Finite Group Databasebeta

← #68341

Subgroups — #68341

Order 384.

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

normal subgroups only

This group has 234 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#5680C211yesmaximal, characteristic
2#4707C211yesmaximal, characteristic, complemented
3D3×#97C211yesmaximal, characteristic, complemented
4C3×#329C211yesmaximal, characteristic, complemented
5#4540C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4465C211yesmaximal, characteristic, complemented
8#1524—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#946C411yescharacteristic
11Q8×Dic3C411yescharacteristic
12#938C411yescharacteristic
13D4×Dic3C411yescharacteristic
14C2×ΓL(1,𝔽25)—21no—
15C2×D3×C8—21no—
16#883—21no—
17#67⋊C4—21no—
18#879—21no—
19#877—21no—
20C42×D3C2211yescharacteristic
21#870C2211yescharacteristic
22C3×#97C2211yescharacteristic
23C3×#97C2211yescharacteristic
24#815—21no—
25C8×Dic3—21no—
26#803C2211yescharacteristic
27#803C2211yescharacteristic
28#529—31no—
29C4×M16—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—
62ΓL(1,𝔽25)—24no—
63D3×C8—24no—
64C2×M16—61no—
65C22×C8—61no—
66#126—61no—
67#125—31no—
68#124C2×D311yescharacteristic
69#123—61no—
70C22⋊Q8—31no—
71#121—31no—
72C4×Q8—31no—
73C4×D4—31no—
74C4×D4—61no—
75#117—61no—
76C2×C42—31no—
77#97—31no—
78#97—31no—
79#97—61no—
80#97C2×D311yescharacteristic
81#97C2×D311yescharacteristic
82C4×C8—61no—
83C23×C3—21no—
84C22×D3C22⋊C411yescharacteristic
85C3×Q8—41no—
86C3×D4—41no—
87C2×C3×C4—21no—
88C2×C3×C4C22×C411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91#67—42no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—42no—
95C2×Dic3C22⋊C411yescharacteristic
96C2×Dic3C22⋊C411yescharacteristic
97C2×Dic3C22⋊C411yescharacteristic
98C4×D3—22no—
99C4×D3—22no—
100C4×D3—41no—
101C3×C8—22no—
102C3×C8—22no—
103C3⋊C8—22no—
104C3⋊C8—22no—
105C2×Q8C4×D311yescharacteristic
106C2×D4—61no—
107C2×D4C4×D311yescharacteristic
108C22×C4—31no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4—61no—
112M16—64no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—64no—
116C2×C8—61no—
117C2×C8—61no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C42—31no—
130C42—31no—
131C42—31no—
132C42—61no—
133C42C22×D311yescharacteristic
134C22×C3—41no—
135C22×C3—41no—
136C22×C3—41no—
137C22×C3C2×C22⋊C411yescharacteristic
138C2×D3—21no—
139C2×D3—21no—
140C2×D3#10412yes—
141C3×C4—21no—
142C3×C4—21no—
143C3×C4—21no—
144C3×C4—41no—
145Dic3—21no—
146Dic3—21no—
147Dic3—41no—
148Dic3—41no—
149Dic3#10412yes—
150C23—21no—
151C23—31no—
152Q8—41no—
153D4—41no—
154D4—122no—
155C2×C4—21no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—62no—
160C2×C4—62no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—121no—
164C2×C4—122no—
165C2×C4C2×C4×D311yescharacteristic
166C2×C4D3×D411yescharacteristic
167C2×C4D3×D411yescharacteristic
168C8—22no—
169C8—22no—
170C8—62no—
171C8—62no—
172C2×C3—81no—
173C2×C3#41311yescharacteristic
174C2×C3C2×#10411yescharacteristic
175C2×C3C2×#10411yescharacteristic
176D3—22no—
177C22—32no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—61no—
182C22—61no—
183C22D3×C22⋊C411yescharacteristic
184C4—21no—
185C4—21no—
186C4—21no—
187C4—32no—
188C4—41no—
189C4—61no—
190C4—61no—
191C4—121no—
192C4—121no—
193C3#152411yescharacteristic, complemented, Sylow
194C2—62no—
195C2—81no—
196C2#474511yescharacteristic
197C2D3×#10411yescharacteristic
198C2D3×#10411yescharacteristic
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#1524
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. #4707C2semidirect
3. D3×#97C2semidirect
4. C3×#329C2semidirect
5. #4540C2semidirect
6. #4465C2semidirect
7. #4465C2semidirect
34. #329D3semidirect
193. C3#1524semidirect

Maximal subgroups

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

subgroupindex
#56802
#47072
D3×#972
C3×#3292
#45402
#44652
#15243