Finite Group Databasebeta

← #68345

Subgroups — #68345

Order 384.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#5668C211yesmaximal, characteristic
2#4705C211yesmaximal, characteristic, complemented
3D3×#97C211yesmaximal, characteristic, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5#4540C211yesmaximal, characteristic, complemented
6#4464C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1530—31nomaximal, Sylow
9C2×#239C411yescharacteristic
10C2×#237C411yescharacteristic
11C3×#124C2211yescharacteristic
12#944C411yescharacteristic
13#940C411yescharacteristic
14C2×D3×C8—21no—
15#884—21no—
16#884—21no—
17#67⋊C4—21no—
18#67⋊C4—21no—
19#875C2211yescharacteristic
20C42×D3C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23(C2×D3)⋊C8—21no—
24#815—21no—
25Dic3⋊C8—21no—
26#803C2211yescharacteristic
27#803C2211yescharacteristic
28#532—31no—
29#432—31no—
30C2×#97—31no—
31#329—31no—
32#329—31no—
33#329—31no—
34#329D311yescharacteristic, complemented
35C2×C3×Q8C2×C411yescharacteristic
36C2×C3×D4C2×C411yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C2×#67—21no—
40C22×Dic3—21no—
41#239—42no—
42#237—42no—
43C2×D12—21no—
44C2×D12C2×C411yescharacteristic
45C2×C4×D3—21no—
46C2×C4×D3C2×C411yescharacteristic
47C2×C4×D3D411yescharacteristic
48C2×C4×D3D411yescharacteristic
49C2×Dic6C2×C411yescharacteristic
50C2×C3×C8—21no—
51C2×C3×C8—21no—
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C42C2311yescharacteristic
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64D3×C8—24no—
65C2×#41—31no—
66C2×#41—31no—
67C22×C8—61no—
68C4⋊Q8—31no—
69#127—31no—
70#124—31no—
71#124C2×D311yescharacteristic
72#121—61no—
73#121—61no—
74C4×D4—61no—
75C4×D4—61no—
76C2×C42—31no—
77C4⋊C8—61no—
78C22⋊C8—61no—
79#97—31no—
80#97—31no—
81#97—61no—
82#97C2×D311yescharacteristic
83#97C2×D311yescharacteristic
84C23×C3—21no—
85C22×D3—21no—
86C22×D3C22⋊C411yescharacteristic
87C3×Q8—41no—
88C3×D4—41no—
89C2×C3×C4—21no—
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93#67—42no—
94#67—42no—
95#67—42no—
96C2×Dic3—21no—
97C2×Dic3—42no—
98C2×Dic3C22⋊C411yescharacteristic
99C2×Dic3C22⋊C411yescharacteristic
100C2×Dic3C22⋊C411yescharacteristic
101D12—41no—
102D12—42no—
103C4×D3—22no—
104C4×D3—22no—
105C4×D3—41no—
106C4×D3—42no—
107Dic6—41no—
108C3×C8—22no—
109C3×C8—41no—
110C3⋊C8—22no—
111C3⋊C8—41no—
112#41—122no—
113#41—122no—
114C2×Q8—31no—
115C2×Q8C4×D311yescharacteristic
116C2×D4—31no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4—61no—
121C2×D4C4×D311yescharacteristic
122C22×C4—31no—
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—61no—
126C22×C4—61no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—64no—
130C2×C8—61no—
131C2×C8—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C42—31no—
139C42—31no—
140C42—31no—
141C42C22×D311yescharacteristic
142C22×C3—41no—
143C22×C3—41no—
144C22×C3—41no—
145C22×C3C2×C22⋊C411yescharacteristic
146C2×D3—21no—
147C2×D3—21no—
148C2×D3—41no—
149C2×D3—41no—
150C2×D3—41no—
151C2×D3#10012yes—
152C3×C4—21no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—41no—
156Dic3—21no—
157Dic3—21no—
158Dic3—41no—
159Dic3#10412yes—
160C23—21no—
161C23—31no—
162C23—61no—
163Q8—41no—
164Q8—121no—
165D4—41no—
166D4—121no—
167D4—122no—
168D4—122no—
169D4—122no—
170D4—122no—
171C2×C4—21no—
172C2×C4—31no—
173C2×C4—31no—
174C2×C4—31no—
175C2×C4—62no—
176C2×C4—62no—
177C2×C4—61no—
178C2×C4—121no—
179C2×C4—122no—
180C2×C4—122no—
181C2×C4C2×C4×D311yescharacteristic
182C2×C4D3×D411yescharacteristic
183C2×C4D3×D411yescharacteristic
184C8—22no—
185C8—41no—
186C8—62no—
187C8—121no—
188C2×C3—81no—
189C2×C3C2×#10011yescharacteristic
190C2×C3C2×#10411yescharacteristic
191C2×C3#42111yescharacteristic
192D3—22no—
193D3—81no—
194C22—32no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—61no—
199C22—61no—
200C22—121no—
201C22—121no—
202C22—121no—
203C22D3×C22⋊C411yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—32no—
208C4—41no—
209C4—61no—
210C4—61no—
211C4—121no—
212C3#153011yescharacteristic, complemented, Sylow
213C2—62no—
214C2—81no—
215C2—241no—
216C2D3×#10011yescharacteristic
217C2D3×#10411yescharacteristic
218C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3C2×C22⋊C4
prime radical#5668C2
cosocleC3×C42C23
hypercenter#329D3
hypocenterC3#1530
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #4705C2semidirect
3. D3×#97C2semidirect
4. C3×#329C2semidirect
5. #4540C2semidirect
6. #4464C2semidirect
7. #4464C2semidirect
34. #329D3semidirect
212. C3#1530semidirect

Maximal subgroups

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

subgroupindex
#56682
#47052
D3×#972
C3×#3292
#45402
#44642
#15303