Finite Group Databasebeta

← #30651437

Subgroups — #30651437

Order 16256.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550548C211yesmaximal, characteristic, complemented
2#22550242C212yesmaximal, complemented
3#22550224C211yesmaximal, characteristic, complemented
4#22550173C212yesmaximal, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550071C211yesmaximal, characteristic, complemented
7#22550071C212yesmaximal, complemented
8#22550007C211yesmaximal, characteristic, complemented
9#22549986C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic, complemented
11#554×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—21no—
13C2×#14523233C2211yescharacteristic
14C2×#14523233C2212yes—
15#16861206C2211yescharacteristic
16#16861206C2212yes—
17#16861184—21no—
18#16861184—22no—
19C4×D508C2211yescharacteristic
20C4×D508C2212yes—
21#16861142C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861130C2211yescharacteristic
26#16861129C2212yes—
27#16861129C2212yes—
28#16861129C2212yes—
29#16861129C2211yescharacteristic
30#16861129C2212yes—
31#16861120C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#125×C127C2212yes—
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—22no—
37C4×Q8×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39C4×D4×C127C2212yes—
40C2×C4⋊C4×C127—21no—
41#14523249—44no—
42C2×D508—22no—
43C2×D508—21no—
44C2×D508—21no—
45C2×D508—22no—
46C2×D508C2312yes—
47C2×D508C2311yescharacteristic
48C2×C4×D127—22no—
49C2×C4×D127—21no—
50#14523233—22no—
51#14523233—24no—
52C2×C127⋊C8C2311yescharacteristic
53C2×C127⋊C8C2312yes—
54C2×C127⋊C8C2311yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56(C2×D127)⋊C4—21no—
57(C2×D127)⋊C4—22no—
58C2×Q8×C127—21no—
59C2×Q8×C127D411yescharacteristic
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—22no—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127D412yes—
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C22⋊C4×C127—21no—
71C22⋊C4×C127—22no—
72C42×C127C2312yes—
73C42×C127C2311yescharacteristic
74C22×D127—22no—
75C22×D127—21no—
76D508—21no—
77D508—22no—
78D508—22no—
79D508—21no—
80D508—44no—
81D508—42no—
82D508—82no—
83D508—81no—
84C2×Dic127—21no—
85C4×D127—44no—
86C4×D127—42no—
87C127⋊C8—21no—
88C127⋊C8—22no—
89C127⋊C8—21no—
90C23×C127—21no—
91Q8×C127—22no—
92Q8×C127—81no—
93D4×C127—21no—
94D4×C127#4112yescomplemented
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127—82no—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2411yescharacteristic
104C22×C127—22no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127C22×D411yescharacteristic
108C2×D127—42no—
109C2×D127—41no—
110C2×D127—42no—
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—42no—
114C4×C127—22no—
115C4×C127—22no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—42no—
119C4×C127AGL(1,ℤ/8)12yes—
120C4×C127C2×#4111yescharacteristic
121C4×C1272−1+411yescharacteristic
122Dic127—41no—
123C2×C127—42no—
124C2×C127#54811yescharacteristic
125C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C127#57611yescharacteristic
127D127—82no—
128D127—81no—
129#3366—1271nomaximal, Sylow
130C127#336611yescharacteristic, complemented, Sylow
131#554D12711yescharacteristic, complemented
132#550—1271no—
133#490—1272no—
134#486—1271no—
135#478—1272no—
136#476—1271no—
137#459—1272no—
138#459—1271no—
139#446—1271no—
140#442—1272no—
141C4×D8—1271no—
142C2×#41—2541no—
143C2×D8—1271no—
144C2×D8—1272no—
145C4⋊Q8C2×D12711yescharacteristic
146#127—1271no—
147#127—1272no—
148#125C2×D12712yes—
149C22⋊Q8—21no—
150C22⋊Q8—22no—
151#121—2541no—
152#121—2542no—
153C4×Q8C2×D12711yescharacteristic
154C4×D4—1271no—
155C4×D4—1272no—
156C4×D4C2×D12711yescharacteristic
157C4×D4C2×D12712yes—
158C2×C4⋊C4—21no—
159#107—1271no—
160#106—1272no—
161C4⋊C8—1271no—
162C4⋊C8—1272no—
163#102—1271no—
164#102—1271no—
165#102—1272no—
166#102—1272no—
167#102—1272no—
168#102—1272no—
169#97—1272no—
170C4×C8—1271no—
171#41—5084no—
172C2×Q8—21no—
173C2×Q8C22⋊D12711yescharacteristic
174C2×D4—1271no—
175C2×D4—1272no—
176C2×D4—2541no—
177C2×D4—2542no—
178C2×D4—2542no—
179C2×D4—2541no—
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2542no—
184C22×C4—2541no—
185D8—2542no—
186D8—2544no—
187C2×C8—1271no—
188C2×C8—1271no—
189C2×C8—1272no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—1271no—
194C4⋊C4C22⋊D12711yescharacteristic
195C4⋊C4C22⋊D12712yes—
196C4⋊C4C22×D12712yes—
197C4⋊C4C22×D12711yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—2541no—
201C22⋊C4—2542no—
202C42C22×D12711yescharacteristic
203C42C22×D12712yes—
204C23—21no—
205C23—2542no—
206C23—2541no—
207Q8—22no—
208Q8—81no—
209D4—21no—
210D4—10162no—
211D4—10161no—
212D4—5084no—
213D4—5082no—
214D4—2541no—
215D4—2542no—
216D4—2541no—
217D4—2542no—
218D4#1452324912yescomplemented
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—5084no—
224C2×C4—5082no—
225C2×C4—2541no—
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12712yes—
228C2×C4C2×C22⋊D12712yes—
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4C23×D12711yescharacteristic
231C8—2541no—
232C8—2542no—
233C8—2541no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—5082no—
238C22—5081no—
239C22—5082no—
240C22—5081no—
241C22—5081no—
242C22—5082no—
243C22C22×C22⋊D12711yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—42no—
249C4—5081no—
250C4#1686123412yes—
251C4C2×#1452324911yescharacteristic
252C4#1686125711yescharacteristic
253C2—42no—
254C2—10162no—
255C2—10161no—
256C2#2255053711yescharacteristic
257C2C2×#1686123411yescharacteristic
258C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#554×C127C2
socleC22×C127C22×D4
hypercenter#554D127
hypocenterC127#3366
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #22550548C2semidirect
2. #22550242C2semidirect
3. #22550224C2semidirect
4. #22550173C2semidirect
5. #22550163C2semidirect
6. #22550071C2semidirect
7. #22550071C2semidirect
8. #22550007C2semidirect
9. #22549986C2semidirect
10. C4×#14523233C2semidirect
11. #554×C127C2semidirect
94. D4×C127#41semidirect
130. C127#3366semidirect
131. #554D127semidirect
218. D4#14523249semidirect

Maximal subgroups

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

subgroupindex
#225505482
#225502422
#225502242
#225501732
#225501632
#225500712
#225500072
#225499862
C4×#145232332
#554×C1272
#3366127