Finite Group Databasebeta

← #80993

Subgroups — #80993

Order 384.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#5727C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×#474C211yesmaximal, characteristic, complemented
4#5093C211yesmaximal, characteristic, complemented
5C4×#213C211yesmaximal, characteristic
6#4889C212yesmaximal, complemented
7#4875C212yesmaximal, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4787C212yesmaximal
11#4765C212yesmaximal, complemented
12#3367—31nomaximal, Sylow
13C2×D3×Q8—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#946—21no—
17#936C2212yes—
18C2×#213C2211yescharacteristic
19#926—21no—
20#924—21no—
21C2×#204C2212yes—
22#896—22no—
23#893C2212yes—
24#888C2212yes—
25#887C2212yes—
26#879—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#107C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34#816C2212yes—
35#815C2212yes—
36Dic3⋊C8C2212yes—
37#810C2212yes—
38#810C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43D4×Q8—31no—
44#489—32no—
45#487—31no—
46#477—32no—
47#474D311yescharacteristic, complemented
48#463—32no—
49#461—31no—
50#446—31no—
51#440—32no—
52C4×D8—31no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56D3×Q8—44no—
57C2×D12C2311yescharacteristic
58C2×C4×D3—21no—
59C2×C4×D3—22no—
60C2×Dic6—22no—
61C2×Dic6D411yescharacteristic
62C2×Dic6C2312yes—
63C2×C3×C8C2312yes—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2311yescharacteristic
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—21no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#217—22no—
71#213—22no—
72(C2×D3)⋊C4—21no—
73(C2×D3)⋊C4—22no—
74C4⋊Dic3—21no—
75C4⋊Dic3—21no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2311yescharacteristic
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85#204—24no—
86C22×Q8—61no—
87C2×SD16—32no—
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—31no—
104#107C2×D312yes—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—32no—
108#103—32no—
109#103—32no—
110#102—31no—
111#102—31no—
112#102C2×D312yes—
113#97—32no—
114C4×C8—31no—
115C23×C3—21no—
116C22×D3—21no—
117C3×Q8—42no—
118C3×D4—21no—
119C3×D4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—42no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3C2×D412yes—
129C2×Dic3C2×D412yes—
130D12—21no—
131D12#4112yes—
132C4×D3—44no—
133C4×D3—42no—
134Dic6—22no—
135Dic6—22no—
136Dic6—22no—
137Dic6—44no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8—21no—
141C2×Q8—21no—
142C2×Q8—31no—
143C2×Q8—32no—
144C2×Q8—62no—
145C2×Q8—124no—
146C2×D4—31no—
147C2×D4C22×D311yescharacteristic
148C22×C4—21no—
149C22×C4—62no—
150C22×C4—61no—
151SD16—64no—
152D8—62no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8C22×D312yes—
156C4⋊C4—21no—
157C4⋊C4—31no—
158C4⋊C4—32no—
159C4⋊C4—31no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4C22×D311yescharacteristic
165C4⋊C4C22×D312yes—
166C22⋊C4—21no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—61no—
171C42—32no—
172C42—62no—
173C42C22×D311yescharacteristic
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3C22×D411yescharacteristic
178C2×D3—22no—
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—42no—
194Q8—62no—
195Q8—62no—
196Q8—62no—
197Q8—124no—
198D4—21no—
199D4—21no—
200D4—32no—
201D4—61no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—42no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—124no—
210C2×C4—122no—
211C2×C4D3×D411yescharacteristic
212C2×C4D3×D411yescharacteristic
213C2×C4C23×D311yescharacteristic
214C8—22no—
215C8—61no—
216C8—61no—
217C2×C3—81no—
218C2×C3#54811yescharacteristic
219C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C3#57811yescharacteristic
221D3—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—62no—
226C22—121no—
227C22—121no—
228C22C2×D3×D411yescharacteristic
229C4—21no—
230C4—41no—
231C4—42no—
232C4—62no—
233C4—62no—
234C4—122no—
235C4—121no—
236C4#91112yes—
237C4#100711yescharacteristic
238C4D3×#4111yescharacteristic
239C3#336711yescharacteristic, complemented, Sylow
240C2—81no—
241C2—122no—
242C2#563311yescharacteristic
243C2C2×#91111yescharacteristic
244C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radicalC2×#213C22
hypercenter#474D3
hypocenterC3#3367
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5727C2semidirect
2. #5542C2semidirect
3. C3×#474C2semidirect
4. #5093C2semidirect
6. #4889C2semidirect
7. #4875C2semidirect
8. #4841C2semidirect
9. #4831C2semidirect
11. #4765C2semidirect
47. #474D3semidirect
239. C3#3367semidirect

Maximal subgroups

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

subgroupindex
#57272
#55422
C3×#4742
#50932
C4×#2132
#48892
#48752
#48412
#48312
#47872
#47652
#33673