Finite Group Databasebeta

← #85405

Subgroups — #85405

Order 384.

199 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 32 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
1C4⋊(#41⋊C3)C211yesmaximal, characteristic
2#5426C211yesmaximal, characteristic, complemented
3#5425C211yesmaximal, characteristic, complemented
4#5425C211yesmaximal, characteristic, complemented
5#5424C211yesmaximal, characteristic, complemented
6#5389C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3317—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C2×#41⋊C3C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#889—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#552—31no—
18#550D311yescharacteristic
19#490—31no—
20#482—31no—
21#481—31no—
22#479—31no—
23#468—31no—
24#465—31no—
25#460—31no—
26#452—31no—
27#452—31no—
28#441—31no—
29#435—31no—
30#417—31no—
31#416—31no—
32#41⋊C3—21no—
33#41⋊C3—21no—
34C2×SL(2,𝔽3)C2311yescharacteristic
35#226—21no—
36C3×C4⋊C4—41no—
37C4⋊Dic3—41no—
38Dic3⋊C4—41no—
39Dic3⋊C4—41no—
40Dic3⋊C4—41no—
41Dic3⋊C4—41no—
42C4×Dic3—41no—
43C2×#41C2×D311yescharacteristic
44C2×#41C2×D311yescharacteristic
45C2×Q16—31no—
46C2×SD16—31no—
47C2×SD16—31no—
48C2×M16—31no—
49C22×C8—31no—
50#127—31no—
51#126—61no—
52#125—31no—
53#125—61no—
54#123—61no—
55C22⋊Q8—31no—
56C22⋊Q8—31no—
57#121—31no—
58#121—31no—
59C4×Q8—31no—
60C4×Q8C2×D311yescharacteristic
61C4×D4—31no—
62#117—31no—
63#117—61no—
64C2×C4⋊C4—31no—
65#107—31no—
66#106—31no—
67#106—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#102—31no—
74#102—31no—
75#102—31no—
76#102—31no—
77#102—31no—
78#102—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81#97—31no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3—41no—
89SL(2,𝔽3)#4111yescharacteristic
90#41—21no—
91#41—21no—
92#41—61no—
93#41—61no—
94C2×Q8—31no—
95C2×Q8C22×D311yescharacteristic
96C2×D4—31no—
97C2×D4—31no—
98C2×D4—31no—
99C2×D4—61no—
100C22×C4—31no—
101C22×C4—31no—
102C22×C4—61no—
103Q16—61no—
104Q16—61no—
105SD16—121no—
106SD16—121no—
107M16—121no—
108C2×C8—31no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—31no—
112C2×C8—121no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4S411yescharacteristic
124C22⋊C4—31no—
125C22⋊C4—31no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C42—31no—
130C42—61no—
131C42—61no—
132C22×C3—41no—
133C3×C4—81no—
134C3×C4—81no—
135C3×C4—81no—
136Dic3—42no—
137Dic3—81no—
138Dic3—81no—
139Dic3—81no—
140C23—31no—
141C23—31no—
142Q8—31no—
143Q8—61no—
144Q8—61no—
145Q8#23711yescharacteristic
146D4—61no—
147D4—61no—
148D4—61no—
149D4—61no—
150D4—121no—
151D4—241no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—122no—
163C2×C4C2×S411yescharacteristic
164C2×C4C2×S411yescharacteristic
165C2×C4C2×S411yescharacteristic
166C8—61no—
167C8—61no—
168C8—61no—
169C8—61no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—41no—
173C22—61no—
174C22—61no—
175C22—61no—
176C22—61no—
177C22—61no—
178C22—61no—
179C22C22×S411yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—31no—
184C4—31no—
185C4—61no—
186C4—121no—
187C4—121no—
188C4—121no—
189C4—121no—
190C3—41noSylow
191C2—121no—
192C2—121no—
193C2#591411yescharacteristic
194C2#592311yescharacteristic
195C2#592411yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
prime radicalC4⋊(#41⋊C3)C2
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #5426C2semidirect
3. #5425C2semidirect
4. #5425C2semidirect
5. #5424C2semidirect
6. #5389C2semidirect
7. #5388C2semidirect

Maximal subgroups

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

subgroupindex
C4⋊(#41⋊C3)2
#54262
#54252
#54242
#53892
#53882
#33173
#8894