Finite Group Databasebeta

← #85377

Subgroups — #85377

Order 384.

211 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 34 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#5426C211yesmaximal, characteristic, complemented
3#5425C211yesmaximal, characteristic, complemented
4#5419C211yesmaximal, characteristic, complemented
5C2×#859C211yesmaximal, characteristic, complemented
6#5388C211yesmaximal, characteristic, complemented
7#5386C211yesmaximal, characteristic, complemented
8#3316—31nomaximal, Sylow
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C2×#226C2211yescharacteristic
12#878—41nomaximal
13C4×SL(2,𝔽3)C2211yescharacteristic
14#859C2211yescharacteristic
15#859C2211yescharacteristic
16#859C2211yescharacteristic
17#554—31no—
18#548D311yescharacteristic
19#489—31no—
20#483—31no—
21#481—31no—
22#475—31no—
23#470—31no—
24#465—31no—
25#464—31no—
26#452—31no—
27#451—31no—
28#441—31no—
29#435—31no—
30#417—31no—
31C2×#103—31no—
32C22×Dic3—41no—
33C2×Dic6—41no—
34#41⋊C3—21no—
35C2×SL(2,𝔽3)—21no—
36C2×SL(2,𝔽3)C2311yescharacteristic
37#226—21no—
38C3×C22⋊C4—41no—
39#217—41no—
40C4⋊Dic3—41no—
41Dic3⋊C4—41no—
42Dic3⋊C4—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#125—61no—
52#124—31no—
53C22⋊Q8—31no—
54C22⋊Q8—31no—
55C22⋊Q8—31no—
56C22⋊Q8—61no—
57C22⋊Q8—61no—
58#121—31no—
59C4×Q8—31no—
60C4×Q8C2×D311yescharacteristic
61C4×D4—31no—
62C4×D4—61no—
63C2×C4⋊C4—31no—
64C2×C4⋊C4—31no—
65#106—31no—
66#106—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—
89C2×Dic3—82no—
90Dic6—82no—
91SL(2,𝔽3)#4111yescharacteristic
92#41—21no—
93#41—61no—
94C2×Q8—21no—
95C2×Q8—31no—
96C2×Q8—31no—
97C2×Q8—61no—
98C2×Q8—61no—
99C2×Q8C22×D311yescharacteristic
100C2×D4—31no—
101C2×D4—31no—
102C22×C4—31no—
103C22×C4—31no—
104C22×C4—61no—
105C22×C4—61no—
106Q16—61no—
107Q16—61no—
108Q16—121no—
109SD16—121no—
110M16—121no—
111C2×C8—31no—
112C2×C8—31no—
113C2×C8—31no—
114C2×C8—31no—
115C2×C8—121no—
116C4⋊C4—31no—
117C4⋊C4—31no—
118C4⋊C4—31no—
119C4⋊C4—31no—
120C4⋊C4—31no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C22⋊C4—31no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4S411yescharacteristic
132C42—31no—
133C42—61no—
134C22×C3—42no—
135C22×C3—41no—
136C22×C3—81no—
137C22×C3—81no—
138C3×C4—81no—
139C3×C4—81no—
140Dic3—42no—
141Dic3—81no—
142Dic3—81no—
143Dic3—81no—
144C23—31no—
145C23C2×S411yescharacteristic
146Q8—31no—
147Q8—61no—
148Q8—61no—
149Q8—61no—
150Q8—61no—
151Q8—241no—
152Q8#23711yescharacteristic
153D4—61no—
154D4—61no—
155D4—62no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—61no—
160C2×C4—61no—
161C2×C4—61no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—61no—
167C2×C4—241no—
168C2×C4—241no—
169C2×C4C2×S411yescharacteristic
170C2×C4C2×S411yescharacteristic
171C8—61no—
172C8—61no—
173C8—61no—
174C8—61no—
175C2×C3—41no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—82no—
179C22—21no—
180C22—21no—
181C22—61no—
182C22—61no—
183C22—61no—
184C22#98412yes—
185C22C22×S411yescharacteristic
186C4—21no—
187C4—21no—
188C4—31no—
189C4—31no—
190C4—61no—
191C4—61no—
192C4—121no—
193C4—121no—
194C4—121no—
195C4—121no—
196C3—41noSylow
197C2—22no—
198C2—121no—
199C2#591411yescharacteristic
200C2C2×#98411yescharacteristic
201C2#592611yescharacteristic
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. #5426C2semidirect
3. #5425C2semidirect
4. #5419C2semidirect
5. C2×#859C2semidirect
6. #5388C2semidirect
7. #5386C2semidirect

Maximal subgroups

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

subgroupindex
#548⋊C32
#54262
#54252
#54192
C2×#8592
#53882
#53862
#33163
#8784