Finite Group Databasebeta

← #30648555

Subgroups — #30648555

Order 16256.

398 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 141 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#22550350C211yesmaximal, characteristic
2#22550173C212yesmaximal, complemented
3#22550173C212yesmaximal, complemented
4#22550163C212yesmaximal, complemented
5#22550153C212yesmaximal, complemented
6#22549916C211yesmaximal, characteristic, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549903C211yesmaximal, characteristic, complemented
9#22549880C211yesmaximal, characteristic
10C2×#16861129C211yesmaximal, characteristic
11#544×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4×D508C2212yes—
16C4×D508C2212yes—
17#16861171—21no—
18#16861171—21no—
19#16861168C2211yescharacteristic
20C22⋊C4×D127—22no—
21#16861142C2211yescharacteristic, complemented
22#16861142C2212yescomplemented
23#16861142C2211yescharacteristic, complemented
24#16861140C2211yescharacteristic, complemented
25#16861140C2212yescomplemented
26#16861140C2211yescharacteristic, complemented
27C4⋊(C127⋊C8)C2212yescomplemented
28C4⋊(C127⋊C8)C2212yescomplemented
29#16861129C2212yes—
30#16861129C2212yes—
31#16861129C2212yes—
32#16861129C2212yes—
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127C2212yescomplemented
35#125×C127—21no—
36#125×C127C2212yescomplemented
37C22⋊Q8×C127—22no—
38#117×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41C23×D127—21no—
42C2×D508—21no—
43C2×D508—22no—
44C2×D508Q814yescomplemented
45C2×D508C2312yes—
46C2×C4×D127—22no—
47C2×C4×D127—22no—
48#14523232—22no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2312yes—
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55C22⋊Dic127—21no—
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—22no—
58C4×Dic127—21no—
59C2×Q8×C127—22no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—22no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127D411yescharacteristic
68C4⋊C4×C127C2312yes—
69C4⋊C4×C127C2311yescharacteristic
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C22⋊C4×C127D412yes—
73C42×C127—21no—
74C42×C127C2312yes—
75C22×D127—22no—
76C22×D127—24no—
77C22×D127—81no—
78C22×D127—81no—
79D508—24no—
80D508—22no—
81D508C2×Q814yes—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C4×D127—82no—
85C4×D127—82no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88C127⋊C8—22no—
89C23×C127C2×D411yescharacteristic
90Q8×C127—82no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—81no—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×Q812yes—
99C2×C4×C127#4112yes—
100C2×C4×C127C2411yescharacteristic
101C22×C127—21no—
102C22×C127—21no—
103C22×C127C22⋊Q812yes—
104C22×C127C22×D411yescharacteristic
105C2×D127—24no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—84no—
109C4×C127—22no—
110C4×C127—22no—
111C4×C127—42no—
112C4×C127—41no—
113C4×C127—41no—
114C4×C127C22⋊Q812yes—
115C4×C127C22×Q811yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—41no—
118Dic127—41no—
119C2×C127—22no—
120C2×C127C2×C22⋊Q811yescharacteristic
121C2×C127#57611yescharacteristic
122C2×C127#57711yescharacteristic
123D127—44no—
124#3081—1271nomaximal, Sylow
125C127#308111yescharacteristic, complemented, Sylow
126#544D12711yescharacteristic, complemented
127#518—1271no—
128#478—1272no—
129#478—1272no—
130#476—1272no—
131#474—1272no—
132#428—1271no—
133#427—1271no—
134#424—1271no—
135#418—1271no—
136C2×#102—1271no—
137C22×D4—1271no—
138C2×M16—1271no—
139C22×C8—1271no—
140C4⋊Q8—21no—
141C4⋊Q8C2×D12712yescomplemented
142#125—21no—
143#125C2×D12712yescomplemented
144C22⋊Q8—22no—
145C4×D4—1272no—
146C4×D4—1272no—
147C4×D4—2541no—
148C4×D4—2541no—
149#117—1271no—
150#117C2×D12711yescharacteristic
151#117C2×D12711yescharacteristic
152C2×C4⋊C4C2×D12711yescharacteristic
153C2×C22⋊C4—2542no—
154#107—1271no—
155#107—1271no—
156#107—1272no—
157#106—1271no—
158#106—1271no—
159#106—1272no—
160C4⋊C8—1272no—
161C4⋊C8—1272no—
162#102—1272no—
163#102—1272no—
164#102—1272no—
165#102—1272no—
166C24—2541no—
167C2×Q8—22no—
168C2×D4—1274no—
169C2×D4—1272no—
170C2×D4—2541no—
171C2×D4—2542no—
172C22×C4—21no—
173C22×C4—2542no—
174C22×C4—2542no—
175C22×C4C22×D12711yescharacteristic
176M16—2542no—
177C2×C8—1271no—
178C2×C8—1272no—
179C2×C8—1271no—
180C2×C8—2542no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—1271no—
186C4⋊C4—1271no—
187C4⋊C4C22⋊D12711yescharacteristic
188C4⋊C4C22⋊D12711yescharacteristic
189C4⋊C4C22×D12712yes—
190C4⋊C4C22×D12711yescharacteristic
191C4⋊C4C22×D12711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—2541no—
194C22⋊C4—2542no—
195C22⋊C4—2542no—
196C22⋊C4C22⋊D12712yes—
197C42—21no—
198C42—2541no—
199C42C22×D12712yes—
200C23—10161no—
201C23—10161no—
202C23—2544no—
203C23—2542no—
204C23C2×C22⋊D12711yescharacteristic
205Q8—82no—
206D4—1274no—
207D4—2544no—
208D4—2542no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—10162no—
214C2×C4—10162no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4C2×C22⋊D12712yes—
218C2×C4C2×C22⋊D12712yes—
219C2×C4C2×C22⋊D12711yescharacteristic
220C2×C4Q8×D12712yes—
221C2×C4#1452324912yes—
222C2×C4C23×D12711yescharacteristic
223C8—2541no—
224C8—2541no—
225C8—2542no—
226C22—21no—
227C22—21no—
228C22—10164no—
229C22—5082no—
230C22—5082no—
231C22—2544no—
232C22#1686118512yes—
233C22C22×C22⋊D12711yescharacteristic
234C4—22no—
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—5081no—
240C4—5081no—
241C4#1686118512yes—
242C4C2×Q8×D12711yescharacteristic
243C4C2×#1452324911yescharacteristic
244C2—22no—
245C2—5084no—
246C2C2×#1686118511yescharacteristic
247C2#2255067911yescharacteristic
248C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#544×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#544D127
hypocenterC127#3081
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #22550173C2semidirect
3. #22550173C2semidirect
4. #22550163C2semidirect
5. #22550153C2semidirect
6. #22549916C2semidirect
7. #22549912C2semidirect
8. #22549903C2semidirect
11. #544×C127C2semidirect
21. #16861142C22semidirect
22. #16861142C22semidirect
23. #16861142C22semidirect
24. #16861140C22semidirect
25. #16861140C22semidirect
26. #16861140C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. C4⋊(C127⋊C8)C22semidirect
34. C4⋊Q8×C127C22semidirect
36. #125×C127C22semidirect
44. C2×D508Q8semidirect
125. C127#3081semidirect
126. #544D127semidirect
141. C4⋊Q8C2×D127semidirect
143. #125C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503502
#225501732
#225501632
#225501532
#225499162
#225499122
#225499032
#225498802
C2×#168611292
#544×C1272
#3081127