Finite Group Databasebeta

← #30639024

Subgroups — #30639024

Order 16256.

408 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#14523231C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D127C211yesmaximal, characteristic
3C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
4C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
5#22549630C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C411yescharacteristic
9C22×C4×D127C2211yescharacteristic
10C2×#14523231—22no—
11C2×#14523231—22no—
12C2×#14523231—22no—
13C22×C127⋊C8C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15C4×C22⋊D127—22no—
16C4×C22⋊D127—22no—
17C2×Dic127⋊C4C411yescharacteristic
18C2×C22⋊Dic127C411yescharacteristic
19C2×(C2×D127)⋊C4C411yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21C22⋊(C127⋊C8)—22no—
22(C2×D127)⋊C8—22no—
23C23×C4×C127C2211yescharacteristic
24C22×C8×C127C2211yescharacteristic
25C22×C8×C127C2211yescharacteristic
26C22⋊C8×C127—22no—
27C23×D127C2×C411yescharacteristic
28C2×C22⋊D127—22no—
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31C22×Dic127C2×C411yescharacteristic
32C22×Dic127C2×C411yescharacteristic
33C22×Dic127C2×C411yescharacteristic
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127—21no—
38C2×C4×D127D412yes—
39C2×C4×D127D412yes—
40#14523231—28no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8D412yescomplemented
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49(C2×D127)⋊C4—22no—
50C4×Dic127—21no—
51C4×Dic127—21no—
52C24×C127C2×C411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2×C411yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127—21no—
66C2×C8×C127D412yescomplemented
67C22×D127—21no—
68C22×D127—21no—
69C22×D127—21no—
70C22×D127—21no—
71C22×D127—21no—
72C22×D127—21no—
73C22×D127C22⋊C412yes—
74C22⋊D127—28no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127C22⋊C412yes—
83C2×Dic127M1612yescomplemented
84C4×D127—22no—
85C4×D127—22no—
86C4×D127—22no—
87C4×D127—22no—
88C127⋊C8—22no—
89C127⋊C8—41no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—21no—
96C23×C127M1612yescomplemented
97C23×C127C22×C411yescharacteristic
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127#4111yescharacteristic
110C2×C4×C127#4111yescharacteristic
111C8×C127—22no—
112C8×C127—41no—
113C22×C127—22no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127—41no—
120C22×C127C2×C22⋊C411yescharacteristic
121C22×C127C4×D411yescharacteristic
122C22×C127C4×D411yescharacteristic
123C22×C127C2×M1611yescharacteristic
124C22×C127C2×M1611yescharacteristic
125C22×C127#13111yescharacteristic
126C22×C127#13111yescharacteristic
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—22no—
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—41no—
133C2×D127—41no—
134C4×C127—41no—
135C4×C127#12011yescharacteristic
136C4×C127#12111yescharacteristic
137C4×C127#12111yescharacteristic
138C4×C127#12311yescharacteristic
139Dic127—22no—
140Dic127—22no—
141Dic127—41no—
142C2×C127—42no—
143C2×C127#38611yescharacteristic
144C2×C127#40711yescharacteristic
145C2×C127#40811yescharacteristic
146C2×C127#43511yescharacteristic
147C2×C127#43511yescharacteristic
148C2×C127#43611yescharacteristic
149C2×C127#43611yescharacteristic
150D127—42no—
151#1870—1271nomaximal, Sylow
152C127#187011yescharacteristic, complemented, Sylow
153C22×M16—1271no—
154C2×C4×D4—1271no—
155C2×C22⋊C8—1271no—
156C2×C22⋊C8—1271no—
157C2×C22⋊C8D12711yescharacteristic, complemented
158#336—1271no—
159#336—1271no—
160C22×D4—1271no—
