Finite Group Databasebeta

← #79557

Subgroups — #79557

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#5555C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×#436C211yesmaximal, characteristic, complemented
4#5134C212yesmaximal, complemented
5#5133C212yesmaximal, complemented
6#4846C211yesmaximal, characteristic, complemented
7#4841C211yesmaximal, characteristic, complemented
8#4734C212yesmaximal, complemented
9#4722C212yesmaximal, complemented
10C4×D24C211yesmaximal, characteristic, complemented
11#4681C211yesmaximal, characteristic, complemented
12#3330—31nomaximal, Sylow
13C2×#237—21no—
14C2×#235C2212yes—
15C2×C3×M16C2212yes—
16C3×C4×D4C2211yescharacteristic
17#938—21no—
18#930C2212yes—
19C2×C4⋊Dic3C2212yes—
20#924C2212yes—
21C2×D24C2211yescharacteristic
22C2×#204C2212yes—
23#897—21no—
24D3×C4⋊C4—21no—
25#886—22no—
26#883—22no—
27C4×D12C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30#869C2211yescharacteristic
31C3×C4⋊C8C2211yescharacteristic
32C3×C22⋊C8C2212yes—
33C3×C4×C8C2211yescharacteristic
34#821C2211yescharacteristic
35#821C2212yes—
36#821C2211yescharacteristic
37#818C2212yes—
38#818C2211yescharacteristic
39#816C2212yes—
40#816C2212yes—
41#816C2212yes—
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×D8—31no—
51#436D311yescharacteristic, complemented
52#417—32no—
53C2×C3×D4D411yescharacteristic
54C22×C3×C4C2312yes—
55C2×#67—22no—
56C2×#67—22no—
57C22×Dic3—22no—
58#237—44no—
59#235—24no—
60#235—42no—
61C2×D12C2311yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×C4×D3—21no—
65C2×C4×D3—22no—
66C2×Dic6—21no—
67C2×Dic6C2312yes—
68C3×M16—42no—
69C2×C3×C8C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C2×C3×C8C2312yes—
72C3×C4⋊C4D411yescharacteristic
73C3×C22⋊C4D412yes—
74C3×C42C2311yescharacteristic
75#217—22no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—22no—
79C4⋊Dic3—21no—
80C4⋊Dic3—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3—22no—
83C4⋊Dic3C2312yes—
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—22no—
87D24—22no—
88#204—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—32no—
92C2×D8—31no—
93C2×M16C2×D312yes—
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×D4—31no—
104C4×D4C2×D311yescharacteristic
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107#107—31no—
108#107—32no—
109C4⋊C8C2×D311yescharacteristic
110#103—32no—
111#103—32no—
112#103—32no—
113#102—31no—
114#102—32no—
115#102—31no—
116C22⋊C8C2×D312yes—
117C4×C8C2×D311yescharacteristic
118C23×C3C2×D412yes—
119C22×D3—21no—
120C22×D3—21no—
121C3×D4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128#67—44no—
129#67—44no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—44no—
135D12—21no—
136D12—21no—
137D12—21no—
138D12#4112yescomplemented
139C4×D3—44no—
140C4×D3—42no—
141C4×D3—42no—
142Dic6—22no—
143Dic6—22no—
144Dic6—42no—
145C3×C8—21no—
146C3×C8—22no—
147C3×C8—21no—
148#41—64no—
149#41—122no—
150#41—124no—
151C2×Q8—32no—
152C2×Q8—61no—
153C2×D4—31no—
154C2×D4—31no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4D1211yescharacteristic
158C22×C4—62no—
159C22×C4—61no—
160C22×C4—62no—
161C22×C4—61no—
162C22×C4C22×D312yes—
163SD16—122no—
164D8—62no—
165M16—42no—
166C2×C8C22×D311yescharacteristic
167C2×C8C22×D311yescharacteristic
168C2×C8C22×D312yes—
169C4⋊C4—31no—
170C4⋊C4—32no—
171C4⋊C4—31no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4D1211yescharacteristic
178C22⋊C4—62no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—62no—
182C22⋊C4D1212yes—
183C42C22×D311yescharacteristic
184C22×C3—22no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3C22×D411yescharacteristic
188C2×D3—22no—
189C2×D3—41no—
190C2×D3—41no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—22no—
195C3×C4—21no—
196C3×C4AGL(1,ℤ/8)12yes—
197C3×C4C2×#4111yescharacteristic
198C3×C4H(5,𝔽2)11yescharacteristic
199Dic3—42no—
200Dic3—42no—
201Dic3—41no—
202Dic3—41no—
203C23—61no—
204C23—61no—
205C23C2×D1212yes—
206Q8—62no—
207Q8—62no—
208Q8—122no—
209D4—22no—
210D4—32no—
211D4—61no—
212D4—61no—
213D4—61no—
214D4—124no—
215D4—124no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—124no—
223C2×C4—124no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4C2×D1211yescharacteristic
227C2×C4C2×D1212yes—
228C2×C4C2×D1211yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—21no—
231C8—22no—
232C8—21no—
233C2×C3—42no—
234C2×C3#54611yescharacteristic
235C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C3#57811yescharacteristic
237D3—42no—
238D3—81no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—62no—
243C22—121no—
244C22—121no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C22×D1211yescharacteristic
249C4—21no—
250C4—22no—
251C4—121no—
252C4—122no—
253C4—121no—
254C4—122no—
255C4#90812yes—
256C4#100411yescharacteristic
257C4D3×#4111yescharacteristic
258C3#333011yescharacteristic, complemented, Sylow
259C2—42no—
260C2—122no—
261C2—241no—
262C2#555411yescharacteristic
263C2C2×#90811yescharacteristic
264C2#575311yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
hypercenter#436D3
hypocenterC3#3330
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #5555C2semidirect
2. #5551C2semidirect
3. C3×#436C2semidirect
4. #5134C2semidirect
5. #5133C2semidirect
6. #4846C2semidirect
7. #4841C2semidirect
8. #4734C2semidirect
9. #4722C2semidirect
10. C4×D24C2semidirect
11. #4681C2semidirect
51. #436D3semidirect
138. D12#41semidirect
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×#4362
#51342
#51332
#48462
#48412
#47342
#47222
C4×D242
#46812
#33303