Finite Group Databasebeta

← #83706

Subgroups — #83706

Order 384.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×Q82C211yesmaximal, characteristic, complemented
2#5579C211yesmaximal, characteristic
3#5087C212yesmaximal
4#5084C211yesmaximal, characteristic
5#5081C212yesmaximal, complemented
6#5078C211yesmaximal, characteristic, complemented
7#5026C212yesmaximal
8C4×#215C211yesmaximal, characteristic
9#5024C211yesmaximal, characteristic, complemented
10#5022C212yesmaximal, complemented
11#5021C211yesmaximal, characteristic, complemented
12#3385—31nomaximal, Sylow
13C3×C4⋊Q8—21no—
14C3×C4⋊Q8—22no—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4⋊Q8C2212yes—
17C3×C4×Q8—21no—
18C3×C4×Q8C2211yescharacteristic
19C3×C4×Q8C2212yes—
20C3×C4×Q8C2211yescharacteristic
21#946—21no—
22C2×#215C2211yescharacteristic
23#896—22no—
24D3×C4⋊C4—21no—
25C4×D12C2211yescharacteristic
26C4×D12C2212yes—
27#870C2212yes—
28#869C2211yescharacteristic
29#835C2212yes—
30#835C2211yescharacteristic
31#809C2211yescharacteristic
32#809C2212yes—
33#808C2212yes—
34#808C2211yescharacteristic
35#807C2212yes—
36#807C2212yes—
37#807C2212yes—
38C4⋊(C3⋊C8)C2211yescharacteristic
39C4⋊(C3⋊C8)C2212yes—
40#803C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42Q82D311yescharacteristic, complemented
43#554—31no—
44#501—32no—
45#500—31no—
46#479—32no—
47#477—31no—
48#476—31no—
49#474—32no—
50#446—31no—
51#440—32no—
52C4×SD16—31no—
53C2×C3×Q8—22no—
54C2×C3×Q8—21no—
55C2×C3×Q8D411yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C2×D12C2311yescharacteristic
58C2×C4×D3—21no—
59C2×C4×D3—22no—
60C2×Dic6—21no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4D411yescharacteristic
66C3×C4⋊C4D412yes—
67C3×C4⋊C4C2311yescharacteristic
68C3×C4⋊C4C2312yes—
69C3×C42—22no—
70C3×C42—21no—
71C3×C42C2311yescharacteristic
72C3×C42C2312yes—
73#215Q814yescomplemented
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—21no—
77C4⋊Dic3C2312yes—
78C4⋊Dic3C2311yescharacteristic
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C2×C3⋊C8C2311yescharacteristic
82C2×C3⋊C8C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×SD16—31no—
85C4⋊Q8—21no—
86C4⋊Q8—22no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D312yes—
89C4⋊Q8C2×D311yescharacteristic
90#125—32no—
91C22⋊Q8—61no—
92C22⋊Q8—62no—
93C4×Q8—21no—
94C4×Q8C2×D311yescharacteristic
95C4×Q8C2×D312yes—
96C4×Q8C2×D311yescharacteristic
97C4×D4—31no—
98C4×D4—32no—
99C2×C4⋊C4—61no—
100#107—32no—
101#107—32no—
102#107—32no—
103#106—31no—
104#106—32no—
105C4⋊C8—32no—
106C4⋊C8—31no—
107#103—31no—
108#103—32no—
109#102—31no—
110#102—32no—
111#97—32no—
112C4×C8—31no—
113C22×D3—21no—
114C3×Q8—22no—
115C3×Q8—21no—
116C3×Q8—81no—
117C3×Q8—82no—
118C3×Q8C2×Q812yes—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4C2×D412yes—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—22no—
129D12—21no—
130D12C2×Q812yes—
131C4×D3—42no—
132C4×D3—82no—
133Dic6—81no—
134C3⋊C8—21no—
135C3⋊C8—22no—
136C3⋊C8C2×Q812yes—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—61no—
140C2×Q8#6711yescharacteristic
141C2×Q8C22×D311yescharacteristic
142C2×D4—31no—
143C22×C4—61no—
144C22×C4—62no—
145SD16—34no—
146C2×C8—31no—
147C2×C8—31no—
148C2×C8—32no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—31no—
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4—62no—
158C4⋊C4#6711yescharacteristic
159C4⋊C4#6712yes—
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D312yes—
162C22⋊C4—61no—
163C22⋊C4—62no—
164C42—22no—
165C42—21no—
166C42C22×D311yescharacteristic
167C42C22×D312yes—
168C22×C3C22×D411yescharacteristic
169C2×D3—22no—
170C2×D3—41no—
171C2×D3—41no—
172C3×C4—22no—
173C3×C4—22no—
174C3×C4—21no—
175C3×C4—22no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4#13712yes—
179C3×C4C22×Q811yescharacteristic
180C3×C42−1+411yescharacteristic
181Dic3—41no—
182Dic3—42no—
183C23—61no—
184Q8—22no—
185Q8—21no—
186Q8—82no—
187Q8—81no—
188Q8—241no—
189Q8D3×Q812yes—
190D4—32no—
191D4—61no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—61no—
196C2×C4—62no—
197C2×C4—122no—
198C2×C4—242no—
199C2×C4C2×#6712yes—
200C2×C4C2×#6711yescharacteristic
201C2×C4C2×#6712yes—
202C2×C4C2×#6711yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—32no—
205C8—61no—
206C8—62no—
207C2×C3D4×Q811yescharacteristic
208C2×C3C2×#13711yescharacteristic
209C2×C3#57611yescharacteristic
210D3—42no—
211C22—62no—
212C22—121no—
213C22—121no—
214C22C22×#6711yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—121no—
222C4—122no—
223C4#94212yes—
224C4C2×D3×Q811yescharacteristic
225C4#100711yescharacteristic
226C3#338511yescharacteristic, complemented, Sylow
227C2—122no—
228C2C2×#94211yescharacteristic
229C2Q8×#6711yescharacteristic
230C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×Q82C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenterQ82D3
hypocenterC3#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
1. C3×Q82C2semidirect
5. #5081C2semidirect
6. #5078C2semidirect
9. #5024C2semidirect
10. #5022C2semidirect
11. #5021C2semidirect
42. Q82D3semidirect
73. #215Q8semidirect
226. C3#3385semidirect

Maximal subgroups

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

subgroupindex
C3×Q822
#55792
#50872
#50842
#50812
#50782
#50262
C4×#2152
#50242
#50222
#50212
#33853