Finite Group Databasebeta

← #30651433

Subgroups — #30651433

Order 16256.

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

normal subgroups only

This group has 396 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#22550586C211yesmaximal, characteristic, complemented
2#22550240C212yesmaximal, complemented
3#22550222C211yesmaximal, characteristic, complemented
4#22550176C212yesmaximal, complemented
5#22550166C211yesmaximal, characteristic, complemented
6#22550068C212yesmaximal, complemented
7#22550068C211yesmaximal, characteristic, complemented
8#22550007C211yesmaximal, characteristic, complemented
9#22549986C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic, complemented
11#550×C127C211yesmaximal, characteristic, complemented
12C2×#14523233C2211yescharacteristic
13C2×#14523233C2212yes—
14#16861209C2211yescharacteristic
15#16861201C2212yes—
16#16861187—22no—
17#16861185—21no—
18C4×D508C2211yescharacteristic
19C4×D508C2212yes—
20C4⋊C4×D127—21no—
21#16861142C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861130C2212yes—
26#16861130C2212yes—
27#16861130C2212yes—
28#16861130C2211yescharacteristic
29#16861130C2212yes—
30#16861129C2211yescharacteristic
31#16861120C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C2×#41×C127—21no—
34#127×C127C2211yescharacteristic
35#127×C127C2212yes—
36#121×C127—22no—
37#121×C127—21no—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40C4×D4×C127C2212yes—
41C2×Dic254—21no—
42C2×D508C2311yescharacteristic
43C2×C4×D127—21no—
44C2×C4×D127—22no—
45#14523233—22no—
46#14523233—24no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—21no—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55(C2×D127)⋊C4—22no—
56(C2×D127)⋊C4—21no—
57#41×C127—44no—
58C2×Q8×C127D411yescharacteristic
59C2×D4×C127—22no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—22no—
63C2×D4×C127C2311yescharacteristic
64C2×D4×C127C2312yes—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C4⋊C4×C127D412yes—
68C4⋊C4×C127D411yescharacteristic
69C4⋊C4×C127C2311yescharacteristic
70C22⋊C4×C127—21no—
71C22⋊C4×C127—22no—
72C42×C127C2312yes—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—81no—
76D508—21no—
77D508#4112yescomplemented
78C2×Dic127—21no—
79C2×Dic127—22no—
80C4×D127—42no—
81C4×D127—82no—
82C127⋊C8—21no—
83C127⋊C8—22no—
84C127⋊C8—21no—
85C23×C127—22no—
86C23×C127—21no—
87Q8×C127—22no—
88D4×C127—21no—
89D4×C127—22no—
90D4×C127—21no—
91D4×C127—22no—
92D4×C127—44no—
93D4×C127—42no—
94D4×C127—82no—
95D4×C127—81no—
96C2×C4×C127—21no—
97C2×C4×C127—44no—
98C2×C4×C127—42no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2411yescharacteristic
104C22×C127—42no—
105C22×C127—41no—
106C22×C127—42no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127C22×D411yescharacteristic
111C2×D127—22no—
112C2×D127—41no—
113C2×D127—41no—
114C4×C127—22no—
115C4×C127—22no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127AGL(1,ℤ/8)12yes—
119C4×C127C2×#4111yescharacteristic
120C4×C1272−1+411yescharacteristic
121Dic127—41no—
122Dic127—42no—
123C2×C127—82no—
124C2×C127—81no—
125C2×C127#54811yescharacteristic
126C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C127#57611yescharacteristic
128D127—42no—
129#3366—1271nomaximal, Sylow
130C127#336611yescharacteristic, complemented, Sylow
131#554—1271no—
132#550D12711yescharacteristic, complemented
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—21no—
143C2×D8—1271no—
144C2×D8—1272no—
145C4⋊Q8—1271no—
146#127C2×D12711yescharacteristic
147#127C2×D12712yes—
148#125—1272no—
149C22⋊Q8—2541no—
150C22⋊Q8—2542no—
151#121—21no—
152#121—22no—
153C4×Q8C2×D12711yescharacteristic
154C4×D4—1271no—
155C4×D4—1272no—
156C4×D4C2×D12711yescharacteristic
157C4×D4C2×D12712yes—
158C2×C4⋊C4—2541no—
159#107—1271no—
160#106—1272no—
161C4⋊C8—1271no—
162C4⋊C8—1272no—
163#102—1271no—
164#102—1272no—
165#102—1272no—
166#102—1272no—
167#102—1272no—
168#102—1271no—
169#97—1272no—
170C4×C8—1271no—
171#41—44no—
172C2×Q8—2541no—
173C2×Q8C22⋊D12711yescharacteristic
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—22no—
178C2×D4—1271no—
179C2×D4C22×D12711yescharacteristic
180C2×D4C22×D12712yes—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—2542no—
184C22×C4—2541no—
185D8—2542no—
186D8—2544no—
187C2×C8—1271no—
188C2×C8—1271no—
189C2×C8—1272no—
190C4⋊C4—1272no—
191C4⋊C4—1271no—
192C4⋊C4—2542no—
193C4⋊C4—2542no—
194C4⋊C4—2541no—
195C4⋊C4C22⋊D12711yescharacteristic
196C4⋊C4C22⋊D12712yes—
197C4⋊C4C22×D12711yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—2541no—
201C22⋊C4—2542no—
202C42C22×D12711yescharacteristic
203C42C22×D12712yes—
204C23—22no—
205C23—21no—
206C23—2541no—
207Q8—22no—
208Q8—10161no—
209D4—21no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—44no—
214D4—42no—
215D4—82no—
216D4—81no—
217D4—1272no—
218D4—2541no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—10162no—
223C2×C4—5082no—
224C2×C4—2541no—
225C2×C4—2542no—
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—42no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—42no—
240C22—5081no—
241C22—5081no—
242C22—2542no—
243C22C22×C22⋊D12711yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—5081no—
249C4—5082no—
250C4#1686123512yes—
251C4#41×D12711yescharacteristic
252C4#1686125611yescharacteristic
253C2—82no—
254C2—81no—
255C2—5082no—
256C2#2255053311yescharacteristic
257C2C2×#1686123511yescharacteristic
258C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#550×C127C2
socleC22×C127C22×D4
hypercenter#550D127
hypocenterC127#3366
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #22550586C2semidirect
2. #22550240C2semidirect
3. #22550222C2semidirect
4. #22550176C2semidirect
5. #22550166C2semidirect
6. #22550068C2semidirect
7. #22550068C2semidirect
8. #22550007C2semidirect
9. #22549986C2semidirect
10. C4×#14523233C2semidirect
11. #550×C127C2semidirect
77. D508#41semidirect
130. C127#3366semidirect
132. #550D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502402
#225502222
#225501762
#225501662
#225500682
#225500072
#225499862
C4×#145232332
#550×C1272
#3366127