Finite Group Databasebeta

← #30639534

Subgroups — #30639534

Order 16256.

436 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16861205C211yesmaximal, characteristic
2C2×#16861131C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic, complemented
5C2×C8×Dic127C211yesmaximal, characteristic, complemented
6#22549745C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127—22no—
10C22×C127⋊C8C2211yescharacteristic
11#16861205—22no—
12#16861205C414yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—22no—
15C2×C4×Dic127C2211yescharacteristic
16#16861131—22no—
17#16861130—22no—
18#16861129—22no—
19C8×Dic127—22no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C23×D127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C2×C22⋊D127—28no—
32C2×C22⋊D127—44no—
33C2×C22⋊D127—44no—
34C22×Dic127—21no—
35C22×Dic127D412yes—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8D412yescomplemented
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41Dic127⋊C4—44no—
42C4×Dic127C2×C412yes—
43C4×Dic127D412yes—
44C24×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127C2×C412yes—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127D412yescomplemented
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C22×D127—22no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127—42no—
62C22×D127—42no—
63D508—22no—
64D508—22no—
65D508—41no—
66D508—41no—
67C22⋊D127—416no—
68C2×Dic127—41no—
69C2×Dic127—42no—
70C2×Dic127C22⋊C414yes—
71C2×Dic127D814yes—
72C2×Dic127SD1614yes—
73C127⋊C8—22no—
74C23×C127—22no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—42no—
78C23×C127—42no—
79C23×C127C2×D411yescharacteristic
80D4×C127—22no—
81D4×C127—22no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—42no—
86C2×C4×C127C22×C411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C2×C4×C127#4111yescharacteristic
92C8×C127—22no—
93C22×C127—42no—
94C22×C127—44no—
95C22×C127—82no—
96C22×C127—81no—
97C22×C127—81no—
98C22×C127C2×C22⋊C411yescharacteristic
99C22×C127#12411yescharacteristic
100C22×C127#12711yescharacteristic
101C22×C127C2×D812yes—
102C22×C127C2×SD1612yes—
103C2×D127—42no—
104C2×D127—44no—
105C2×D127—82no—
106C2×D127—81no—
107C2×D127—81no—
108C4×C127—81no—
109C4×C127C4×D411yescharacteristic
110C4×C127C4×D411yescharacteristic
111C4×C127#12111yescharacteristic
112C4×C127#12111yescharacteristic
113Dic127—81no—
114Dic127#10218yescomplemented
115C2×C127—82no—
116C2×C127#39011yescharacteristic
117C2×C127C2×#10212yes—
118C2×C127#48612yes—
119C2×C127#49211yescharacteristic
120C2×C127#49311yescharacteristic
121D127—82no—
122#1957—1271nomaximal, Sylow
123C127#195711yescharacteristic, complemented, Sylow
124C2×#127—1271no—
125C2×#102—1271no—
126C2×#102—1271no—
127C2×#102—1271no—
128C2×#102D12711yescharacteristic, complemented
129C2×C4×C8—1271no—
130#384—1271no—
131C22×D4—1271no—
132C22×D4—2542no—
133C22×D4C2×D12711yescharacteristic
134C22×C8—1271no—
135C22×C8C2×D12711yescharacteristic
136#127—1274no—
137#127—2542no—
138C2×C4⋊C4—1271no—
139C2×C4⋊C4—2542no—
140C2×C4⋊C4C2×D12711yescharacteristic
141C2×C42—1271no—
142#102—22no—
143#102—2542no—
144#102—2542no—
145#102—2542no—
146C4×C8—2542no—
147C24—21no—
148C24—2541no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—1272no—
155C2×D4—5084no—
156C2×D4—5084no—
157C2×D4—2548no—
158C2×D4—2541no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4—2541no—
162C2×D4—2541no—
163C2×D4C4×D12712yes—
164C22×C4—21no—
165C22×C4—1272no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—1272no—
171C2×C8—2541no—
172C2×C8—2541no—
173C2×C8C22⋊D12712yescomplemented
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—5084no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C42—1272no—
180C42—1272no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—42no—
185C23—42no—
186C23—5081no—
187C23—5081no—
188C23—5082no—
189C23—5082no—
190C23—2542no—
191C23D4×D12711yescharacteristic
192D4—22no—
193D4—22no—
194D4—41no—
195D4—41no—
196D4—50816no—
197D4—5081no—
198D4—5081no—
199D4—2542no—
200D4—2542no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—1274no—
204C2×C4—1274no—
205C2×C4—1274no—
206C2×C4—5081no—
207C2×C4—5082no—
208C2×C4C2×C4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4D4×D12711yescharacteristic
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4#1452324711yescharacteristic
213C2×C4#1452324811yescharacteristic
214C8—22no—
215C8—2542no—
216C22—42no—
217C22—44no—
218C22—82no—
219C22—81no—
220C22—81no—
221C22—10162no—
222C22—10161no—
223C22—10161no—
224C22—5082no—
225C22—5084no—
226C22C22⋊C4×D12711yescharacteristic
227C22#1686119511yescharacteristic
228C22#1686120511yescharacteristic
229C22D8×D12712yes—
230C22SD16×D12712yes—
231C4—81no—
232C4—1278no—
233C4—10161no—
234C4(C22⋊D127)⋊C411yescharacteristic
235C4C4×C22⋊D12711yescharacteristic
236C4#1686118111yescharacteristic
237C4#1686118211yescharacteristic
238C2—82no—
239C2—10162no—
240C2#2254977611yescharacteristic
241C2#102×D12712yes—
242C2#2255022112yes—
243C2#2255024711yescharacteristic
244C2#2255025111yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861205C2
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1957
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861131C2semidirect
3. C2×#16861130C2semidirect
4. C2×#16861129C2semidirect
5. C2×C8×Dic127C2semidirect
6. #22549745C2semidirect
7. C2×#102×C127C2semidirect
12. #16861205C4semidirect
38. C2×C127⋊C8D4semidirect
55. C2×C8×C127D4semidirect
114. Dic127#102semidirect
123. C127#1957semidirect
128. C2×#102D127semidirect
173. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×#168611312
C2×#168611302
C2×#168611292
C2×C8×Dic1272
#225497452
C2×#102×C1272
#1957127