Finite Group Databasebeta

← #69252

Subgroups — #69252

Order 384.

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

normal subgroups only

This group has 226 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, complemented
2#5193C211yesmaximal, characteristic, complemented
3#4908C211yesmaximal, characteristic, complemented
4#4865C211yesmaximal, characteristic, complemented
5C3×#360C211yesmaximal, characteristic, complemented
6#4514C211yesmaximal, characteristic, complemented
7#4488C211yesmaximal, characteristic, complemented
8#2190—31nomaximal, Sylow
9D3×#41—21no—
10C3×#135C2211yescharacteristic
11#950—21no—
12#918—21no—
13#915—21no—
14#914—21no—
15#911—21no—
16#904C2211yescharacteristic
17C2×D24C2211yescharacteristic, complemented
18C2×ΓL(1,𝔽25)C2211yescharacteristic
19#895—21no—
20C3×M32C2211yescharacteristic, complemented
21C3×#106C2211yescharacteristic, complemented
22#828—21no—
23#826—21no—
24#812C2211yescharacteristic, complemented
25#809—21no—
26#800—21no—
27D48—21no—
28#575—31no—
29#509—31no—
30#509—31no—
31#469—31no—
32#360—31no—
33#360D311yescharacteristic, complemented
34#349—31no—
35C3×#41—21no—
36#239—21no—
37D3×Q8—21no—
38#237—21no—
39#237—41no—
40D3×D4—21no—
41D3×D4—41no—
42#235—21no—
43#235—41no—
44C2×D12—21no—
45C2×D12—41no—
46C2×C4×D3—41no—
47C2×C4×D3D411yescharacteristic
48C3×Q16D411yescharacteristic, complemented
49C3×SD16—21no—
50C3×D8D411yescharacteristic, complemented
51C2×C3×C8C2311yescharacteristic
52C3×C4⋊C4—21no—
53#216—21no—
54#215—21no—
55#214—21no—
56#213—21no—
57(C2×D3)⋊C4—41no—
58C2×C3⋊C8D411yescharacteristic
59Dic12D411yescharacteristic, complemented
60D24—21no—
61D24—21no—
62D24D411yescharacteristic, complemented
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×D8—31no—
77C2×M16—31no—
78#121—61no—
79SD32—61no—
80SD32—61no—
81D16—61no—
82D16—61no—
83M32—31no—
84M32C2×D311yescharacteristic, complemented
85#106C2×D311yescharacteristic, complemented
86#102—61no—
87C22×D3—41no—
88C22×D3—41no—
89C22×D3D811yescharacteristic
90C3×Q8—21no—
91C3×D4—21no—
92C3×D4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4C2×D411yescharacteristic
96#67—41no—
97#67—41no—
98#67—41no—
99C2×Dic3—41no—
100C2×Dic3D811yescharacteristic
101D12—21no—
102D12—41no—
103D12—41no—
104D12—41no—
105D12—82no—
106C4×D3—21no—
107C4×D3—21no—
108C4×D3—41no—
109C4×D3—41no—
110C4×D3—41no—
111Dic6—21no—
112Dic6—41no—
113C3×C8C2×D411yescharacteristic
114C3×C8C2×D411yescharacteristic
115C3⋊C8—21no—
116#41—21no—
117#41—61no—
118#41—61no—
119#41—61no—
120#41—121no—
121#41—121no—
122C2×Q8—61no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—121no—
126C2×D4—121no—
127C22×C4—31no—
128C22×C4—121no—
129Q16—31no—
130Q16—61no—
131Q16D1211yescharacteristic, complemented
132SD16—21no—
133SD16—61no—
134SD16—61no—
135SD16—61no—
136D8—31no—
137D8—61no—
138D8—61no—
139D8—61no—
140D8D1211yescharacteristic, complemented
141M16—61no—
142M16—61no—
143C2×C8—31no—
144C2×C8C22×D311yescharacteristic
145C4⋊C4—21no—
146C22⋊C4—121no—
147C16—21no—
148C16—61no—
149C22×C3—41no—
150C22×C3C2×D811yescharacteristic
151C2×D3—21no—
152C2×D3—41no—
153C2×D3—41no—
154C2×D3—82no—
155C2×D3—81no—
156C2×D3—81no—
157C2×D3—81no—
158C3×C4—41no—
159C3×C4—81no—
160C3×C4#12011yescharacteristic
161C3×C4AGL(1,ℤ/8)11yescharacteristic
162Dic3—21no—
163Dic3—41no—
164C23—31no—
165C23—121no—
166C23—121no—
167Q8—21no—
168Q8—61no—
169Q8—121no—
170D4—21no—
171D4—41no—
172D4—61no—
173D4—121no—
174D4—121no—
175D4—121no—
176D4—121no—
177D4—121no—
178D4—121no—
179D4—242no—
180C2×C4—31no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4D3×D411yescharacteristic
190C8—61no—
191C8C2×D1211yescharacteristic
192C8D3×D411yescharacteristic
193C2×C3—21no—
194C2×C3—81no—
195C2×C3#44711yescharacteristic
196D3—41no—
197D3—81no—
198D3—161no—
199C22—41no—
200C22—61no—
201C22—121no—
202C22—121no—
203C22—241no—
204C22—241no—
205C22—242no—
206C22—241no—
207C22D3×D811yescharacteristic
208C4—41no—
209C4—61no—
210C4—81no—
211C4—121no—
212C4#88211yescharacteristic
213C4#91411yescharacteristic
214C3#219011yescharacteristic, complemented, Sylow
215C2—21no—
216C2—81no—
217C2—121no—
218C2—241no—
219C2—481no—
220C2#477311yescharacteristic
subgroupG/H
centerC2#4773
commutator, cosocleC2×C3×C8C23
FrattiniC2×C8C22×D3
FittingC3×#360C2
socleC2×C3#447
hypercenter#360D3
hypocenterC3#2190
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #5768C2semidirect
2. #5193C2semidirect
3. #4908C2semidirect
4. #4865C2semidirect
5. C3×#360C2semidirect
6. #4514C2semidirect
7. #4488C2semidirect
17. C2×D24C22semidirect
20. C3×M32C22semidirect
21. C3×#106C22semidirect
24. #812C22semidirect
33. #360D3semidirect
48. C3×Q16D4semidirect
50. C3×D8D4semidirect
59. Dic12D4semidirect
62. D24D4semidirect
84. M32C2×D3semidirect
85. #106C2×D3semidirect
131. Q16D12semidirect
140. D8D12semidirect
214. C3#2190semidirect

Maximal subgroups

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

subgroupindex
#57682
#51932
#49082
#48652
C3×#3602
#45142
#44882
#21903