Finite Group Databasebeta

← #85379

Subgroups — #85379

Order 384.

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

normal subgroups only

This group has 200 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
1#548⋊C3C211yesmaximal, characteristic
2#5425C211yesmaximal, characteristic, complemented
3#5424C211yesmaximal, characteristic, complemented
4#5420C211yesmaximal, characteristic, complemented
5#5419C211yesmaximal, characteristic, complemented
6#5391C211yesmaximal, characteristic, complemented
7#5388C211yesmaximal, characteristic, complemented
8#3313—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#879—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#556—31no—
18#548D311yescharacteristic
19#489—31no—
20#485—31no—
21#482—31no—
22#477—31no—
23#468—31no—
24#467—31no—
25#464—31no—
26#452—31no—
27#451—31no—
28#441—31no—
29#435—31no—
30#419—31no—
31#416—31no—
32#41⋊C3—21no—
33C2×SL(2,𝔽3)—21no—
34C2×SL(2,𝔽3)C2311yescharacteristic
35#226—21no—
36C3×C22⋊C4—41no—
37#217—41no—
38#217—41no—
39C4⋊Dic3—41no—
40Dic3⋊C4—41no—
41Dic3⋊C4—41no—
42C4×Dic3—41no—
43C2×#41C2×D311yescharacteristic
44C22×Q8C2×D311yescharacteristic
45C2×Q16—31no—
46C2×Q16—31no—
47C2×SD16—31no—
48C2×M16—31no—
49C22×C8—31no—
50C4⋊Q8—31no—
51#126—61no—
52#126—61no—
53#124—31no—
54#124—61no—
55C22⋊Q8—31no—
56C22⋊Q8—31no—
57C22⋊Q8—31no—
58#121—31no—
59C4×Q8—31no—
60C4×Q8—61no—
61C4×Q8C2×D311yescharacteristic
62C4×D4—31no—
63#117—31no—
64#117—31no—
65#107—31no—
66#107—31no—
67#106—31no—
68C4⋊C8—31no—
69#103—31no—
70#103—31no—
71#103—31no—
72#103—31no—
73#103—31no—
74#103—31no—
75#103—31no—
76#102—31no—
77#102—31no—
78#102—31no—
79C22⋊C8—31no—
80C22⋊C8—31no—
81#97—31no—
82C23×C3—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—61no—
92C2×Q8—21no—
93C2×Q8—31no—
94C2×Q8—31no—
95C2×Q8—61no—
96C2×Q8—61no—
97C2×Q8C22×D311yescharacteristic
98C2×D4—31no—
99C2×D4—31no—
100C22×C4—31no—
101C22×C4—31no—
102Q16—61no—
103Q16—61no—
104Q16—121no—
105SD16—121no—
106M16—121no—
107C2×C8—31no—
108C2×C8—31no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—121no—
112C4⋊C4—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C4⋊C4—31no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C4⋊C4—61no—
120C4⋊C4—61no—
121C22⋊C4—31no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C22⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4S411yescharacteristic
128C42—31no—
129C42—61no—
130C42—61no—
131C42—61no—
132C22×C3—41no—
133C22×C3—81no—
134C22×C3—81no—
135C22×C3—81no—
136C3×C4—81no—
137C3×C4—81no—
138Dic3—81no—
139Dic3—81no—
140Dic3—81no—
141Dic3—81no—
142C23—31no—
143C23C2×S411yescharacteristic
144Q8—31no—
145Q8—61no—
146Q8—61no—
147Q8—61no—
148Q8—61no—
149Q8—122no—
150Q8#23711yescharacteristic
151D4—61no—
152D4—61no—
153D4—121no—
154C2×C4—31no—
155C2×C4—31no—
156C2×C4—31no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4C2×S411yescharacteristic
166C2×C4C2×S411yescharacteristic
167C8—61no—
168C8—61no—
169C8—61no—
170C8—61no—
171C2×C3—41no—
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—161no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—61no—
179C22—61no—
180C22—61no—
181C22C22×S411yescharacteristic
182C4—21no—
183C4—21no—
184C4—31no—
185C4—31no—
186C4—61no—
187C4—61no—
188C4—121no—
189C4—121no—
190C4—121no—
191C4—121no—
192C3—41noSylow
193C2—41no—
194C2—121no—
195C2#591411yescharacteristic
196C2#592311yescharacteristic
197C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#548D3
prime radical#548⋊C3C2
hypercenterC22⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #5425C2semidirect
3. #5424C2semidirect
4. #5420C2semidirect
5. #5419C2semidirect
6. #5391C2semidirect
7. #5388C2semidirect

Maximal subgroups

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

subgroupindex
#548⋊C32
#54252
#54242
#54202
#54192
#53912
#53882
#33133
#8794