Finite Group Databasebeta

← #70045

Subgroups — #70045

Order 384.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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#5832C211yesmaximal, characteristic
2#5819C211yesmaximal, characteristic, complemented
3C3×#410C211yesmaximal, characteristic, complemented
4#5005C212yesmaximal, complemented
5#4532C212yesmaximal, complemented
6#1884—31nomaximal, Sylow
7C2×C3×#41C2211yescharacteristic
8C2×#235C2211yescharacteristic
9C3×#117C2212yes—
10#948—24no—
11#947C411yescharacteristic, complemented
12#944C411yescharacteristic, complemented
13#940C411yescharacteristic, complemented
14#939—22no—
15#936C411yescharacteristic, complemented
16C4×#67—22no—
17C2×C4×Dic3C2211yescharacteristic
18C2×#208—22no—
19C2×#208C2212yes—
20C3×#99—22no—
21#805—24no—
22#568—31no—
23#532—31no—
24#421—32no—
25#410D311yescharacteristic, complemented
26#339—32no—
27C3×#41—22no—
28C3×#41—22no—
29C2×C3×Q8C2×C411yescharacteristic
30C2×C3×D4C2×C411yescharacteristic
31C2×C3×D4D412yes—
32C22×C3×C4D412yes—
33C22×C3×C4C2311yescharacteristic
34C2×#67—22no—
35C2×#67—21no—
36C22×Dic3—21no—
37#235—22no—
38#235—22no—
39C2×D12C2×C411yescharacteristic
40C2×C4×D3—21no—
41C2×Dic6C2×C411yescharacteristic
42C3×C4⋊C4—22no—
43C3×C22⋊C4—22no—
44C3×C42—22no—
45#217—22no—
46(C2×D3)⋊C4—22no—
47Dic3⋊C4—22no—
48C4×Dic3—21no—
49C4×Dic3C2×C411yescharacteristic
50C4×Dic3C2×C411yescharacteristic
51#208—24no—
52#208—22no—
53#208D414yescomplemented
54C2×C3⋊C8—24no—
55C2×C3⋊C8—22no—
56C2×#41—31no—
57C2×#41C2×D311yescharacteristic
58#131—64no—
59C2×M16—32no—
60C2×M16—62no—
61C4⋊Q8—31no—
62#127—31no—
63#124—31no—
64#124—31no—
65#121—62no—
66C4×D4—62no—
67#117C2×D312yes—
68C2×C42—31no—
69#104—64no—
70#99—22no—
71C23×C3C22⋊C412yes—
72C23×C3C2×D411yescharacteristic
73C22×D3—21no—
74C3×Q8—22no—
75C3×D4—24no—
76C3×D4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—42no—
80C2×C3×C4C22⋊C412yes—
81C2×C3×C4C22×C411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D412yes—
84C2×C3×C4#4112yes—
85#67—44no—
86#67—42no—
87C2×Dic3—21no—
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—42no—
91D12—22no—
92C4×D3—41no—
93C4×D3—41no—
94Dic6—22no—
95C3⋊C8—24no—
96#41—22no—
97#41—22no—
98#41—62no—
99#41—62no—
100C2×Q8—31no—
101C2×Q8C4×D311yescharacteristic
102C2×D4—31no—
103C2×D4—62no—
104C2×D4—61no—
105C2×D4C4×D311yescharacteristic
106C2×D4#6712yes—
107C22×C4—61no—
108C22×C4—61no—
109C22×C4D1212yes—
110C22×C4C22×D311yescharacteristic
111M16—34no—
112M16—64no—
113M16—62no—
114C2×C8—64no—
115C2×C8—62no—
116C4⋊C4—22no—
117C4⋊C4—62no—
118C22⋊C4—22no—
119C22⋊C4—62no—
120C22⋊C4—62no—
121C42—22no—
122C42—31no—
123C42—31no—
124C42—61no—
125C22×C3—22no—
126C22×C3—41no—
127C22×C3—42no—
128C22×C3C2×C22⋊C411yescharacteristic
129C22×C3#12112yes—
130C2×D3—42no—
131C2×D3—41no—
132C3×C4—22no—
133C3×C4—82no—
134C3×C4C4×D412yes—
135C3×C4#12011yescharacteristic
136C3×C4#12311yescharacteristic
137Dic3—22no—
138Dic3—41no—
139Dic3—41no—
140C23—61no—
141C23(C2×D3)⋊C412yes—
142C23C2×#6711yescharacteristic
143Q8—22no—
144Q8—62no—
145D4—24no—
146D4—22no—
147D4—62no—
148D4—124no—
149D4—122no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—122no—
157C2×C4—121no—
158C2×C4—121no—
159C2×C4(C2×D3)⋊C412yes—
160C2×C4C2×C4×D311yescharacteristic
161C2×C4C2×D1211yescharacteristic
162C2×C4D3×D412yes—
163C2×C4#23712yes—
164C8—64no—
165C2×C3—21no—
166C2×C3—22no—
167C2×C3—42no—
168C2×C3#38611yescharacteristic
169D3—81no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—122no—
174C22—121no—
175C22C2×(C2×D3)⋊C411yescharacteristic
176C22#93912yes—
177C4—22no—
178C4—62no—
179C4—82no—
180C4—121no—
181C4—121no—
182C4#67⋊C412yes—
183C4#88211yescharacteristic
184C4#88611yescharacteristic
185C3#188411yescharacteristic, complemented, Sylow
186C2—21no—
187C2—22no—
188C2—42no—
189C2—241no—
190C2#495811yescharacteristic
subgroupG/H
centerC4#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#410C2
socleC2×C3#386
prime radical#5832C2
cosocleC22×C3×C4C23
hypercenter#410D3
hypocenterC3#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #5819C2semidirect
3. C3×#410C2semidirect
4. #5005C2semidirect
5. #4532C2semidirect
11. #947C4semidirect
12. #944C4semidirect
13. #940C4semidirect
15. #936C4semidirect
25. #410D3semidirect
53. #208D4semidirect
185. C3#1884semidirect

Maximal subgroups

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

subgroupindex
#58322
#58192
C3×#4102
#50052
#45322
#18843