Finite Group Databasebeta

← #10690137

Subgroups — #10690137

Order 768.

416 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 416 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
1C3×#3135C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic
3#71827C212yesmaximal, complemented
4#71805C211yesmaximal, characteristic
5#67662C212yesmaximal, complemented
6#9005—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#532C2211yescharacteristic
9C3×#491—21no—
10C3×#490—21no—
11C3×#489C414yes—
12C3×#416—22no—
13C2×C3×#97C2211yescharacteristic
14#5234—22no—
15#5011—24no—
16#4926C2212yes—
17#4925C2212yes—
18#4546—21no—
19#4544—21no—
20#4535—24no—
21#3135D311yescharacteristic
22#2980—31no—
23#1543—31no—
24#1497—32no—
25#1302—32no—
26C2×C3×#41D412yes—
27C22×C3×C8—21no—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8D411yescharacteristic
30C3×#127D411yescharacteristic
31C3×#125—21no—
32C3×#124C2×C412yes—
33C3×C22⋊Q8—22no—
34C3×#121—22no—
35C3×C4×Q8—22no—
36C3×C4×D4—22no—
37C3×#117—21no—
38C2×C3×C42C2311yescharacteristic
39C2×#208—22no—
40C22×C3⋊C8—22no—
41C3×#103—24no—
42C3×#102—24no—
43C3×#97—21no—
44C3×#97C2×C412yes—
45#830—24no—
46#830—22no—
47#830—22no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)D414yes—
50#803—22no—
51C4×C3⋊C8—22no—
52#533C2×D311yescharacteristic
53#532C2×D311yescharacteristic
54#491—21no—
55#490—21no—
56#489Dic314yes—
57#435—64no—
58#432—32no—
59#431—32no—
60#416—22no—
61#408—62no—
62C2×#97C2×D311yescharacteristic
63#333—61no—
64#331—61no—
65Q8⋊C8—64no—
66C3×#41—42no—
67C3×#41—42no—
68C2×C3×Q8—22no—
69C2×C3×Q8C22⋊C412yes—
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4C22⋊C412yes—
73C22×C3×C4—22no—
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4112yes—
76C2×C3×C8—22no—
77C2×C3×C8—42no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—42no—
82C3×C22⋊C4—24no—
83C3×C22⋊C4—41no—
84C3×C42—41no—
85C3×C42C22×C411yescharacteristic
86C3×C42C2×D412yes—
87C3×C42C2×D411yescharacteristic
88#208—82no—
89C2×C3⋊C8—24no—
90C2×C3⋊C8—42no—
91C2×C3⋊C8—82no—
92C2×#41#6712yes—
93C2×M16—62no—
94C22×C8—21no—
95C22×C8—62no—
96C4⋊Q8C2×Dic312yes—
97C4⋊Q8#6711yescharacteristic
98#127#6711yescharacteristic
99#125—21no—
100#124C2×Dic312yes—
101C22⋊Q8—22no—
102#121—22no—
103C4×Q8—22no—
104C4×D4—22no—
105#117—21no—
106C2×C42C22×D311yescharacteristic
107C4⋊C8—34no—
108C4⋊C8—62no—
109#103—24no—
110#102—24no—
111C22⋊C8—64no—
112C22⋊C8—62no—
113C22⋊C8—62no—
114#97—21no—
115#97—62no—
116#97C2×Dic312yes—
117C4×C8—62no—
118C23×C3—22no—
119C23×C3#12311yescharacteristic
120C3×Q8—42no—
121C3×Q8—44no—
122C3×D4—42no—
123C3×D4—82no—
124C3×D4—82no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4C2×C22⋊C411yescharacteristic
132C2×C3×C4C4×D412yes—
133C2×C3×C4#12011yescharacteristic
134C2×C3×C4#12112yes—
135C3×C8—42no—
136C3⋊C8—44no—
137C3⋊C8—82no—
138#41—42no—
139#41—42no—
140C2×Q8—22no—
141C2×Q8#21712yes—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4#21712yes—
145C22×C4—22no—
146C22×C4#23712yes—
147C22×C4C2×#6711yescharacteristic
148M16—242no—
149C2×C8—22no—
150C2×C8—42no—
151C2×C8—64no—
152C2×C8—122no—
153C2×C8—242no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C22⋊C4—24no—
159C22⋊C4—41no—
160C42—41no—
161C42D3×D412yes—
162C42C22×Dic311yescharacteristic
163C42C2×#6711yescharacteristic
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—82no—
174C3×C4#45512yes—
175C23—22no—
176C23#93511yescharacteristic
177Q8—42no—
178Q8—44no—
179D4—42no—
180D4—82no—
181D4—82no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4D4×Dic312yes—
189C2×C4#93711yescharacteristic
190C2×C4#93912yes—
191C2×C4C2×#21711yescharacteristic
192C8—42no—
193C8—124no—
194C8—242no—
195C2×C3—41no—
196C2×C3—82no—
197C2×C3#188112yes—
198C2×C3#188511yescharacteristic
199C22—21no—
200C22—22no—
201C22—42no—
202C22—44no—
203C22#522011yescharacteristic
204C4—21no—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—82no—
209C4#506112yes—
210C3#900511yescharacteristic, complemented, Sylow
211C2—41no—
212C2—82no—
213C2#7172311yescharacteristic
214C2#7211612yes—
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3135C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3135D3
hypocenterC3#9005
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
3. #71827C2semidirect
5. #67662C2semidirect
210. C3#9005semidirect

Maximal subgroups

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

subgroupindex
C3×#31352
#832792
#718272
#718052
#676622
#90053