Finite Group Databasebeta

← #85535

Subgroups — #85535

Order 384.

298 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C2×2−1+4⋊C3C211yesmaximal, characteristic
2C22×#226C211yesmaximal, characteristic, complemented
3#5425C212yesmaximal, complemented
4#5420C211yesmaximal, characteristic, complemented
5#5419C212yesmaximal, complemented
6#3011—31nomaximal, Sylow
72−1+4⋊C3—22no—
8C2×#41⋊C3C2211yescharacteristic
9C22×SL(2,𝔽3)C2211yescharacteristic
10C22×SL(2,𝔽3)C2211yescharacteristic
11C2×#226—22no—
12C2×#226C2212yes—
13#935—41nomaximal
14#859C2212yes—
15C2×2−1+4D311yescharacteristic
16C2×#137—31no—
17C22×Q16—31no—
18#536—31no—
19#452—32no—
20#452—32no—
21#451—32no—
22#451—32no—
23#419—31no—
24#416—31no—
25#408—31no—
26C2×C3×D4—41no—
27C22×Dic3—41no—
28#41⋊C3—21no—
29C2×SL(2,𝔽3)—21no—
30C2×SL(2,𝔽3)D412yes—
31C2×SL(2,𝔽3)C2311yescharacteristic
32#226—22no—
33#217—42no—
34#217—41no—
35Dic3⋊C4—42no—
362−1+4—22no—
372−1+4—62no—
38C2×#41—31no—
39C2×#41—31no—
40C2×#41—31no—
41C2×#41C2×D311yescharacteristic
42C22×Q8—31no—
43C22×Q8—31no—
44C22×Q8C2×D311yescharacteristic
45C22×Q8C2×D311yescharacteristic
46#137—64no—
47C2×Q16—32no—
48C2×Q16—32no—
49C2×Q16—32no—
50C2×Q16—62no—
51C2×Q16—62no—
52C2×SD16—32no—
53C2×M16—31no—
54C22×C8—31no—
55C4⋊Q8—61no—
56#124—61no—
57#123—61no—
58#123—61no—
59C22⋊Q8—32no—
60C22⋊Q8—32no—
61#117—31no—
62#103—32no—
63#103—32no—
64#103—32no—
65#102—32no—
66C22⋊C8—32no—
67C22⋊C8—32no—
68C23×C3—41no—
69C23×C3—41no—
70C3×D4—82no—
71C2×C3×C4—41no—
72C2×Dic3—42no—
73C2×Dic3—42no—
74C2×Dic3—82no—
75SL(2,𝔽3)C2×D411yescharacteristic
76#41—21no—
77#41—34no—
78#41—62no—
79#41—62no—
80#41—62no—
81#41—62no—
82#41—62no—
83#41—61no—
84C2×Q8—21no—
85C2×Q8—31no—
86C2×Q8—31no—
87C2×Q8—31no—
88C2×Q8—32no—
89C2×Q8—32no—
90C2×Q8—61no—
91C2×Q8—62no—
92C2×Q8—62no—
93C2×Q8—62no—
94C2×Q8—62no—
95C2×Q8—61no—
96C2×Q8#6712yes—
97C2×Q8C22×D311yescharacteristic
98C2×D4—31no—
99C2×D4—31no—
100C2×D4—31no—
101C2×D4S411yescharacteristic
102C22×C4—31no—
103C22×C4—31no—
104C22×C4—31no—
105C22×C4—61no—
106C22×C4—61no—
107Q16—64no—
108Q16—62no—
109Q16—64no—
110Q16—62no—
111SD16—64no—
112M16—62no—
113C2×C8—32no—
114C2×C8—32no—
115C2×C8—62no—
116C4⋊C4—32no—
117C4⋊C4—62no—
118C4⋊C4—62no—
119C22⋊C4—62no—
120C22⋊C4—61no—
121C22⋊C4—62no—
122C42—61no—
123C22×C3—42no—
124C22×C3—41no—
125C22×C3—82no—
126C22×C3—81no—
127C22×C3—82no—
128C3×C4—81no—
129Dic3—82no—
130Dic3—82no—
131C23—31no—
132C23C2×S411yescharacteristic
133C23C2×S411yescharacteristic
134Q8—32no—
135Q8—31no—
136Q8—32no—
137Q8—62no—
138Q8—61no—
139Q8—61no—
140Q8—62no—
141Q8—62no—
142Q8—122no—
143Q8C2×#6711yescharacteristic
144D4—22no—
145D4—32no—
146D4—62no—
147D4—61no—
148D4—62no—
149C2×C4—32no—
150C2×C4—32no—
151C2×C4—31no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—64no—
159C2×C4—62no—
160C2×C4—62no—
161C2×C4—122no—
162C2×C4C2×S411yescharacteristic
163C8—62no—
164C8—62no—
165C2×C3—42no—
166C2×C3—41no—
167C2×C3—82no—
168C2×C3—161no—
169C22—22no—
170C22—22no—
171C22—21no—
172C22—62no—
173C22—61no—
174C22GL(2,ℤ/4)12yes—
175C22C22×S411yescharacteristic
176C4—21no—
177C4—31no—
178C4—31no—
179C4—32no—
180C4—61no—
181C4—122no—
182C4—122no—
183C3—41noSylow
184C2—22no—
185C2—41no—
186C2—121no—
187C2#592712yes—
188C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC2×2−1+4D3
prime radicalC2×2−1+4⋊C3C2
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#226C2semidirect
3. #5425C2semidirect
4. #5420C2semidirect
5. #5419C2semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
C22×#2262
#54252
#54202
#54192
#30113
#9354