Finite Group Databasebeta

← #80416

Subgroups — #80416

Order 384.

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

normal subgroups only

This group has 372 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#5732C211yesmaximal, characteristic, complemented
2D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5165C212yesmaximal
5#5164C212yesmaximal, complemented
6#5079C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic
8#4846C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4794C212yesmaximal, complemented
11#4793C212yesmaximal, complemented
12#3367—31nomaximal, Sylow
13C22×Dic6—21no—
14C2×C3×SD16C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#947C2212yes—
18Q8×Dic3C2212yes—
19D4×Dic3C2212yes—
20C2×#213C2211yescharacteristic
21#924—21no—
22#898—22no—
23#897—21no—
24#892—21no—
25#888C2212yes—
26#878—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#832C2211yescharacteristic
35#821C2212yes—
36#818C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43D4×Q8—31no—
44#489—32no—
45#487—31no—
46#477—32no—
47#474—31no—
48#463—32no—
49#461D311yescharacteristic, complemented
50#446—31no—
51#440—32no—
52C4×D8—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×D12C2311yescharacteristic
58C2×C4×D3—21no—
59C2×Dic6—21no—
60C2×Dic6—22no—
61C2×Dic6—44no—
62C2×Dic6D411yescharacteristic
63C3×SD16—24no—
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4—21no—
67C3×C4⋊C4C2311yescharacteristic
68C3×C22⋊C4—21no—
69C3×C42C2311yescharacteristic
70#217—22no—
71#213—22no—
72(C2×D3)⋊C4—21no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3—22no—
77C4⋊Dic3D411yescharacteristic
78C4⋊Dic3C2311yescharacteristic
79C4⋊Dic3C2312yes—
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—22no—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86C22×Q8—61no—
87C2×SD16C2×D312yes—
88C2×D8—31no—
89C4⋊Q8—32no—
90C4⋊Q8—31no—
91C4⋊Q8C2×D311yescharacteristic
92#126—62no—
93#124—32no—
94C22⋊Q8—62no—
95C22⋊Q8—61no—
96C22⋊Q8—61no—
97C4×Q8—31no—
98C4×Q8—32no—
99C4×D4—32no—
100C4×D4—31no—
101C4×D4C2×D311yescharacteristic
102#117—61no—
103#107—32no—
104#107—31no—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—32no—
108#103—32no—
109#103C2×D312yes—
110#102—31no—
111#102—32no—
112#102—31no—
113#97—32no—
114C4×C8—31no—
115C23×C3—21no—
116C22×D3—21no—
117C3×Q8—22no—
118C3×Q8—22no—
119C3×D4—21no—
120C3×D4#4112yes—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—42no—
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2411yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—44no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132D12—21no—
133D12—21no—
134C4×D3—42no—
135Dic6—22no—
136Dic6—44no—
137Dic6—42no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8—21no—
141C2×Q8—31no—
142C2×Q8—62no—
143C2×Q8—61no—
144C2×Q8—124no—
145C2×Q8C22×D312yes—
146C2×D4—31no—
147C2×D4C22×D311yescharacteristic
148C22×C4—21no—
149C22×C4—62no—
150C22×C4—61no—
151SD16—24no—
152D8—62no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8C22×D312yes—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—32no—
161C4⋊C4—31no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4C22×D311yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—61no—
171C42—32no—
172C42—62no—
173C42C22×D311yescharacteristic
174C22×C3—22no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3C22×D411yescharacteristic
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—41no—
181C3×C4—21no—
182C3×C4—42no—
183C3×C4—41no—
184C3×C4AGL(1,ℤ/8)12yes—
185C3×C4C2×#4111yescharacteristic
186C3×C42−1+411yescharacteristic
187Dic3—22no—
188Dic3—22no—
189Dic3—42no—
190Dic3—41no—
191C23—21no—
192C23—61no—
193Q8—22no—
194Q8—22no—
195Q8—62no—
196Q8—124no—
197Q8—122no—
198D4—21no—
199D4—61no—
200D4—61no—
201D4#23712yes—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—42no—
207C2×C4—61no—
208C2×C4—62no—
209C2×C4—124no—
210C2×C4—122no—
211C2×C4D3×D411yescharacteristic
212C2×C4D3×D411yescharacteristic
213C2×C4C23×D311yescharacteristic
214C8—22no—
215C8—61no—
216C8—61no—
217C2×C3—42no—
218C2×C3#54811yescharacteristic
219C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C3#57811yescharacteristic
221D3—81no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—121no—
226C22—121no—
227C22—121no—
228C22C2×D3×D411yescharacteristic
229C4—21no—
230C4—41no—
231C4—42no—
232C4—62no—
233C4—62no—
234C4—122no—
235C4—121no—
236C4#91412yes—
237C4C2×#23711yescharacteristic
238C4#101011yescharacteristic
239C3#336711yescharacteristic, complemented, Sylow
240C2—42no—
241C2—241no—
242C2#559811yescharacteristic
243C2C2×#91411yescharacteristic
244C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×#213C22
hypercenter#461D3
hypocenterC3#3367
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. D4×Dic6C2semidirect
3. C3×#461C2semidirect
5. #5164C2semidirect
6. #5079C2semidirect
8. #4846C2semidirect
9. #4831C2semidirect
10. #4794C2semidirect
11. #4793C2semidirect
49. #461D3semidirect
239. C3#3367semidirect

Maximal subgroups

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

subgroupindex
#57322
D4×Dic62
C3×#4612
#51652
#51642
#50792
C4×#2132
#48462
#48312
#47942
#47932
#33673