Finite Group Databasebeta

← #30648559

Subgroups — #30648559

Order 16256.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550507C211yesmaximal, characteristic, complemented
2#22550350C211yesmaximal, characteristic
3#22550179C212yesmaximal, complemented
4#22550175C212yesmaximal, complemented
5#22550165C212yesmaximal, complemented
6#22550155C212yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549904C211yesmaximal, characteristic, complemented
9#22549882C211yesmaximal, characteristic
10C2×#16861129C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861208—21no—
15#16861208C2212yescomplemented
16#16861200—21no—
17#16861200C2212yescomplemented
18#16861188—22no—
19C4×D508—21no—
20C4×D508—21no—
21#16861171C2212yes—
22#16861171C2212yes—
23#16861168C2211yescharacteristic
24#16861168C2211yescharacteristic
25C22⋊C4×D127—22no—
26#16861142C2211yescharacteristic, complemented
27#16861142C2211yescharacteristic, complemented
28#16861140C2211yescharacteristic, complemented
29#16861140C2211yescharacteristic, complemented
30Dic127⋊C8C2212yescomplemented
31Dic127⋊C8C2212yescomplemented
32#16861131C2212yes—
33#16861131C2212yes—
34#16861129C2212yes—
35#16861129C2212yes—
36C2×M16×C127C2211yescharacteristic
37#117×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#107×C127C2212yescomplemented
40#106×C127C2212yescomplemented
41C23×D127—21no—
42C2×Dic254—22no—
43C2×D508—21no—
44C2×D508—22no—
45C2×D508Q814yescomplemented
46C2×D508C2312yes—
47C2×C4×D127—22no—
48C2×C4×D127—22no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—21no—
60C22⋊Dic127D412yes—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—21no—
64C4×Dic127C2312yes—
65C22×C4×C127—21no—
66C22×C4×C127C2311yescharacteristic
67M16×C127—22no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C42×C127—21no—
75C22×D127—22no—
76C22×D127—24no—
77C22×D127—81no—
78C22×D127—81no—
79Dic254—82no—
80D508—24no—
81D508—22no—
82D508C2×Q814yes—
83C2×Dic127—21no—
84C2×Dic127—21no—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C4×D127—82no—
88C4×D127—82no—
89C127⋊C8—21no—
90C127⋊C8—21no—
91C23×C127C2×D411yescharacteristic
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—81no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×Q812yes—
98C2×C4×C127#4112yes—
99C2×C4×C127C2411yescharacteristic
100C8×C127—22no—
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—42no—
110C4×C127—41no—
111C4×C127—41no—
112C4×C127C22⋊Q812yes—
113C4×C127C22×Q811yescharacteristic
114C4×C127C2×#4111yescharacteristic
115Dic127—22no—
116Dic127—22no—
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#544—1271no—
127#518—1271no—
128#478—1272no—
129#478—1272no—
130#476—1272no—
131#474—1272no—
132#428D12711yescharacteristic, complemented
133#427—1271no—
134#424—1271no—
135#418—1271no—
136C2×#102—1271no—
137C22×D4—1271no—
138C2×M16C2×D12711yescharacteristic
139C22×C8—1271no—
140C4⋊Q8—1272no—
141C4⋊Q8—2541no—
142#125—1272no—
143#125—2541no—
144C22⋊Q8—2542no—
145C4×D4—1272no—
146C4×D4—1272no—
147C4×D4—2541no—
148C4×D4—2541no—
149#117—1271no—
150#117—1271no—
151#117C2×D12711yescharacteristic
152C2×C4⋊C4C2×D12711yescharacteristic
153C2×C22⋊C4—2542no—
154#107—1271no—
155#107—1271no—
156#107C2×D12712yescomplemented
157#106—1271no—
158#106—1271no—
159#106C2×D12712yescomplemented
160C4⋊C8—1272no—
161C4⋊C8—1272no—
162#102—1272no—
163#102—1272no—
164#102—1272no—
165#102—1272no—
166C24—2541no—
167C2×Q8—2542no—
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—22no—
177C2×C8—1271no—
178C2×C8—1271no—
179C2×C8—2542no—
180C2×C8C22×D12712yes—
181C4⋊C4—21no—
182C4⋊C4—1271no—
183C4⋊C4—1271no—
184C4⋊C4—1271no—
185C4⋊C4—1271no—
186C4⋊C4—2542no—
187C4⋊C4—2542no—
188C4⋊C4—2542no—
189C4⋊C4C22×D12712yes—
190C4⋊C4C22×D12711yescharacteristic
191C4⋊C4C22×D12711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—1272no—
194C22⋊C4—2542no—
195C22⋊C4—2542no—
196C22⋊C4—2541no—
197C42—21no—
198C42—1272no—
199C42—2541no—
200C23—10161no—
201C23—10161no—
202C23—2544no—
203C23—2542no—
204C23D4×D12711yescharacteristic
205Q8—10162no—
206D4—1274no—
207D4—2544no—
208D4—2542no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—81no—
213C2×C4—1272no—
214C2×C4—1272no—
215C2×C4—10162no—
216C2×C4—10162no—
217C2×C4—2541no—
218C2×C4—2541no—
219C2×C4D4×D12711yescharacteristic
220C2×C4Q8×D12712yes—
221C2×C4#1452324812yes—
222C2×C4C23×D12711yescharacteristic
223C8—22no—
224C8—2541no—
225C8—2541no—
226C22—21no—
227C22—21no—
228C22—10164no—
229C22—5082no—
230C22—5082no—
231C22—2544no—
232C22#1686118712yes—
233C22C2×D4×D12711yescharacteristic
234C4—42no—
235C4—41no—
236C4—41no—
237C4—5081no—
238C4—5081no—
239C4—2542no—
240C4—2542no—
241C4#1686118712yes—
242C4C2×Q8×D12711yescharacteristic
243C4C2×#1452324811yescharacteristic
244C2—22no—
245C2—5084no—
246C2C2×#1686118711yescharacteristic
247C2#2255068011yescharacteristic
248C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#428D127
hypocenterC127#3081
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
3. #22550179C2semidirect
4. #22550175C2semidirect
5. #22550165C2semidirect
6. #22550155C2semidirect
7. #22549912C2semidirect
8. #22549904C2semidirect
11. #428×C127C2semidirect
15. #16861208C22semidirect
17. #16861200C22semidirect
26. #16861142C22semidirect
27. #16861142C22semidirect
28. #16861140C22semidirect
29. #16861140C22semidirect
30. Dic127⋊C8C22semidirect
31. Dic127⋊C8C22semidirect
39. #107×C127C22semidirect
40. #106×C127C22semidirect
45. C2×D508Q8semidirect
125. C127#3081semidirect
132. #428D127semidirect
156. #107C2×D127semidirect
159. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225503502
#225501792
#225501752
#225501652
#225501552
#225499122
#225499042
#225498822
C2×#168611292
#428×C1272
#3081127