Finite Group Databasebeta

← #30639923

Subgroups — #30639923

Order 16256.

316 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16861140C211yesmaximal, characteristic, complemented
2C2×#16861133C211yesmaximal, characteristic, complemented
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
5#22549736C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×C22⋊Dic127—21no—
13#16861140—22no—
14#16861133—22no—
15#16861133—22no—
16C22⋊(C127⋊C8)—22no—
17C127⋊#95—21no—
18C22×Q8×C127C2211yescharacteristic
19C23×C4×C127C2211yescharacteristic
20C22⋊Q8×C127—22no—
21C22⋊Q8×C127—22no—
22C2×C4⋊C4×C127—21no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127—21no—
25C22×Dic127—21no—
26C22×Dic127—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8D412yes—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C22⋊Dic127—42no—
38C24×C127D411yescharacteristic
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127D412yes—
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—24no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127D412yes—
58C22⋊C4×C127—24no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C127⋊C8—22no—
66C127⋊C8—41no—
67C23×C127—21no—
68C23×C127—21no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127SD1612yes—
74C23×C127C2×D411yescharacteristic
75Q8×C127—22no—
76Q8×C127—22no—
77Q8×C127—41no—
78Q8×C127—41no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C22×C127—22no—
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127#12011yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12311yescharacteristic
108C22×C127C2×SD1611yescharacteristic
109C22×C127#13511yescharacteristic
110C22×C127#13711yescharacteristic
111C22×C127#13711yescharacteristic
112C4×C127—41no—
113C4×C127—42no—
114C4×C127—42no—
115C4×C127#12111yescharacteristic
116C4×C127#12111yescharacteristic
117C4×C127#12311yescharacteristic
118C4×C127#12411yescharacteristic
119Dic127—81no—
120Dic127—81no—
121C2×C127—42no—
122C2×C127#39411yescharacteristic
123C2×C127#44811yescharacteristic
124C2×C127#45211yescharacteristic
125C2×C127#46511yescharacteristic
126C2×C127#47011yescharacteristic
127C2×C127#48311yescharacteristic
128C2×C127#48511yescharacteristic
129#2035—1271nomaximal, Sylow
130C127#203511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8D12711yescharacteristic
132C2×#106—1271no—
133C2×#103—1271no—
134C2×#103—1271no—
135C2×C22⋊C8—1271no—
136#380—1271no—
137#340—1271no—
138C22×Q8C2×D12711yescharacteristic
139C23×C4C2×D12711yescharacteristic
140C22×C8—1271no—
141C22×C8—1271no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4—1271no—
147C2×C4⋊C4C2×D12711yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—2541no—
150#106—2542no—
151#103—2542no—
152#103—2542no—
153C22⋊C8—2542no—
154#95—2541no—
155C24C22⋊D12711yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D12712yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2541no—
171C22×C4—2541no—
172C22×C4C22⋊D12711yescharacteristic
173C22×C4C22×D12711yescharacteristic
174C2×C8—1272no—
175C2×C8—2541no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C4⋊C4—24no—
181C4⋊C4—21no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C4⋊C4—2541no—
186C4⋊C4C22⋊D12712yes—
187C22⋊C4—24no—
188C22⋊C4—5082no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1452323512yes—
196C23C2×C22⋊D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4D4×D12711yescharacteristic
221C2×C4C2×C22⋊D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324711yescharacteristic
225C2×C4#1452324711yescharacteristic
226C8—5081no—
227C8—2542no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#1686117811yescharacteristic
236C22#1686118111yescharacteristic
237C22#1686119011yescharacteristic
238C22C2×#1452323511yescharacteristic
239C22#1686122911yescharacteristic
240C22#1686123911yescharacteristic
241C22#1686124011yescharacteristic
242C4—42no—
243C4—41no—
244C4—42no—
245C4—10161no—
246C4—10161no—
247C4#1686118011yescharacteristic
248C4#1686118111yescharacteristic
249C4#1686119011yescharacteristic
250C4#1686119611yescharacteristic
251C2—42no—
252C2#2254979011yescharacteristic
253C2#2255001511yescharacteristic
254C2#2255003811yescharacteristic
255C2#2255010711yescharacteristic
256C2#2255013211yescharacteristic
257C2#2255020411yescharacteristic
258C2#2255021411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC2×C22⋊Q8D127
hypocenterC127#2035
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
2. C2×#16861133C2semidirect
3. C2×#16861133C2semidirect
6. #22549638C2semidirect
130. C127#2035semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×#168611332
C2×C22⋊(C127⋊C8)2
#225497362
#225496382
C2×C22⋊Q8×C1272
#2035127