Finite Group Databasebeta

← #70225

Subgroups — #70225

Order 384.

332 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#944C211yesmaximal, characteristic
2C2×C3×#103C211yesmaximal, characteristic
3C2×#835C211yesmaximal, characteristic
4C2×#816C211yesmaximal, characteristic
5C2×C8×Dic3C211yesmaximal, characteristic
6#4975C211yesmaximal, characteristic
7C2×#810C211yesmaximal, characteristic
8#1956—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13#944—22no—
14#944C414yescomplemented
15C2×C4⋊Dic3C2211yescharacteristic
16C2×Dic3⋊C4—22no—
17C2×Dic3⋊C4—22no—
18C2×C4×Dic3C2211yescharacteristic
19C22×C3⋊C8C2211yescharacteristic
20C3×#103—22no—
21#835—22no—
22#816—22no—
23C8×Dic3—22no—
24#810—22no—
25C2×C4⋊Q8—31no—
26C2×#103—31no—
27C2×#103—31no—
28C2×#103—31no—
29C2×#103D311yescharacteristic
30C2×C4×C8—31no—
31#384—31no—
32C2×C3×Q8—21no—
33C2×C3×Q8—21no—
34C2×C3×Q8—21no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8C2×C412yes—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4C2311yescharacteristic
41C22×Dic3—21no—
42C22×Dic3—21no—
43C22×Dic3D412yes—
44C2×Dic6—21no—
45C2×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6C2×C412yes—
50C2×C3×C8—21no—
51C2×C3×C8—21no—
52C2×C3×C8D412yes—
53C3×C4⋊C4—21no—
54C3×C4⋊C4—21no—
55C4⋊Dic3—21no—
56C4⋊Dic3—21no—
57Dic3⋊C4—28no—
58Dic3⋊C4—44no—
59C4×Dic3C2×C412yes—
60C4×Dic3D412yes—
61C2×C3⋊C8—21no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8D412yes—
64C22×Q8—31no—
65C22×Q8C2×D311yescharacteristic
66C22×C8—31no—
67C22×C8C2×D311yescharacteristic
68C4⋊Q8—34no—
69C4⋊Q8—62no—
70C2×C4⋊C4—31no—
71C2×C4⋊C4—62no—
72C2×C4⋊C4—62no—
73C2×C4⋊C4C2×D311yescharacteristic
74C2×C42—31no—
75#103—22no—
76#103—62no—
77#103—62no—
78#103—62no—
79C4×C8—62no—
80C23×C3C2×D411yescharacteristic
81C3×Q8—22no—
82C3×Q8—22no—
83C3×Q8—41no—
84C3×Q8—41no—
85C2×C3×C4—22no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—42no—
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4#4111yescharacteristic
95C2×C3×C4#4111yescharacteristic
96C2×Dic3—22no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3—42no—
101C2×Dic3C22⋊C414yes—
102C2×Dic3SD1614yes—
103C2×Dic3Q1614yes—
104Dic6—22no—
105Dic6—22no—
106Dic6—41no—
107Dic6—41no—
108C3×C8—22no—
109C3⋊C8—22no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—32no—
116C2×Q8—61no—
117C2×Q8—61no—
118C2×Q8—61no—
119C2×Q8—61no—
120C2×Q8—61no—
121C2×Q8C4×D312yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—32no—
125C22×C4—61no—
126C22×C4—61no—
127C22×C4C22×D311yescharacteristic
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—32no—
131C2×C8—61no—
132C2×C8—61no—
133C2×C8#6712yes—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—68no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—124no—
140C42—32no—
141C42—32no—
142C22×C3C2×C22⋊C411yescharacteristic
143C22×C3#12411yescharacteristic
144C22×C3#12711yescharacteristic
145C22×C3C2×SD1612yes—
146C22×C3C2×Q1612yes—
147C3×C4—42no—
148C3×C4—81no—
149C3×C4C4×D411yescharacteristic
150C3×C4C4×D411yescharacteristic
151C3×C4#12111yescharacteristic
152C3×C4#12111yescharacteristic
153Dic3—42no—
154Dic3—81no—
155Dic3#10318yescomplemented
156C23D3×D411yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161Q8—62no—
162Q8—62no—
163Q8—121no—
164Q8—121no—
165C2×C4—22no—
166C2×C4—34no—
167C2×C4—34no—
168C2×C4—34no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—62no—
174C2×C4—121no—
175C2×C4—121no—
176C2×C4—122no—
177C2×C4—121no—
178C2×C4C2×C4×D311yescharacteristic
179C2×C4#23511yescharacteristic
180C2×C4D3×D411yescharacteristic
181C2×C4D3×D411yescharacteristic
182C2×C4#23711yescharacteristic
183C2×C4C2×#6711yescharacteristic
184C8—22no—
185C8—62no—
186C2×C3#39011yescharacteristic
187C2×C3C2×#10312yes—
188C2×C3#48712yes—
189C2×C3#49211yescharacteristic
190C2×C3#49411yescharacteristic
191C22D3×C22⋊C411yescharacteristic
192C22#88511yescharacteristic
193C22D3×SD1612yes—
194C22D3×Q1612yes—
195C22#94011yescharacteristic
196C4—38no—
197C4—42no—
198C4—81no—
199C4—122no—
200C4—241no—
201C4#67⋊C411yescharacteristic
202C4#88411yescharacteristic
203C4C4×#6711yescharacteristic
204C4#93911yescharacteristic
205C3#195611yescharacteristic, complemented, Sylow
206C2#479212yes—
207C2D3×#10312yes—
208C2#495711yescharacteristic
209C2#517511yescharacteristic
210C2#518311yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#103C2
socle, prime radicalC23×C3C2×D4
cosocleC22×C3×C4C23
hypercenterC2×#103D3
hypocenterC3#1956
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
14. #944C4semidirect
155. Dic3#103semidirect
205. C3#1956semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×C3×#1032
C2×#8352
C2×#8162
C2×C8×Dic32
#49752
C2×#8102
#19563