Finite Group Databasebeta

← #68346

Subgroups — #68346

Order 384.

218 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 59 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#5680C211yesmaximal, characteristic
2#4705C211yesmaximal, characteristic, complemented
3D3×#97C211yesmaximal, characteristic, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5#4540C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4465C211yesmaximal, characteristic, complemented
8#1532—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#946C411yescharacteristic
11Q8×Dic3C411yescharacteristic
12#938C411yescharacteristic
13D4×Dic3C411yescharacteristic
14C2×D3×C8—21no—
15#883—21no—
16#67⋊C4—21no—
17#879—21no—
18#877—21no—
19C42×D3C2211yescharacteristic
20#870C2211yescharacteristic
21C3×#97C2211yescharacteristic
22C3×#97C2211yescharacteristic
23(C2×D3)⋊C8—21no—
24#815—21no—
25Dic3⋊C8—21no—
26#803C2211yescharacteristic
27#803C2211yescharacteristic
28#529—31no—
29#432—31no—
30C2×#97—31no—
31#330—31no—
32#330—31no—
33#329—31no—
34#329D311yescharacteristic, complemented
35C2×C3×Q8C2×C411yescharacteristic
36C2×C3×D4C2×C411yescharacteristic
37C2×#67—21no—
38C22×Dic3—21no—
39C2×C4×D3C2×C411yescharacteristic
40C2×C4×D3D411yescharacteristic
41C2×C4×D3D411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C22⋊C4—21no—
45C3×C22⋊C4—21no—
46C3×C42C2311yescharacteristic
47#217—21no—
48#217—21no—
49(C2×D3)⋊C4—21no—
50C4⋊Dic3—21no—
51C4⋊Dic3C2×C411yescharacteristic
52C4⋊Dic3C2×C411yescharacteristic
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62D3×C8—24no—
63C22×C8—61no—
64#126—61no—
65#125—31no—
66#124C2×D311yescharacteristic
67#123—61no—
68C22⋊Q8—31no—
69#121—31no—
70C4×Q8—31no—
71C4×D4—31no—
72C4×D4—61no—
73#117—61no—
74C2×C42—31no—
75C4⋊C8—61no—
76C22⋊C8—61no—
77#97—31no—
78#97—31no—
79#97—61no—
80#97C2×D311yescharacteristic
81#97C2×D311yescharacteristic
82C23×C3—21no—
83C22×D3C22⋊C411yescharacteristic
84C3×Q8—41no—
85C3×D4—41no—
86C2×C3×C4—21no—
87C2×C3×C4C22×C411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90#67—42no—
91C2×Dic3—21no—
92C2×Dic3—21no—
93C2×Dic3—42no—
94C2×Dic3C22⋊C411yescharacteristic
95C2×Dic3C22⋊C411yescharacteristic
96C2×Dic3C22⋊C411yescharacteristic
97C4×D3—22no—
98C4×D3—41no—
99C4×D3—41no—
100C3×C8—22no—
101C3×C8—41no—
102C3⋊C8—22no—
103C3⋊C8—41no—
104C2×Q8C4×D311yescharacteristic
105C2×D4—61no—
106C2×D4C4×D311yescharacteristic
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—61no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—64no—
114C2×C8—61no—
115C2×C8—61no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C42—31no—
128C42—31no—
129C42—31no—
130C42—61no—
131C42C22×D311yescharacteristic
132C22×C3—41no—
133C22×C3—41no—
134C22×C3—41no—
135C22×C3C2×C22⋊C411yescharacteristic
136C2×D3—21no—
137C2×D3—21no—
138C2×D3#10412yes—
139C3×C4—21no—
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—41no—
143Dic3—21no—
144Dic3—21no—
145Dic3—21no—
146Dic3—41no—
147Dic3—41no—
148C23—21no—
149C23—31no—
150Q8—41no—
151D4—41no—
152D4—122no—
153C2×C4—21no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—62no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—121no—
161C2×C4—121no—
162C2×C4—122no—
163C2×C4C2×C4×D311yescharacteristic
164C2×C4D3×D411yescharacteristic
165C2×C4D3×D411yescharacteristic
166C8—22no—
167C8—41no—
168C8—62no—
169C8—121no—
170C2×C3—81no—
171C2×C3#41311yescharacteristic
172C2×C3C2×#10411yescharacteristic
173C2×C3#42111yescharacteristic
174D3—22no—
175C22—32no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—61no—
180C22—61no—
181C22D3×C22⋊C411yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—61no—
187C4—61no—
188C4—61no—
189C4—121no—
190C4—121no—
191C3#153211yescharacteristic, complemented, Sylow
192C2—62no—
193C2—81no—
194C2#474511yescharacteristic
195C2D3×#10411yescharacteristic
196C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3C2×C22⋊C4
prime radical#938C4
cosocleC3×C42C23
hypercenter#329D3
hypocenterC3#1532
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. #4465C2semidirect
7. #4465C2semidirect
34. #329D3semidirect
191. C3#1532semidirect

Maximal subgroups

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

subgroupindex
#56802
#47052
D3×#972
C3×#3292
#45402
#44652
#15323