Finite Group Databasebeta

← #79613

Subgroups — #79613

Order 384.

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

normal subgroups only

This group has 420 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#5555C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×C4×D8C211yesmaximal, characteristic, complemented
4#5158C212yesmaximal, complemented
5#5019C211yesmaximal, characteristic, complemented
6#5007C211yesmaximal, characteristic, complemented
7#4884C212yesmaximal, complemented
8#4780C212yesmaximal, complemented
9#4771C212yesmaximal, complemented
10#4688C211yesmaximal, characteristic, complemented
11#4681C211yesmaximal, characteristic, complemented
12#3330—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×#235—21no—
15C2×C3×D8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#938C2212yes—
19C2×#214C2212yes—
20#930—21no—
21C2×C4⋊Dic3—21no—
22#924—21no—
23C2×ΓL(1,𝔽25)C2212yes—
24#897C2212yes—
25D3×C4⋊C4C2212yes—
26#886—22no—
27#883—22no—
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30#869C2211yescharacteristic
31C3×#107C2211yescharacteristic
32C3×#102C2211yescharacteristic
33C3×#102C2211yescharacteristic
34C3×C4×C8C2211yescharacteristic
35#832C2212yes—
36(C2×D3)⋊C8C2212yes—
37#816C2212yes—
38#816C2212yes—
39#810C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#547—31no—
43#547—31no—
44#487—31no—
45#484—32no—
46#474—31no—
47#468—32no—
48#461—31no—
49#449—32no—
50C4×D8D311yescharacteristic, complemented
51#436—31no—
52#417—32no—
53C2×C3×D4C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C2×#67—22no—
58C2×#67—22no—
59C22×Dic3—22no—
60#237—24no—
61#237—42no—
62#235—44no—
63C2×D12D411yescharacteristic
64C2×C4×D3—22no—
65C2×C4×D3C2312yes—
66C2×Dic6—21no—
67C2×Dic6C2312yes—
68C3×D8—22no—
69C2×C3×C8C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C3×C4⋊C4C2311yescharacteristic
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C22⋊C4—21no—
75C3×C42C2311yescharacteristic
76#217—22no—
77#214—42no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—22no—
84C4⋊Dic3D411yescharacteristic
85C4⋊Dic3C2312yes—
86Dic3⋊C4—22no—
87C2×C3⋊C8C2312yes—
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—32no—
92C2×D8C2×D311yescharacteristic
93C2×M16—32no—
94C4⋊Q8—31no—
95C4⋊Q8—31no—
96#123—62no—
97#123—62no—
98C22⋊Q8—32no—
99C22⋊Q8—61no—
100#121—32no—
101#121—61no—
102C4×D4—31no—
103C4×D4C2×D311yescharacteristic
104C4×D4C2×D311yescharacteristic
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107#107—32no—
108#107C2×D311yescharacteristic
109C4⋊C8—31no—
110#103—32no—
111#103—32no—
112#103—32no—
113#102—32no—
114#102C2×D311yescharacteristic
115#102C2×D311yescharacteristic
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic
118C23×C3—21no—
119C23×C3—21no—
120C22×D3C2×D412yes—
121C3×D4—21no—
122C3×D4—21no—
123C3×D4—21no—
124C3×D4#4112yescomplemented
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—44no—
133#67—44no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—44no—
137C2×Dic3C2×D412yes—
138D12—22no—
139C4×D3—22no—
140C4×D3—22no—
141C4×D3—44no—
142Dic6—22no—
143Dic6—22no—
144Dic6—42no—
145C3×C8—21no—
146C3×C8—21no—
147C3⋊C8—22no—
148#41—64no—
149#41—124no—
150#41—122no—
151C2×Q8—32no—
152C2×Q8—61no—
153C2×D4—31no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4C22×D311yescharacteristic
157C2×D4C22×D311yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—32no—
161C22×C4—62no—
162C22×C4—62no—
163SD16—122no—
164D8—22no—
165M16—122no—
166C2×C8—32no—
167C2×C8C22×D311yescharacteristic
168C2×C8C22×D311yescharacteristic
169C4⋊C4—31no—
170C4⋊C4—32no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4C22×D311yescharacteristic
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—32no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C42C22×D311yescharacteristic
184C22×C3—22no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3C22×D411yescharacteristic
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—22no—
194C3×C4—21no—
195C3×C4—41no—
196C3×C4—41no—
197C3×C4AGL(1,ℤ/8)12yes—
198C3×C4C2×#4111yescharacteristic
199C3×C4H(5,𝔽2)11yescharacteristic
200Dic3—22no—
201Dic3—42no—
202Dic3—42no—
203C23—21no—
204C23—21no—
205C23—32no—
206Q8—62no—
207Q8—62no—
208Q8—122no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—62no—
213D4—124no—
214D4—124no—
215D4#23512yescomplemented
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—32no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—124no—
226C2×C4—124no—
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—21no—
231C8—21no—
232C8—62no—
233C2×C3—42no—
234C2×C3—81no—
235C2×C3#54611yescharacteristic
236C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
237C2×C3#57811yescharacteristic
238D3—42no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—62no—
246C22—62no—
247C22—62no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—62no—
253C4—122no—
254C4—122no—
255C4#91112yes—
256C4C2×#23511yescharacteristic
257C4#100911yescharacteristic
258C3#333011yescharacteristic, complemented, Sylow
259C2—42no—
260C2—81no—
261C2—122no—
262C2#555011yescharacteristic
263C2C2×#91111yescharacteristic
264C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3330
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5555C2semidirect
2. #5551C2semidirect
3. C3×C4×D8C2semidirect
4. #5158C2semidirect
5. #5019C2semidirect
6. #5007C2semidirect
7. #4884C2semidirect
8. #4780C2semidirect
9. #4771C2semidirect
10. #4688C2semidirect
11. #4681C2semidirect
50. C4×D8D3semidirect
124. C3×D4#41semidirect
215. D4#235semidirect
258. C3#3330semidirect

Maximal subgroups

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

subgroupindex
#55552
#55512
C3×C4×D82
#51582
#50192
#50072
#48842
#47802
#47712
#46882
#46812
#33303