Finite Group Databasebeta

← #71522

Subgroups — #71522

Order 384.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#5768C211yesmaximal, characteristic
2C3×#510C211yesmaximal, characteristic, complemented
3#5189C211yesmaximal, characteristic, complemented
4#5179C211yesmaximal, characteristic, complemented
5#4566C211yesmaximal, characteristic, complemented
6#4518C211yesmaximal, characteristic, complemented
7#4514C211yesmaximal, characteristic, complemented
8#2191—31nomaximal, Sylow
9D3×#41—21no—
10C3×#135C2211yescharacteristic
11C2×C3×Q16C2211yescharacteristic, complemented
12#950—21no—
13#946—21no—
14#918—21no—
15#915—21no—
16#914—21no—
17#911—21no—
18#904C2211yescharacteristic
19C2×ΓL(1,𝔽25)C2211yescharacteristic
20C3×Q32—21no—
21C3×SD32—21no—
22C3×M32C2211yescharacteristic, complemented
23#835—21no—
24#829—21no—
25#828—21no—
26#817C2211yescharacteristic, complemented
27#812C2211yescharacteristic, complemented
28#575—31no—
29#510—31no—
30#510D311yescharacteristic, complemented
31#470—31no—
32#360—31no—
33#360—31no—
34#349—31no—
35C3×#41—21no—
36C2×C3×Q8—21no—
37#239—21no—
38D3×Q8—21no—
39#237—21no—
40#237—41no—
41D3×D4—21no—
42D3×D4—41no—
43#235—21no—
44#235—41no—
45C2×C4×D3—41no—
46C2×C4×D3D411yescharacteristic
47C3×Q16—21no—
48C3×Q16—21no—
49C3×Q16D411yescharacteristic
50C3×SD16—21no—
51C3×D8D411yescharacteristic
52C2×C3×C8C2311yescharacteristic
53#216—21no—
54#215—21no—
55#214—21no—
56#213—21no—
57(C2×D3)⋊C4—41no—
58C4⋊Dic3—21no—
59Dic3⋊C4—41no—
60C2×C3⋊C8D411yescharacteristic
61Dic12D411yescharacteristic
62D24D411yescharacteristic
63#204—21no—
64ΓL(1,𝔽25)—21no—
65ΓL(1,𝔽25)—21no—
66C3×C16—21no—
67C3⋊C16—21no—
68C2×#41—61no—
69#137—61no—
70#137—61no—
71AGL(1,ℤ/8)—61no—
72AGL(1,ℤ/8)—61no—
73#135—31no—
74#135—61no—
75#135C2×D311yescharacteristic
76C2×Q16C2×D311yescharacteristic, complemented
77C2×M16—31no—
78C22⋊Q8—61no—
79Q32—21no—
80Q32—61no—
81SD32—21no—
82SD32—61no—
83M32—31no—
84M32C2×D311yescharacteristic, complemented
85#106—31no—
86#103—61no—
87C22×D3—41no—
88C22×D3D811yescharacteristic
89C3×Q8—21no—
90C3×Q8—41no—
91C3×Q8—41no—
92C3×D4—21no—
93C3×D4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4C2×D411yescharacteristic
97#67—41no—
98#67—41no—
99#67—41no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3D811yescharacteristic
103D12—21no—
104D12—41no—
105C4×D3—21no—
106C4×D3—21no—
107C4×D3—41no—
108C4×D3—41no—
109C4×D3—41no—
110Dic6—21no—
111Dic6—41no—
112C3×C8C2×D411yescharacteristic
113C3×C8C2×D411yescharacteristic
114C3⋊C8—21no—
115#41—21no—
116#41—61no—
117#41—61no—
118#41—61no—
119#41—121no—
120#41—121no—
121C2×Q8—21no—
122C2×Q8—61no—
123C2×D4—61no—
124C2×D4—121no—
125C22×C4—31no—
126C22×C4—121no—
127Q16—21no—
128Q16—21no—
129Q16—31no—
130Q16—61no—
131Q16#6711yescharacteristic
132SD16—21no—
133SD16—61no—
134SD16—61no—
135SD16—61no—
136D8—31no—
137D8—61no—
138D8#6711yescharacteristic
139M16—61no—
140M16—61no—
141C2×C8—31no—
142C2×C8C22×D311yescharacteristic
143C4⋊C4—61no—
144C4⋊C4—121no—
145C22⋊C4—121no—
146C16—21no—
147C16—61no—
148C22×C3—41no—
149C22×C3C2×D811yescharacteristic
150C2×D3—21no—
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—82no—
154C3×C4—41no—
155C3×C4—81no—
156C3×C4#12011yescharacteristic
157C3×C4AGL(1,ℤ/8)11yescharacteristic
158Dic3—21no—
159Dic3—41no—
160Dic3—81no—
161C23—31no—
162C23—121no—
163Q8—21no—
164Q8—41no—
165Q8—41no—
166Q8—61no—
167Q8—121no—
168D4—21no—
169D4—41no—
170D4—61no—
171D4—121no—
172D4—121no—
173D4—121no—
174D4—121no—
175C2×C4—31no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—61no—
179C2×C4—61no—
180C2×C4—121no—
181C2×C4—121no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—121no—
185C2×C4D3×D411yescharacteristic
186C8—61no—
187C8D3×D411yescharacteristic
188C8C2×#6711yescharacteristic
189C2×C3—21no—
190C2×C3—81no—
191C2×C3#44711yescharacteristic
192D3—41no—
193D3—81no—
194C22—41no—
195C22—61no—
196C22—121no—
197C22—121no—
198C22—242no—
199C22D3×D811yescharacteristic
200C4—41no—
201C4—61no—
202C4—81no—
203C4—121no—
204C4—241no—
205C4#91111yescharacteristic
206C4#93711yescharacteristic
207C3#219111yescharacteristic, complemented, Sylow
208C2—21no—
209C2—81no—
210C2—121no—
211C2—241no—
212C2#515611yescharacteristic
subgroupG/H
centerC2#5156
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#510C2
socleC2×C3#447
prime radical#5768C2
hypercenter#510D3
hypocenterC3#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C3×#510C2semidirect
3. #5189C2semidirect
4. #5179C2semidirect
5. #4566C2semidirect
6. #4518C2semidirect
7. #4514C2semidirect
11. C2×C3×Q16C22semidirect
22. C3×M32C22semidirect
26. #817C22semidirect
27. #812C22semidirect
30. #510D3semidirect
76. C2×Q16C2×D3semidirect
84. M32C2×D3semidirect
207. C3#2191semidirect

Maximal subgroups

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

subgroupindex
#57682
C3×#5102
#51892
#51792
#45662
#45182
#45142
#21913