161C23×C4—1271no—
162C23×C4C2×D12711yescharacteristic
163C2×M16—2542no—
164C2×M16—2542no—
165C2×M16—2542no—
166C22×C8—1271no—
167C22×C8—1271no—
168C22×C8C2×D12711yescharacteristic
169C22×C8C2×D12711yescharacteristic
170C4×D4—2542no—
171C4×D4—2542no—
172C2×C4⋊C4—1271no—
173C2×C22⋊C4—1271no—
174C2×C22⋊C4—1271no—
175C2×C42—1271no—
176C22⋊C8—22no—
177C22⋊C8—2542no—
178C22⋊C8—2542no—
179C24—1271no—
180C24C4×D12711yescharacteristic
181C2×D4—2542no—
182C2×D4—2542no—
183C2×D4—2542no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—1271no—
191C22×C4—1271no—
192C22×C4—1272no—
193C22×C4—1272no—
194C22×C4—1271no—
195C22×C4—2541no—
196C22×C4—2541no—
197C22×C4—2541no—
198C22×C4—2541no—
199C22×C4C4×D12711yescharacteristic
200C22×C4C22×D12711yescharacteristic
201M16—2548no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—1272no—
208C2×C8—2541no—
209C2×C8—2541no—
210C2×C8—2541no—
211C2×C8—2541no—
212C2×C8—2541no—
213C2×C8C22⋊D12712yescomplemented
214C4⋊C4—2542no—
215C22⋊C4—2542no—
216C22⋊C4—2542no—
217C42—2541no—
218C42—2541no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—1272no—
226C23—2541no—
227C23—2541no—
228C23—2541no—
229C23—2541no—
230C23—2541no—
231C23—2541no—
232C23#1452323112yescomplemented
233C23C2×C4×D12711yescharacteristic
234D4—2548no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—1272no—
243C2×C4—1272no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4—2541no—
247C2×C4—2541no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4—2541no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4D4×D12711yescharacteristic
256C2×C4D4×D12711yescharacteristic
257C2×C4D4×D12711yescharacteristic
258C2×C4C2×C22⋊D12711yescharacteristic
259C2×C4#1452324711yescharacteristic
260C2×C4#1452324811yescharacteristic
261C8—22no—
262C8—41no—
263C8—5081no—
264C8—2542no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—5081no—
273C22—5081no—
274C22—5081no—
275C22—5081no—
276C22—2542no—
277C22—2542no—
278C22—2542no—
279C22C22⋊C4×D12711yescharacteristic
280C22(C22⋊D127)⋊C411yescharacteristic
281C22C4×C22⋊D12711yescharacteristic
282C22M16×D12711yescharacteristic
283C22C2×#1452323111yescharacteristic
284C22#1686121511yescharacteristic
285C22#1686121711yescharacteristic
286C4—41no—
287C4—5081no—
288C4—2542no—
289C4—2542no—
290C4#1686117711yescharacteristic
291C4#1686118011yescharacteristic
292C4#1686118211yescharacteristic
293C4#1686119111yescharacteristic
294C2—42no—
295C2—5082no—
296C2#2254975211yescharacteristic
297C2#2254983811yescharacteristic
298C2#2254984111yescharacteristic
299C2#2254994411yescharacteristic
300C2#2254994711yescharacteristic
301C2#2254995111yescharacteristic
302C2#2254995211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×D127
commutator, socleC23×C127C22×C4
FittingC2×C22⋊C8×C127C2
prime radicalC22×C22⋊D127C4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1870
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#14523231C2semidirect
3. C2×C22⋊(C127⋊C8)C2semidirect
4. C2×(C2×D127)⋊C8C2semidirect
5. #22549630C2semidirect
6. #22549630C2semidirect
7. C2×C22⋊C8×C127C2semidirect
46. C2×C127⋊C8D4semidirect
66. C2×C8×C127D4semidirect
83. C2×Dic127M16semidirect
96. C23×C127M16semidirect
152. C127#1870semidirect
157. C2×C22⋊C8D127semidirect
213. C2×C8C22⋊D127semidirect
232. C23#14523231semidirect

Maximal subgroups

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

subgroupindex
C22×#145232312
C2×C4×C22⋊D1272
C2×C22⋊(C127⋊C8)2
C2×(C2×D127)⋊C82
#225496302
C2×C22⋊C8×C1272
#1870127