Finite Group Databasebeta

← #85517

Subgroups — #85517

Order 384.

307 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C22×#41⋊C3C211yesmaximal, characteristic
2C2×#984C211yesmaximal, characteristic, complemented
3#5426C211yesmaximal, characteristic, complemented
4#5425C212yesmaximal, complemented
5#5419C212yesmaximal, complemented
6#3007—31nomaximal, Sylow
7C2×#41⋊C3—22no—
8C2×#41⋊C3C2211yescharacteristic
9C2×#41⋊C3C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11#984—22no—
12C2×#226C2212yescomplemented
13#924—41nomaximal
14#859C2212yescomplemented
15C22×#41D311yescharacteristic
16C2×#137—31no—
17C2×#137—31no—
18C2×C22⋊Q8—31no—
19#452—32no—
20#452—32no—
21#451—32no—
22#450—32no—
23#417—31no—
24#417—31no—
25#407—31no—
26C22×C3×C4—41no—
27C2×Dic6—41no—
28#41⋊C3—21no—
29#41⋊C3D412yes—
30C2×SL(2,𝔽3)—21no—
31C2×SL(2,𝔽3)C2311yescharacteristic
32#226—22no—
33#217—42no—
34C4⋊Dic3—41no—
35Dic3⋊C4—42no—
36C2×#41—22no—
37C2×#41—31no—
38C2×#41—31no—
39C2×#41—62no—
40C2×#41C2×D311yescharacteristic
41C2×#41C2×D311yescharacteristic
42C22×Q8—31no—
43C22×Q8C2×D311yescharacteristic
44C22×D4—31no—
45C23×C4—31no—
46#137—62no—
47#137—64no—
48#137—62no—
49C2×Q16—32no—
50C2×Q16—32no—
51C2×SD16—32no—
52C2×SD16—32no—
53C2×M16—31no—
54C2×M16—31no—
55C22⋊Q8—32no—
56C22⋊Q8—32no—
57C22⋊Q8—61no—
58C22⋊Q8—61no—
59C2×C4⋊C4—31no—
60C2×C4⋊C4—61no—
61C2×C22⋊C4—61no—
62#103—32no—
63#103—32no—
64#102—32no—
65#102—32no—
66C22⋊C8—32no—
67C22⋊C8—32no—
68C23×C3—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—82no—
72C2×Dic3—42no—
73C2×Dic3—42no—
74Dic6—82no—
75SL(2,𝔽3)C2×D411yescharacteristic
76C24—31no—
77#41—21no—
78#41—32no—
79#41—34no—
80#41—62no—
81#41—62no—
82#41—61no—
83#41—62no—
84#41#6712yes—
85C2×Q8—21no—
86C2×Q8—31no—
87C2×Q8—32no—
88C2×Q8—62no—
89C2×Q8—62no—
90C2×Q8—61no—
91C2×Q8—61no—
92C2×Q8C22×D311yescharacteristic
93C2×D4—31no—
94C2×D4—31no—
95C2×D4—61no—
96C2×D4—62no—
97C2×D4—62no—
98C22×C4—31no—
99C22×C4—31no—
100C22×C4—31no—
101C22×C4—31no—
102C22×C4—31no—
103C22×C4—62no—
104C22×C4—62no—
105C22×C4—61no—
106C22×C4—61no—
107C22×C4S411yescharacteristic
108Q16—62no—
109Q16—64no—
110Q16—62no—
111SD16—64no—
112SD16—64no—
113M16—62no—
114M16—62no—
115C2×C8—32no—
116C2×C8—32no—
117C4⋊C4—32no—
118C4⋊C4—62no—
119C4⋊C4—61no—
120C4⋊C4—62no—
121C22⋊C4—62no—
122C22⋊C4—62no—
123C22×C3—42no—
124C22×C3—41no—
125C22×C3—81no—
126C22×C3—81no—
127C3×C4—42no—
128C3×C4—81no—
129Dic3—82no—
130Dic3—82no—
131C23—31no—
132C23—31no—
133C23—62no—
134C23—62no—
135C23—121no—
136C23C2×S411yescharacteristic
137Q8—31no—
138Q8—32no—
139Q8—61no—
140Q8—62no—
141Q8—62no—
142Q8—122no—
143Q8C2×#6711yescharacteristic
144D4—32no—
145D4—32no—
146D4—61no—
147D4—61no—
148D4—64no—
149C2×C4—22no—
150C2×C4—31no—
151C2×C4—32no—
152C2×C4—32no—
153C2×C4—31no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—62no—
157C2×C4—61no—
158C2×C4—61no—
159C2×C4—62no—
160C2×C4—62no—
161C2×C4—62no—
162C2×C4—122no—
163C2×C4—122no—
164C2×C4C2×S411yescharacteristic
165C2×C4C2×S411yescharacteristic
166C8—62no—
167C8—62no—
168C2×C3—41no—
169C2×C3—41no—
170C2×C3—41no—
171C2×C3—82no—
172C22—21no—
173C22—21no—
174C22—32no—
175C22—61no—
176C22—61no—
177C22—61no—
178C22—61no—
179C22—61no—
180C22—124no—
181C22#98412yescomplemented
182C22C22×S411yescharacteristic
183C4—21no—
184C4—31no—
185C4—31no—
186C4—61no—
187C4—122no—
188C4—122no—
189C4GL(2,ℤ/4)12yes—
190C3—41noSylow
191C2—22no—
192C2—62no—
193C2—121no—
194C2C2×#98411yescharacteristic
195C2#592311yescharacteristic
196C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC22×#41D3
prime radicalC22×#41⋊C3C2
hypercenterC22×C4S4
hypocenterSL(2,𝔽3)C2×D4
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. C2×#984C2semidirect
3. #5426C2semidirect
4. #5425C2semidirect
5. #5419C2semidirect
12. C2×#226C22semidirect
14. #859C22semidirect
181. C22#984semidirect

Maximal subgroups

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

subgroupindex
C22×#41⋊C32
C2×#9842
#54262
#54252
#54192
#30073
#9244