Finite Group Databasebeta

← #10690136

Subgroups — #10690136

Order 768.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2980C211yesmaximal, characteristic
2#83282C211yesmaximal, characteristic
3#71805C211yesmaximal, characteristic
4#70803C212yesmaximal, complemented
5#67751C212yesmaximal, complemented
6#9005—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#435—24no—
10C3×#432C2212yes—
11C3×#408—22no—
12#5233—22no—
13#5082—21no—
14#5071—21no—
15#5055C414yes—
16#4925C2212yes—
17C2×#803C2211yescharacteristic
18#4546—21no—
19#4544—21no—
20#4484—24no—
21#3135—31no—
22#2980D311yescharacteristic
23#1543—31no—
24#1497—32no—
25#1302—32no—
26C2×C3×#41D412yes—
27C2×C3×M16—22no—
28C22×C3×C8—22no—
29C3×C4⋊Q8D411yescharacteristic
30C3×#127D411yescharacteristic
31C3×#124C2×C412yes—
32C3×#121—22no—
33C3×C4×D4—22no—
34C2×C3×C42C2311yescharacteristic
35#926—21no—
36#924—22no—
37C22×C3⋊C8—21no—
38#870—21no—
39#869C2×C412yes—
40C4×Dic6—22no—
41C3×C4⋊C8D414yes—
42C3×C22⋊C8—24no—
43C3×C22⋊C8—22no—
44C3×#97—22no—
45#835—24no—
46#832—24no—
47#830—22no—
48C4⋊(C3⋊C8)—22no—
49#803—21no—
50#803C2×C412yes—
51C4×C3⋊C8—22no—
52#533—31no—
53#532C2×D311yescharacteristic
54#491—61no—
55#490—61no—
56#489—34no—
57#435—24no—
58#432C2×D312yes—
59#431—32no—
60#416—62no—
61#408—22no—
62C2×#97—31no—
63#333—61no—
64#331—61no—
65Q8⋊C8—64no—
66C3×#41—42no—
67C3×#41—42no—
68C2×C3×Q8C22⋊C412yes—
69C2×C3×D4—22no—
70C2×C3×D4—22no—
71C2×C3×D4C22⋊C412yes—
72C22×C3×C4—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4#4112yes—
75C2×Dic6—22no—
76C3×M16—82no—
77C2×C3×C8—24no—
78C2×C3×C8—82no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—24no—
81C3×C42C22×C411yescharacteristic
82C3×C42C2×D411yescharacteristic
83C3×C42C2×D412yes—
84#217—41no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87Dic3⋊C4—42no—
88C4×Dic3—41no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8—42no—
91C2×C3⋊C8—42no—
92C2×#41#6712yes—
93C2×M16—22no—
94C22×C8—22no—
95C22×C8—61no—
96C4⋊Q8—32no—
97C4⋊Q8D1211yescharacteristic
98#127D1211yescharacteristic
99#125—61no—
100#124C4×D312yes—
101C22⋊Q8—62no—
102#121—22no—
103C4×Q8—62no—
104C4×D4—22no—
105#117—61no—
106C2×C42C22×D311yescharacteristic
107C4⋊C8—62no—
108C4⋊C8#6714yes—
109#103—64no—
110#102—64no—
111C22⋊C8—24no—
112C22⋊C8—22no—
113C22⋊C8—62no—
114#97—22no—
115#97—32no—
116#97—61no—
117C4×C8—62no—
118C23×C3—22no—
119C23×C3#12311yescharacteristic
120C3×Q8—42no—
121C3×D4—42no—
122C3×D4—82no—
123C3×D4—82no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4C2×C22⋊C411yescharacteristic
130C2×C3×C4C4×D412yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12112yes—
133C2×Dic3—42no—
134Dic6—44no—
135C3×C8—44no—
136C3⋊C8—42no—
137C3⋊C8—82no—
138#41—42no—
139#41—42no—
140C2×Q8—62no—
141C2×Q8(C2×D3)⋊C412yes—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4(C2×D3)⋊C412yes—
145C22×C4—22no—
146C22×C4#23512yes—
147C22×C4C2×#6711yescharacteristic
148M16—82no—
149C2×C8—24no—
150C2×C8—62no—
151C2×C8—82no—
152C2×C8—122no—
153C2×C8—122no—
154C4⋊C4—22no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—122no—
158C22⋊C4—24no—
159C22⋊C4—121no—
160C42—121no—
161C42C2×C4×D311yescharacteristic
162C42C2×D1211yescharacteristic
163C42C2×#6712yes—
164C22×C3—22no—
165C22×C3—21no—
166C22×C3—44no—
167C22×C3—42no—
168C22×C3#38611yescharacteristic
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—21no—
172C3×C4—42no—
173C3×C4#45512yes—
174Dic3—82no—
175C23—22no—
176C23#92911yescharacteristic
177Q8—42no—
178Q8—124no—
179D4—42no—
180D4—82no—
181D4—82no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—122no—
188C2×C4C2×(C2×D3)⋊C411yescharacteristic
189C2×C4C4×#6712yes—
190C2×C4#93012yes—
191C2×C4#95311yescharacteristic
192C8—44no—
193C8—122no—
194C8—242no—
195C2×C3—41no—
196C2×C3—82no—
197C2×C3#188112yes—
198C2×C3#188511yescharacteristic
199C22—22no—
200C22—21no—
201C22—44no—
202C22—42no—
203C22#521311yescharacteristic
204C4—22no—
205C4—22no—
206C4—21no—
207C4—42no—
208C4—242no—
209C4#524212yes—
210C3#900511yescharacteristic, complemented, Sylow
211C2—41no—
212C2—82no—
213C2#7164911yescharacteristic
214C2#7202012yes—
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2980C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#2980D3
hypocenterC3#9005
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
4. #70803C2semidirect
5. #67751C2semidirect
210. C3#9005semidirect

Maximal subgroups

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

subgroupindex
C3×#29802
#832822
#718052
#708032
#677512
#90053