Finite Group Databasebeta

← #72144

Subgroups — #72144

Order 384.

536 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 121 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×C3×2−1+4C211yesmaximal, characteristic
2#5240C211yesmaximal, characteristic, complemented
3#5237C211yesmaximal, characteristic, complemented
4#5236C212yesmaximal, complemented
5#5225C212yesmaximal, complemented
6#1796—31nomaximal, Sylow
7C3×2−1+4—24no—
8C3×2−1+4—24no—
9C2×C3×#41—21no—
10C2×C3×#41—22no—
11C2×C3×#41C412yescomplemented
12C2×C3×#41C2211yescharacteristic
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8—21no—
15C22×C3×Q8C412yescomplemented
16C22×C3×Q8C2211yescharacteristic
17#926C2212yes—
18C2×#208C2212yes—
19#837—24no—
20#836—21no—
21#835—24no—
22#834—22no—
23#833—21no—
24C2×2−1+4D311yescharacteristic
25#421—32no—
26#419—32no—
27#413—31no—
28#410—31no—
29C3×#41—216no—
30C3×#41—28no—
31C3×#41—24no—
32C3×#41—24no—
33C3×#41—22no—
34C3×#41—22no—
35C3×#41—22no—
36C3×#41D414yes—
37C2×C3×Q8—28no—
38C2×C3×Q8—24no—
39C2×C3×Q8—24no—
40C2×C3×Q8—22no—
41C2×C3×Q8—21no—
42C2×C3×Q8—22no—
43C2×C3×Q8C2×C411yescharacteristic
44C2×C3×Q8C2×C412yes—
45C2×C3×Q8D414yes—
46C2×C3×Q8D411yescharacteristic
47C2×C3×D4—22no—
48C2×C3×D4—21no—
49C2×C3×D4C2×C411yescharacteristic
50C2×C3×D4D412yes—
51C2×C3×D4D411yescharacteristic
52C22×C3×C4—22no—
53C22×C3×C4—21no—
54C22×C3×C4—22no—
55C22×C3×C4—21no—
56C22×C3×C4C2×C412yes—
57C22×C3×C4C2311yescharacteristic
58#217—22no—
59C4⋊Dic3—22no—
60C4×Dic3—22no—
61#208—22no—
62#208—22no—
63C2×C3⋊C8—22no—
642−1+4—24no—
652−1+4—24no—
66C2×#41—21no—
67C2×#41—22no—
68C2×#41Dic312yescomplemented
69C2×#41C2×D311yescharacteristic
70C2×#41C2×D311yescharacteristic
71C22×Q8—21no—
72C22×Q8Dic312yescomplemented
73C22×Q8C2×D311yescharacteristic
74C2×M16—32no—
75#117—32no—
76#104—64no—
77#103—64no—
78#101—61no—
79#100—61no—
80#99—62no—
81C23×C3—21no—
82C23×C3C22⋊C412yes—
83C23×C3C2×D411yescharacteristic
84C3×Q8—28no—
85C3×Q8—21no—
86C3×Q8—22no—
87C3×Q8—24no—
88C3×Q8—22no—
89C3×Q8C22⋊C412yes—
90C3×Q8C22⋊C414yes—
91C3×D4—28no—
92C3×D4—24no—
93C3×D4—24no—
94C3×D4—21no—
95C3×D4—22no—
96C3×D4C22⋊C412yes—
97C2×C3×C4—28no—
98C2×C3×C4—28no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—24no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—24no—
105C2×C3×C4C22⋊C412yes—
106C2×C3×C4C22×C411yescharacteristic
107C2×C3×C4C2×D412yes—
108C2×C3×C4C2×D412yes—
109C2×C3×C4C2×D411yescharacteristic
110C2×Dic3—42no—
111C3⋊C8—42no—
112#41—216no—
113#41—28no—
114#41—24no—
115#41—24no—
116#41—22no—
117#41—22no—
118#41—22no—
119#41#6714yes—
120C2×Q8—28no—
121C2×Q8—24no—
122C2×Q8—24no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8C2×Dic311yescharacteristic
127C2×Q8C2×Dic312yes—
128C2×Q8#6714yes—
129C2×Q8#6711yescharacteristic
130C2×D4—22no—
131C2×D4—21no—
132C2×D4C2×Dic311yescharacteristic
133C2×D4#6712yes—
134C2×D4#6711yescharacteristic
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4C2×Dic312yes—
140C22×C4C22×D311yescharacteristic
141M16—62no—
142M16—62no—
143C2×C8—62no—
144C4⋊C4—62no—
145C22⋊C4—62no—
146C42—62no—
147C22×C3—22no—
148C22×C3—22no—
149C22×C3—41no—
150C22×C3—42no—
151C22×C3—42no—
152C22×C3C2×C22⋊C411yescharacteristic
153C22×C3#12012yes—
154C3×C4—24no—
155C3×C4—22no—
156C3×C4—22no—
157C3×C4C2×C22⋊C411yescharacteristic
158C3×C4C2×C22⋊C411yescharacteristic
159C3×C4#12012yes—
160Dic3—82no—
161C23—21no—
162C23#21712yes—
163C23C2×#6711yescharacteristic
164Q8—28no—
165Q8—21no—
166Q8—22no—
167Q8—24no—
168Q8—22no—
169Q8#21712yes—
170Q8#21714yes—
171D4—28no—
172D4—24no—
173D4—24no—
174D4—21no—
175D4—22no—
176D4#21712yes—
177C2×C4—28no—
178C2×C4—28no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—122no—
186C2×C4#21712yes—
187C2×C4C22×Dic311yescharacteristic
188C2×C4C2×#6712yes—
189C2×C4C2×#6712yes—
190C2×C4C2×#6711yescharacteristic
191C8—122no—
192C2×C3—22no—
193C2×C3—21no—
194C2×C3—42no—
195C2×C3—42no—
196C2×C3#37911yescharacteristic
197C22—22no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22C2×#21711yescharacteristic
203C22#95312yes—
204C4—24no—
205C4—22no—
206C4—22no—
207C4—242no—
208C4C2×#21711yescharacteristic
209C4C2×#21711yescharacteristic
210C4#95312yes—
211C3#179611yescharacteristic, complemented, Sylow
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#524711yescharacteristic
subgroupG/H
centerC2#5247
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
Fitting, prime radicalC2×C3×2−1+4C2
socleC2×C3#379
cosocleC22×C3×C4C23
hypercenterC2×2−1+4D3
hypocenterC3#1796
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
2. #5240C2semidirect
3. #5237C2semidirect
4. #5236C2semidirect
5. #5225C2semidirect
11. C2×C3×#41C4semidirect
15. C22×C3×Q8C4semidirect
68. C2×#41Dic3semidirect
72. C22×Q8Dic3semidirect
211. C3#1796semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
#52402
#52372
#52362
#52252
#17963