Finite Group Databasebeta

← #30639140

Subgroups — #30639140

Order 16256.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861192C211yesmaximal, characteristic
2#22550349C211yesmaximal, characteristic
3C2×#16861123C211yesmaximal, characteristic
4#22549739C211yesmaximal, characteristic, complemented
5#22549643C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7#381×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9#16861192—22no—
10#16861192C414yescomplemented
11C4×C22⋊D127—22no—
12(C22⋊D127)⋊C4—22no—
13#16861165—21no—
14C2×Dic127⋊C4—21no—
15C2×C22⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127C2211yescharacteristic
17C2×C22⋊Dic127C2211yescharacteristic
18C2×(C2×D127)⋊C4—21no—
19C22⋊C4×D127—21no—
20#16861123—22no—
21C127⋊#95—22no—
22C23×C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25#95×C127—22no—
26C23×D127D411yescharacteristic
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127C2×C412yes—
32C2×C22⋊D127D412yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C22×Dic127D411yescharacteristic
36C22×Dic127D412yes—
37C2×C4×D127—22no—
38Dic127⋊C4—22no—
39Dic127⋊C4—24no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127C2×C412yescomplemented
46(C2×D127)⋊C4—22no—
47(C2×D127)⋊C4—24no—
48C4×Dic127—22no—
49C24×C127C2311yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2×C412yescomplemented
58C22×C4×C127D412yes—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—42no—
66C22×D127C22⋊C412yes—
67C22⋊D127—22no—
68C22⋊D127—22no—
69C22⋊D127—22no—
70C22⋊D127—22no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—42no—
76C2×Dic127—42no—
77C2×Dic127—42no—
78C2×Dic127—41no—
79C2×Dic127C22⋊C412yes—
80C4×D127—82no—
81C23×C127—42no—
82C23×C127C22×C411yescharacteristic
83C23×C127C2×D411yescharacteristic
84C23×C127C2×D411yescharacteristic
85C23×C127C2×D411yescharacteristic
86C23×C127C2×D411yescharacteristic
87C23×C127#4111yescharacteristic
88C23×C127#4111yescharacteristic
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—22no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127C2×C22⋊C411yescharacteristic
108C22×C127C4×D411yescharacteristic
109C22×C127C4×D411yescharacteristic
110C22×C127#12011yescharacteristic
111C22×C127#12111yescharacteristic
112C22×C127#12111yescharacteristic
113C22×C127#12311yescharacteristic
114C2×D127—22no—
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—81no—
118C4×C127—42no—
119C4×C127—42no—
120C4×C127—81no—
121Dic127—22no—
122Dic127—42no—
123Dic127—42no—
124Dic127—81no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—21no—
130C2×C127—21no—
131C2×C127#38611yescharacteristic
132C2×C127#45812yes—
133D127—42no—
134#1890—1271nomaximal, Sylow
135C127#189011yescharacteristic, complemented, Sylow
136C2×#123—1271no—
137#518—1271no—
138C2×#99—1271no—
139#381—1271no—
140#381D12711yescharacteristic, complemented
141#342—1271no—
142#342—1271no—
143C22×D4—1271no—
144C23×C4C2×D12711yescharacteristic
145#123—1274no—
146#123—2542no—
147C4×D4—2542no—
148C4×D4—2542no—
149#117—2541no—
150C2×C4⋊C4—2541no—
151C2×C22⋊C4—1271no—
152C2×C22⋊C4—1271no—
153C2×C22⋊C4—1271no—
154C2×C22⋊C4—2541no—
155C2×C22⋊C4—2541no—
156C2×C22⋊C4C2×D12711yescharacteristic
157C2×C22⋊C4C2×D12711yescharacteristic
158#99—2542no—
159#95—22no—
160#95—2542no—
161C24—1271no—
162C24C22×D12711yescharacteristic
163C2×D4—1272no—
164C2×D4—1272no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C2×D4—2541no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1271no—
177C22×C4—1272no—
178C22×C4—2542no—
179C22×C4—2541no—
180C22×C4—2541no—
181C22×C4C4×D12712yescomplemented
182C22×C4C22⋊D12712yes—
183C4⋊C4—2544no—
184C4⋊C4—2542no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—1272no—
190C22⋊C4—2544no—
191C22⋊C4—2542no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C22⋊C4—2541no—
197C42—2542no—
198C23—42no—
199C23—1272no—
200C23—5082no—
201C23—2541no—
202C23—2541no—
203C23C2×C4×D12711yescharacteristic
204C23D4×D12711yescharacteristic
205C23D4×D12711yescharacteristic
206C23D4×D12711yescharacteristic
207C23C2×C22⋊D12711yescharacteristic
208C23#1452324711yescharacteristic
209C23#1452324811yescharacteristic
210D4—2542no—
211D4—2542no—
212D4—2542no—
213D4—2542no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—1272no—
226C2×C4—10162no—
227C2×C4—5082no—
228C2×C4—5081no—
229C2×C4—5082no—
230C2×C4—5082no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2541no—
234C2×C4—2541no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—10161no—
243C22—5082no—
244C22—5082no—
245C22—2542no—
246C22C22⋊C4×D12711yescharacteristic
247C22(C22⋊D127)⋊C411yescharacteristic
248C22C4×C22⋊D12711yescharacteristic
249C22#1686117711yescharacteristic
250C22#1686118011yescharacteristic
251C22#1686118211yescharacteristic
252C22#1686119111yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—10161no—
257C4—5082no—
258C4—5082no—
259C4—2542no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—5082no—
267C2#2254975211yescharacteristic
268C2#2255006512yes—
subgroupG/H
centerC22#16861177
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#381×C127C2
socleC22×C127#120
prime radicalC22×C22⋊D127C22
cosocleC24×C127C23
hypercenter#381D127
hypocenterC127#1890
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #22549739C2semidirect
5. #22549643C2semidirect
6. #22549642C2semidirect
7. #381×C127C2semidirect
10. #16861192C4semidirect
45. C22⋊Dic127C2×C4semidirect
57. C22×C4×C127C2×C4semidirect
135. C127#1890semidirect
140. #381D127semidirect
181. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611922
#225503492
C2×#168611232
#225497392
#225496432
#225496422
#381×C1272
#1890127