Finite Group Databasebeta

← #69151

Subgroups — #69151

Order 384.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#5716C211yesmaximal, characteristic
2#4740C212yesmaximal, complemented
3C3×#353C211yesmaximal, characteristic, complemented
4#4545C211yesmaximal, characteristic, complemented
5#4473C212yesmaximal, complemented
6#2126—31nomaximal, Sylow
7C2×#237C411yescharacteristic, complemented
8C2×#237C2211yescharacteristic
9C3×#127C2211yescharacteristic
10#938—22no—
11#936C411yescharacteristic, complemented
12#936C2211yescharacteristic
13D4×Dic3—22no—
14#877—22no—
15C42×D3C411yescharacteristic, complemented
16#869C411yescharacteristic, complemented
17C3×#99C2212yes—
18#834C2212yes—
19#806—22no—
20#532—31no—
21#410—32no—
22#354—32no—
23#353—31no—
24#353D311yescharacteristic, complemented
25C2×C3×D4—22no—
26C2×C3×D4C2×C411yescharacteristic
27C2×C3×D4C2311yescharacteristic
28C2×#67—21no—
29C2×#67D412yes—
30C22×Dic3—21no—
31C22×Dic3D412yes—
32#237—41no—
33#237—41no—
34#237—41no—
35#237—41no—
36C2×C4×D3—21no—
37C2×C4×D3C2×C411yescharacteristic
38C2×Dic6C2×C411yescharacteristic
39C2×Dic6C2×C411yescharacteristic
40C3×C22⋊C4—22no—
41C3×C42C2×C411yescharacteristic
42#217—22no—
43#217—22no—
44#217—22no—
45C4⋊Dic3—22no—
46Dic3⋊C4—42no—
47C4×Dic3—21no—
48C4×Dic3—42no—
49C4×Dic3C2×C411yescharacteristic
50C2×#41—31no—
51C2×#41—31no—
52C4⋊Q8—31no—
53#127C2×D311yescharacteristic
54#124—31no—
55#124—31no—
56#121—62no—
57C4×D4—62no—
58#117—62no—
59C2×C42—31no—
60#99—32no—
61#99—62no—
62#99C2×D312yes—
63C23×C3—21no—
64C23×C3C2×D412yes—
65C22×D3C22⋊C411yescharacteristic
66C3×D4—41no—
67C3×D4—41no—
68C3×D4—44no—
69C2×C3×C4—21no—
70C2×C3×C4—42no—
71C2×C3×C4C22×C411yescharacteristic
72#67—42no—
73#67—42no—
74C2×Dic3—22no—
75C2×Dic3—22no—
76C2×Dic3—21no—
77C2×Dic3—21no—
78C2×Dic3—41no—
79C2×Dic3—41no—
80C2×Dic3—42no—
81C2×Dic3C22⋊C412yes—
82C2×Dic3C22⋊C411yescharacteristic
83C4×D3—21no—
84C4×D3—21no—
85C4×D3—42no—
86Dic6—41no—
87Dic6—41no—
88#41—121no—
89#41—121no—
90#41—121no—
91#41—121no—
92C2×Q8—31no—
93C2×Q8—31no—
94C2×D4—22no—
95C2×D4—32no—
96C2×D4—61no—
97C2×D4C4×D311yescharacteristic
98C2×D4C22×D311yescharacteristic
99C22×C4—32no—
100C22×C4—31no—
101C22×C4—61no—
102C22×C4—61no—
103C4⋊C4—62no—
104C4⋊C4—122no—
105C22⋊C4—22no—
106C22⋊C4—62no—
107C22⋊C4—62no—
108C22⋊C4—62no—
109C42—31no—
110C42—61no—
111C42—122no—
112C42C4×D311yescharacteristic
113C22×C3—22no—
114C22×C3—42no—
115C22×C3—42no—
116C22×C3—41no—
117C22×C3C2×C22⋊C411yescharacteristic
118C2×D3—21no—
119C2×D3—41no—
120C3×C4—22no—
121C3×C4—21no—
122C3×C4—82no—
123Dic3—22no—
124Dic3—41no—
125Dic3—41no—
126Dic3—82no—
127Dic3#9911yescharacteristic
128Dic3#9911yescharacteristic
129C23—21no—
130C23—31no—
131C23D3×D412yes—
132Q8—121no—
133Q8—121no—
134D4—41no—
135D4—41no—
136D4—44no—
137D4—122no—
138D4—122no—
139C2×C4—21no—
140C2×C4—31no—
141C2×C4—32no—
142C2×C4—42no—
143C2×C4—61no—
144C2×C4—61no—
145C2×C4—62no—
146C2×C4—62no—
147C2×C4—61no—
148C2×C4—61no—
149C2×C4—122no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—122no—
153C2×C4C2×C4×D311yescharacteristic
154C2×C3—21no—
155C2×C3—42no—
156C2×C3—81no—
157C2×C3C2×#9911yescharacteristic
158D3—41no—
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—61no—
164C22—121no—
165C22D3×C22⋊C411yescharacteristic
166C4—22no—
167C4—21no—
168C4—31no—
169C4—31no—
170C4—62no—
171C4—82no—
172C4—121no—
173C4—121no—
174C4—242no—
175C3#212611yescharacteristic, complemented, Sylow
176C2—21no—
177C2—42no—
178C2—81no—
179C2—121no—
180C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×D4C22×D3
FittingC3×#353C2
socleC2×C3C2×#99
prime radical#5716C2
cosocleC2×C3×D4C23
hypercenter#353D3
hypocenterC3#2126
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #4740C2semidirect
3. C3×#353C2semidirect
4. #4545C2semidirect
5. #4473C2semidirect
7. C2×#237C4semidirect
11. #936C4semidirect
15. C42×D3C4semidirect
16. #869C4semidirect
24. #353D3semidirect
175. C3#2126semidirect

Maximal subgroups

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

subgroupindex
#57162
#47402
C3×#3532
#45452
#44732
#21263