Finite Group Databasebeta

← #80320

Subgroups — #80320

Order 384.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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#5718C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5152C212yesmaximal, complemented
5#5150C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4764C212yesmaximal, complemented
11#4763C212yesmaximal, complemented
12#3366—31nomaximal, Sylow
13C2×#237—21no—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#940C2212yes—
18#939—22no—
19#938—21no—
20D4×Dic3C2212yes—
21D4×Dic3C2212yes—
22C2×#213C2211yescharacteristic
23C2×C4⋊Dic3—21no—
24#924—21no—
25#890C2212yes—
26#878—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#102C2212yes—
32#832C2212yes—
33#832C2211yescharacteristic
34#832C2212yes—
35#821C2212yes—
36#817C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#554—31no—
43#550—31no—
44#490—32no—
45#486—31no—
46#478—32no—
47#476—31no—
48#459—32no—
49#459D311yescharacteristic, complemented
50#446—31no—
51#442—32no—
52C4×D8—31no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3—22no—
61C22×Dic3—22no—
62#237—44no—
63C2×D12C2311yescharacteristic
64C2×C4×D3—21no—
65C2×Dic6—21no—
66C2×Dic6D411yescharacteristic
67C3×D8—24no—
68C2×C3×C8C2312yes—
69C3×C4⋊C4C2311yescharacteristic
70C3×C22⋊C4—21no—
71C3×C42C2311yescharacteristic
72#217—22no—
73#217—22no—
74#213—22no—
75(C2×D3)⋊C4—21no—
76C4⋊Dic3—21no—
77C4⋊Dic3—22no—
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2312yes—
81Dic3⋊C4—22no—
82Dic3⋊C4D412yes—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86C2×#41—61no—
87C2×D8—31no—
88C2×D8C2×D312yes—
89C4⋊Q8—31no—
90#127—32no—
91#127C2×D311yescharacteristic
92#125—32no—
93C22⋊Q8—62no—
94C22⋊Q8—61no—
95#121—61no—
96#121—62no—
97C4×Q8—31no—
98C4×D4—32no—
99C4×D4—32no—
100C4×D4—31no—
101C4×D4C2×D311yescharacteristic
102C2×C4⋊C4—61no—
103#107—31no—
104#106—32no—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#102—31no—
108#102—32no—
109#102—32no—
110#102—32no—
111#102—31no—
112#102C2×D312yes—
113#97—32no—
114C4×C8—31no—
115C23×C3—21no—
116C23×C3—22no—
117C22×D3—21no—
118C3×D4—22no—
119C3×D4—21no—
120C3×D4—22no—
121C3×D4—42no—
122C3×D4—81no—
123C3×D4#4112yescomplemented
124C2×C3×C4—21no—
125C2×C3×C4—42no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2411yescharacteristic
129#67—44no—
130#67—82no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—44no—
134C2×Dic3—82no—
135C2×Dic3C2×D412yes—
136C2×Dic3C2×D412yes—
137D12—21no—
138D12—21no—
139C4×D3—42no—
140Dic6—22no—
141Dic6—81no—
142C3×C8—22no—
143C3⋊C8—21no—
144C3⋊C8—21no—
145#41—124no—
146C2×Q8—31no—
147C2×Q8—61no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—31no—
151C2×D4—62no—
152C2×D4—62no—
153C2×D4C22×D312yes—
154C2×D4C22×D311yescharacteristic
155C22×C4—21no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4—61no—
159D8—24no—
160D8—62no—
161C2×C8—31no—
162C2×C8—31no—
163C2×C8C22×D312yes—
164C4⋊C4—31no—
165C4⋊C4—32no—
166C4⋊C4—32no—
167C4⋊C4—31no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4C22×D311yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4—61no—
176C42—32no—
177C42C22×D311yescharacteristic
178C22×C3—22no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—42no—
182C22×C3—42no—
183C22×C3—42no—
184C22×C3C22×D411yescharacteristic
185C2×D3—41no—
186C2×D3—41no—
187C2×D3—41no—
188C3×C4—21no—
189C3×C4—41no—
190C3×C4AGL(1,ℤ/8)12yes—
191C3×C4C2×#4111yescharacteristic
192C3×C42−1+411yescharacteristic
193Dic3—22no—
194Dic3—22no—
195Dic3—41no—
196Dic3—42no—
197C23—21no—
198C23—22no—
199C23—61no—
200Q8—62no—
201Q8—241no—
202D4—22no—
203D4—21no—
204D4—22no—
205D4—42no—
206D4—61no—
207D4—61no—
208D4—81no—
209D4—124no—
210D4—242no—
211D4#23712yescomplemented
212C2×C4—21no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—42no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—124no—
219C2×C4—122no—
220C2×C4—242no—
221C2×C4D3×D411yescharacteristic
222C2×C4D3×D411yescharacteristic
223C2×C4C23×D311yescharacteristic
224C8—22no—
225C8—61no—
226C8—61no—
227C2×C3—42no—
228C2×C3—82no—
229C2×C3#54811yescharacteristic
230C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C3#57611yescharacteristic
232D3—81no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—121no—
240C22—121no—
241C22—121no—
242C22C2×D3×D411yescharacteristic
243C4—21no—
244C4—41no—
245C4—62no—
246C4—62no—
247C4—121no—
248C4—122no—
249C4#91112yes—
250C4C2×#23711yescharacteristic
251C4#101011yescharacteristic
252C3#336611yescharacteristic, complemented, Sylow
253C2—42no—
254C2—82no—
255C2—241no—
256C2#559811yescharacteristic
257C2C2×#91111yescharacteristic
258C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3366
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5543C2semidirect
3. C3×#459C2semidirect
4. #5152C2semidirect
5. #5150C2semidirect
6. #5070C2semidirect
7. C4×#213C2semidirect
8. #4842C2semidirect
9. #4831C2semidirect
10. #4764C2semidirect
11. #4763C2semidirect
49. #459D3semidirect
123. C3×D4#41semidirect
211. D4#237semidirect
252. C3#3366semidirect

Maximal subgroups

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

subgroupindex
#57182
#55432
C3×#4592
#51522
#51502
#50702
C4×#2132
#48422
#48312
#47642
#47632
#33663