Finite Group Databasebeta

← #80986

Subgroups — #80986

Order 384.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#5726C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×#474C211yesmaximal, characteristic, complemented
4#5078C211yesmaximal, characteristic, complemented
5C4×#213C211yesmaximal, characteristic, complemented
6#4882C212yesmaximal, complemented
7D3×#107C212yesmaximal, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4775C212yesmaximal, complemented
11#4774C212yesmaximal, complemented
12#3334—31nomaximal, Sylow
13C2×#239—21no—
14C2×#235—21no—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#946—21no—
18#938C2212yes—
19C2×#213C2211yescharacteristic
20#924—21no—
21C2×D3×C8C2212yes—
22#895C2212yes—
23#894—22no—
24D3×C4⋊C4C2212yes—
25D3×C4⋊C4C2212yes—
26#883—22no—
27C4×D12C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30C3×#107C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34#821C2212yes—
35(C2×D3)⋊C8C2212yes—
36#818C2212yes—
37#809C2211yescharacteristic
38#807C2211yescharacteristic
39#807C2212yes—
40#807C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#547—31no—
43#547—31no—
44#500—31no—
45#480—32no—
46#480—32no—
47#474—31no—
48#474D311yescharacteristic, complemented
49#466—32no—
50C4×D8—31no—
51C8×D4—31no—
52C2×#107—32no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57#239—44no—
58#235—44no—
59C2×D12—22no—
60C2×D12D411yescharacteristic
61C2×D12C2311yescharacteristic
62C2×C4×D3—22no—
63C2×C4×D3—22no—
64C2×C4×D3—21no—
65C2×C4×D3C2312yes—
66C2×Dic6—21no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4C2312yes—
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#217—22no—
74#213—22no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4D412yes—
78C4⋊Dic3—21no—
79C4⋊Dic3D411yescharacteristic
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84C2×C3⋊C8C2311yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86D3×C8—24no—
87C2×#41—61no—
88C2×#41—61no—
89C2×D8—31no—
90C22×C8—32no—
91C4⋊Q8—31no—
92C4⋊Q8C2×D311yescharacteristic
93#123—62no—
94#123—62no—
95C22⋊Q8—61no—
96C22⋊Q8—61no—
97#121—32no—
98#121—32no—
99C4×D4—31no—
100C4×D4—31no—
101C4×D4C2×D311yescharacteristic
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104#107—32no—
105#107—31no—
106#107—32no—
107#107—32no—
108#107C2×D312yes—
109C4⋊C8C2×D311yescharacteristic
110#102—31no—
111#102—32no—
112#102—31no—
113#102C2×D312yes—
114C22⋊C8—32no—
115C4×C8—31no—
116C23×C3—21no—
117C22×D3—21no—
118C22×D3C2×D412yes—
119C3×Q8—42no—
120C3×D4—21no—
121C3×D4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128#67—44no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132D12—21no—
133D12—21no—
134D12—44no—
135D12D814yescomplemented
136C4×D3—22no—
137C4×D3—22no—
138C4×D3—44no—
139C4×D3—44no—
140C4×D3—42no—
141Dic6—42no—
142C3×C8—22no—
143C3⋊C8—21no—
144C3⋊C8#4112yescomplemented
145#41—124no—
146#41—124no—
147C2×Q8—21no—
148C2×Q8—61no—
149C2×D4—31no—
150C2×D4—31no—
151C2×D4—62no—
152C2×D4—62no—
153C2×D4C22×D311yescharacteristic
154C22×C4—21no—
155C22×C4—32no—
156C22×C4—62no—
157C22×C4—62no—
158C22×C4—61no—
159D8—62no—
160C2×C8—31no—
161C2×C8—31no—
162C2×C8—64no—
163C2×C8C22×D312yes—
164C4⋊C4—21no—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—32no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4C22×D311yescharacteristic
172C4⋊C4C22×D312yes—
173C22⋊C4—21no—
174C22⋊C4—32no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—61no—
178C42C22×D311yescharacteristic
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3C22×D411yescharacteristic
183C2×D3—22no—
184C2×D3—22no—
185C2×D3—41no—
186C2×D3—41no—
187C2×D3—41no—
188C2×D3C2×D814yes—
189C3×C4—21no—
190C3×C4—41no—
191C3×C4—42no—
192C3×C4C2×D812yes—
193C3×C4C2×#4111yescharacteristic
194C3×C42−1+411yescharacteristic
195Dic3—22no—
196Dic3—42no—
197Dic3—41no—
198C23—21no—
199C23—32no—
200C23—61no—
201Q8—42no—
202Q8—122no—
203D4—21no—
204D4—21no—
205D4—34no—
206D4—61no—
207D4—61no—
208D4—124no—
209D4—124no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—32no—
213C2×C4—42no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—124no—
219C2×C4—124no—
220C2×C4—122no—
221C2×C4D3×D411yescharacteristic
222C2×C4D3×D411yescharacteristic
223C2×C4C23×D311yescharacteristic
224C8—22no—
225C8—32no—
226C8—61no—
227C2×C3—81no—
228C2×C3#54711yescharacteristic
229C2×C3C22×D811yescharacteristic
230C2×C3#57811yescharacteristic
231D3—24no—
232D3—81no—
233C22—34no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—62no—
238C22—62no—
239C22—121no—
240C22—121no—
241C22—121no—
242C22C2×D3×D411yescharacteristic
243C4—21no—
244C4—41no—
245C4—42no—
246C4—62no—
247C4—121no—
248C4—122no—
249C4D3×D812yes—
250C4#100711yescharacteristic
251C4D3×#4111yescharacteristic
252C3#333411yescharacteristic, complemented, Sylow
253C2—64no—
254C2—81no—
255C2—241no—
256C2#563111yescharacteristic
257C2C2×D3×D811yescharacteristic
258C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
hypercenter#474D3
hypocenterC3#3334
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. #5726C2semidirect
2. #5551C2semidirect
3. C3×#474C2semidirect
4. #5078C2semidirect
5. C4×#213C2semidirect
6. #4882C2semidirect
7. D3×#107C2semidirect
8. #4846C2semidirect
9. #4834C2semidirect
10. #4775C2semidirect
11. #4774C2semidirect
48. #474D3semidirect
135. D12D8semidirect
144. C3⋊C8#41semidirect
252. C3#3334semidirect

Maximal subgroups

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

subgroupindex
#57262
#55512
C3×#4742
#50782
C4×#2132
#48822
D3×#1072
#48462
#48342
#47752
#47742
#33343