Finite Group Databasebeta

← #30639142

Subgroups — #30639142

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×#16861193C211yesmaximal, characteristic
2#22550349C211yesmaximal, characteristic
3C2×#16861123C211yesmaximal, characteristic
4#22549737C211yesmaximal, characteristic, complemented
5#22549737C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10#16861193—22no—
11#16861193C414yescomplemented
12C4×C22⋊D127—22no—
13(C22⋊D127)⋊C4—22no—
14#16861165—21no—
15C2×C4⋊Dic127—21no—
16C2×C22⋊Dic127C2211yescharacteristic
17C2×C22⋊Dic127C2211yescharacteristic
18C2×(C2×D127)⋊C4—21no—
19C22⋊C4×D127—21no—
20#16861123—22no—
21C127⋊#95—22no—
22C127⋊#95—22no—
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
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×Dic127—21no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127C2×C412yescomplemented
41C22×Dic127D411yescharacteristic
42C22×Dic127D412yes—
43C2×C4×D127—22no—
44C4⋊Dic127—24no—
45Dic127⋊C4—22no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4—24no—
52C4×Dic127—22no—
53C24×C127C2311yescharacteristic
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127D412yes—
57C22⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127C2×C412yescomplemented
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—22no—
76C2×Dic127—22no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C2×Dic127—42no—
82C2×Dic127—42no—
83C2×Dic127—41no—
84C2×Dic127C22⋊C412yes—
85C4×D127—82no—
86C23×C127—42no—
87C23×C127C22×C411yescharacteristic
88C23×C127C2×D411yescharacteristic
89C23×C127C2×D411yescharacteristic
90C23×C127C2×D411yescharacteristic
91C23×C127C2×D411yescharacteristic
92C23×C127#4111yescharacteristic
93C23×C127#4111yescharacteristic
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—42no—
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#381—1271no—
141#342—1271no—
142#342D12711yescharacteristic, complemented
143C22×D4—1271no—
144C23×C4—1271no—
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—2541no—
154C2×C22⋊C4—2541no—
155C2×C22⋊C4C2×D12711yescharacteristic
156C2×C22⋊C4C2×D12711yescharacteristic
157C2×C22⋊C4C2×D12711yescharacteristic
158#99—2542no—
159#95—2542no—
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—1271no—
172C22×C4—1272no—
173C22×C4—1272no—
174C22×C4—2542no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4—2541no—
181C22×C4—2541no—
182C22×C4D50812yes—
183C4⋊C4—2544no—
184C4⋊C4—2542no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—2544no—
191C22⋊C4—2542no—
192C22⋊C4—2541no—
193C22⋊C4—2541no—
194C22⋊C4—2541no—
195C22⋊C4—2541no—
196C22⋊C4C4×D12712yescomplemented
197C42—2542no—
198C23—42no—
199C23—1272no—
200C23—5082no—
201C23—2541no—
202C23—2541no—
203C23C2×C4×D12711yescharacteristic
204C23D4×D12711yescharacteristic
205C23D4×D12711yescharacteristic
206C23D4×D12711yescharacteristic
207C23C2×D50811yescharacteristic
208C23#1452324811yescharacteristic
209C23#1452324911yescharacteristic
210D4—2542no—
211D4—2542no—
212D4—2542no—
213D4—2542no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—1272no—
221C2×C4—10162no—
222C2×C4—5082no—
223C2×C4—5081no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—5081no—
227C2×C4—5081no—
228C2×C4—5082no—
229C2×C4—2542no—
230C2×C4—2542no—
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#1686117111yescharacteristic
248C22C4×D50811yescharacteristic
249C22#1686117911yescharacteristic
250C22#1686118211yescharacteristic
251C22#1686118411yescharacteristic
252C22#1686119411yescharacteristic
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#2254975411yescharacteristic
268C2#2255006712yes—
subgroupG/H
centerC22#16861179
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
prime radicalC22×C22⋊D127C22
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1890
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #22549737C2semidirect
5. #22549737C2semidirect
6. #22549642C2semidirect
7. #342×C127C2semidirect
11. #16861193C4semidirect
40. C22×Dic127C2×C4semidirect
62. C22⋊C4×C127C2×C4semidirect
135. C127#1890semidirect
142. #342D127semidirect
196. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611932
#225503492
C2×#168611232
#225497372
#225496422
#342×C1272
#1890127