nomaximal, Sylow
7#5470C2212yes—
8C2×C3×C4×C8C2211yescharacteristic
9C2×C3×C4×C8C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5116—22no—
12#4924C2212yes—
13#4699C411yescharacteristic, complemented
14C4×Dic12—22no—
15#4682C411yescharacteristic, complemented
16#4681C412yescomplemented
17C3×#336—22no—
18#4523—24no—
19#4457—22no—
20#4456—24no—
21#3170—31no—
22#1994—32no—
23#1744D311yescharacteristic
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×C4×C8—21no—
43C3×C4×C8C2×C411yescharacteristic
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8D412yes—
46C3×C4×C8D412yes—
47#818—22no—
48#816—24no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)—22no—
51#533—32no—
52#500—31no—
53#494—31no—
54#487—32no—
55#467—62no—
56C4×Q16—62no—
57#423—32no—
58C2×C4×C8C2×D311yescharacteristic
59C2×C4×C8C2×D311yescharacteristic
60C43C2×D311yescharacteristic
61#339—64no—
62#336—22no—
63#334—62no—
64Q8⋊C8—64no—
65C22×C3×C4—22no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4#4111yescharacteristic
69C2×Dic6—22no—
70C2×Dic6—22no—
71C2×C3×C8—22no—
72C2×C3×C8—42no—
73C2×C3×C8—42no—
74C2×C3×C8C22⋊C412yes—
75C2×C3×C8C22⋊C412yes—
76C3×C42—24no—
77C3×C42—24no—
78C3×C42—42no—
79C3×C42C22×C411yescharacteristic
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42#4111yescharacteristic
83#217—42no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88Dic3⋊C4—42no—
89Dic3⋊C4—42no—
90C4×Dic3—42no—
91#208—44no—
92C2×C3⋊C8—42no—
93Dic12—44no—
94C2×Q16—62no—
95C2×M16—62no—
96C22×C8—21no—
97C22×C8#6712yes—
98C4⋊Q8—32no—
99C4⋊Q8—32no—
100#125—62no—
101C22⋊Q8—62no—
102C22⋊Q8—62no—
103C4×Q8—62no—
104C4×Q8—62no—
105#117—62no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42D1212yes—
109C2×C42C22×D311yescharacteristic
110#107—62no—
111C4⋊C8—62no—
112C4⋊C8—62no—
113#103—64no—
114C4×C8—21no—
115C4×C8C4×D311yescharacteristic
116C4×C8C4×D311yescharacteristic
117C4×C8D1212yes—
118C4×C8D1212yes—
119C23×C3#12111yescharacteristic
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—24no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
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×C4#13714yes—
132C2×Dic3—42no—
133C2×Dic3—42no—
134Dic6—44no—
135Dic6—44no—
136C3×C8—22no—
137C3×C8—22no—
138C3×C8—42no—
139C3⋊C8—82no—
140C2×Q8—62no—
141C2×Q8—62no—
142C22×C4—22no—
143C22×C4C2×D1211yescharacteristic
144C22×C4#23511yescharacteristic
145C22×C4C2×#6711yescharacteristic
146Q16—124no—
147M16—124no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—122no—
152C2×C8(C2×D3)⋊C412yes—
153C2×C8(C2×D3)⋊C412yes—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—122no—
159C4⋊C4—122no—
160C22⋊C4—122no—
161C42—24no—
162C42—24no—
163C42—42no—
164C42—122no—
165C42C2×C4×D311yescharacteristic
166C42C2×D1211yescharacteristic
167C42C2×D1211yescharacteristic
168C42#23511yescharacteristic
169C22×C3—21no—
170C22×C3—21no—
171C22×C3#39011yescharacteristic
172C22×C3#47012yes—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—44no—
176C3×C4#42112yes—
177C3×C4#44112yes—
178C3×C4#44312yes—
179C3×C4#49712yes—
180Dic3—82no—
181Dic3—82no—
182C23#93011yescharacteristic
183Q8—124no—
184Q8—124no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4C4×D1211yescharacteristic
193C2×C4C4×D1211yescharacteristic
194C2×C4#87411yescharacteristic
195C2×C4#87511yescharacteristic
196C2×C4#90914yes—
197C2×C4C2×(C2×D3)⋊C411yescharacteristic
198C2×C4#93011yescharacteristic
199C8—22no—
200C8—22no—
201C8—42no—
202C8—242no—
203C2×C3—22no—
204C2×C3#196311yescharacteristic
205C2×C3#197111yescharacteristic
206C2×C3#197411yescharacteristic
207C22—21no—
208C22—21no—
209C22#493911yescharacteristic
210C22#513712yes—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—242no—
215C4—242no—
216C4#470012yes—
217C4#471512yes—
218C4#471612yes—
219C4#500512yes—
220C3#985311yescharacteristic, complemented, Sylow
221C2—22no—
222C2#6944011yescharacteristic
223C2#6979511yescharacteristic
224C2#6982411yescharacteristic
subgroupG/H
centerC2×C4#874
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1744C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1744D3
hypocenterC3#9853
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1744

Complements

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

NH ≅ G/Ntype
13. #4699C4semidirect
15. #4682C4semidirect
16. #4681C4semidirect
220. C3#9853semidirect

Maximal subgroups

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

subgroupindex
#784752
C3×#17442
#697382
#697302
#693962
#98533