Finite Group Databasebeta

← #87520

Subgroups — #87520

Order 384.

267 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 16 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#5950C211yesmaximal, characteristic, complemented
2#5935C211yesmaximal, characteristic, complemented
3#5933C211yesmaximal, characteristic, complemented
4#3018—31nomaximal, Sylow
5#997C2211yescharacteristic
6#985—42nomaximal
7C4×S4—41nomaximal
8#585D311yescharacteristic, complemented
9#575—32no—
10#532—31no—
11#456—32no—
12#455—31no—
13#455—31no—
14#454—32no—
15#453—31no—
16#453—31no—
17C2×#104—32no—
18#413—31no—
19C2×S4—41no—
20#41⋊C3—42no—
21C4×A4—41no—
22C4×A4—81no—
23SL(2,ℤ/4)—41no—
24GL(2,𝔽3)—42no—
25#226—42no—
262−1+4—32no—
27H(5,𝔽2)—31no—
28H(5,𝔽2)—61no—
29H(5,𝔽2)C2×D311yescharacteristic
30C2×#41—31no—
31C2×#41—32no—
32C2×#41—31no—
33C2×#41—61no—
34#137—32no—
35#137—32no—
36#137—62no—
37AGL(1,ℤ/8)—32no—
38AGL(1,ℤ/8)—32no—
39AGL(1,ℤ/8)—62no—
40#135—62no—
41#135—62no—
42C2×M16—32no—
43C4⋊Q8—31no—
44#127—31no—
45#124—31no—
46#124—31no—
47#121—61no—
48#121—61no—
49C4×D4—61no—
50C4×D4—61no—
51C2×C42—31no—
52#104—32no—
53#104—32no—
54#104—32no—
55#104—32no—
56#101—32no—
57#100—31no—
58#100—31no—
59C2×A4—41no—
60C2×A4—81no—
61S4—42no—
62C4×D3—161no—
63SL(2,𝔽3)—42no—
64#41—32no—
65#41—32no—
66#41—32no—
67#41—62no—
68#41—62no—
69#41—62no—
70#41—62no—
71#41—62no—
72#41—62no—
73#41—62no—
74#41S412yescomplemented
75C2×Q8—31no—
76C2×Q8—32no—
77C2×Q8—31no—
78C2×Q8—61no—
79C2×D4—32no—
80C2×D4—31no—
81C2×D4—31no—
82C2×D4—31no—
83C2×D4—31no—
84C2×D4—61no—
85C2×D4—62no—
86C2×D4—61no—
87C2×D4—61no—
88C2×D4—61no—
89C2×D4—61no—
90C2×D4—61no—
91C22×C4—21no—
92C22×C4—31no—
93C22×C4—31no—
94C22×C4—61no—
95C22×C4—61no—
96C22×C4—61no—
97C22×C4S411yescharacteristic
98Q16—62no—
99Q16—62no—
100SD16—62no—
101SD16—62no—
102SD16—62no—
103SD16—62no—
104D8—62no—
105D8—62no—
106M16—32no—
107M16—32no—
108M16—62no—
109C2×C8—62no—
110C4⋊C4—61no—
111C4⋊C4—61no—
112C22⋊C4—61no—
113C22⋊C4—61no—
114C22⋊C4—61no—
115C22⋊C4—61no—
116C42—31no—
117C42—31no—
118C42—61no—
119C2×D3—161no—
120A4—41no—
121A4—81no—
122C3×C4—161no—
123Dic3—161no—
124C23—21no—
125C23—31no—
126C23—31no—
127C23—61no—
128C23—61no—
129C23C2×S411yescharacteristic
130Q8—32no—
131Q8—62no—
132Q8—62no—
133Q8C2×S412yes—
134D4—32no—
135D4—32no—
136D4—62no—
137D4—62no—
138D4—62no—
139D4—62no—
140D4—62no—
141D4—122no—
142D4—122no—
143D4—122no—
144C2×C4—31no—
145C2×C4—32no—
146C2×C4—31no—
147C2×C4—32no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—61no—
151C2×C4—62no—
152C2×C4—61no—
153C2×C4—61no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—61no—
157C2×C4—61no—
158C2×C4—121no—
159C2×C4—122no—
160C2×C4—121no—
161C8—62no—
162C8—62no—
163C2×C3—161no—
164D3—162no—
165C22—31no—
166C22—32no—
167C22—41no—
168C22—61no—
169C22—81no—
170C22—121no—
171C22—121no—
172C22—121no—
173C22—121no—
174C22—121no—
175C22—241no—
176C4—32no—
177C4—31no—
178C4—62no—
179C4—61no—
180C4—121no—
181C4—121no—
182C4#102011yescharacteristic
183C3—161noSylow
184C2—61no—
185C2—62no—
186C2—121no—
187C2—241no—
188C2C2×#102011yescharacteristic
subgroupG/H
center, hypercenterC4#1020
commutator, cosocle, hypocenter#997C22
Frattini, socleC2C2×#1020
Fitting#585D3
LCSG ▷ #997
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #5950C2semidirect
2. #5935C2semidirect
3. #5933C2semidirect
8. #585D3semidirect
74. #41S4semidirect

Maximal subgroups

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

subgroupindex
#59502
#59352
#59332
#30183
#9854
C4×S44