Finite Group Databasebeta

← #30639187

Subgroups — #30639187

Order 16256.

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

normal subgroups only

This group has 378 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
2C2×#16861165C211yesmaximal, characteristic, complemented
3C2×#16861123C211yesmaximal, characteristic
4C2×#16861123C211yesmaximal, characteristic
5#22549737C211yesmaximal, characteristic, complemented
6#22549643C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10#16861193—22no—
11#16861193—21no—
12#16861193—21no—
13#16861165—22no—
14#16861165—22no—
15C2×C4⋊Dic127C412yescomplemented
16C2×Dic127⋊C4—21no—
17C2×C22⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127C2211yescharacteristic
19C2×(C2×D127)⋊C4C412yescomplemented
20C2×C4×Dic127—21no—
21#16861123—22no—
22#16861123—22no—
23C127⋊#95—22no—
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2211yescharacteristic
27C23×D127C2×C411yescharacteristic
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127C2×C411yescharacteristic
41C22×Dic127C2×C412yes—
42C22×Dic127D412yes—
43C22×Dic127D411yescharacteristic
44C22×Dic127D411yescharacteristic
45C4⋊Dic127—22no—
46C4⋊Dic127—22no—
47Dic127⋊C4—24no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—21no—
51C22⋊Dic127D412yes—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—24no—
55C24×C127C2311yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2×C412yes—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127—21no—
64C22⋊C4×C127D412yes—
65C22×D127—21no—
66C22×D127—21no—
67C22×D127—21no—
68C22×D127—42no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—42no—
86C2×Dic127—42no—
87C2×Dic127—41no—
88C2×Dic127C22⋊C412yes—
89C2×Dic127C22⋊C412yes—
90C23×C127—42no—
91C23×C127C22×C411yescharacteristic
92C23×C127C2×D411yescharacteristic
93C23×C127C2×D411yescharacteristic
94C23×C127C2×D411yescharacteristic
95C23×C127C2×D411yescharacteristic
96C23×C127#4111yescharacteristic
97C23×C127#4111yescharacteristic
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—42no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—41no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—22no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127C2×C22⋊C411yescharacteristic
112C22×C127C4×D411yescharacteristic
113C22×C127C4×D411yescharacteristic
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C22×C127#12111yescharacteristic
117C22×C127#12311yescharacteristic
118C2×D127—41no—
119C2×D127—42no—
120C2×D127—81no—
121C2×D127—81no—
122C4×C127—42no—
123C4×C127—42no—
124C4×C127—81no—
125Dic127—22no—
126Dic127—22no—
127Dic127—42no—
128Dic127—41no—
129Dic127—81no—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—21no—
135C2×C127—21no—
136C2×C127#38611yescharacteristic
137C2×C127#41012yes—
138D127—81no—
139#1899—1271nomaximal, Sylow
140C127#189911yescharacteristic, complemented, Sylow
141C2×#123—1271no—
142C2×#117—1271no—
143C2×#99—1271no—
144C2×#99—1271no—
145#381—1271no—
146#342—1271no—
147#342D12711yescharacteristic, complemented
148C22×D4—1271no—
149C23×C4—1271no—
150#123—2542no—
151#123—2541no—
152#123—2541no—
153#117—2542no—
154#117—2542no—
155C2×C4⋊C4—1272no—
156C2×C4⋊C4—2541no—
157C2×C22⋊C4—1271no—
158C2×C22⋊C4—1272no—
159C2×C22⋊C4—1271no—
160C2×C22⋊C4C2×D12711yescharacteristic
161C2×C22⋊C4C2×D12711yescharacteristic
162C2×C22⋊C4C2×D12711yescharacteristic
163C2×C42—2541no—
164#99—2542no—
165#99—2542no—
166#95—2542no—
167C24—1271no—
168C24C22×D12711yescharacteristic
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4—2541no—
174C2×D4—2541no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—1272no—
178C22×C4—1272no—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1271no—
182C22×C4—2541no—
183C22×C4—2541no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4—2541no—
187C22×C4—2541no—
188C22×C4C4×D12712yes—
189C4⋊C4—2542no—
190C4⋊C4—2542no—
191C4⋊C4—2544no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—1272no—
198C22⋊C4—2542no—
199C22⋊C4—2542no—
200C22⋊C4—2541no—
201C22⋊C4—2541no—
202C22⋊C4—2541no—
203C22⋊C4D50812yes—
204C42—2544no—
205C23—42no—
206C23—5082no—
207C23—2541no—
208C23—2541no—
209C23—2541no—
210C23C2×C4×D12711yescharacteristic
211C23D4×D12711yescharacteristic
212C23D4×D12711yescharacteristic
213C23D4×D12711yescharacteristic
214C23C2×D50811yescharacteristic
215C23#1452324811yescharacteristic
216C23#1452324911yescharacteristic
217D4—5081no—
218D4—5081no—
219D4—5081no—
220D4—5081no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—1272no—
228C2×C4—1272no—
229C2×C4—5082no—
230C2×C4—5081no—
231C2×C4—5082no—
232C2×C4—2541no—
233C2×C4—2541no—
234C2×C4—2541no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4—2541no—
242C2×C4—2541no—
243C2×C4—2542no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—10161no—
252C22—10161no—
253C22—5082no—
254C22—5081no—
255C22C22⋊C4×D12711yescharacteristic
256C22#1686117111yescharacteristic
257C22C4×D50811yescharacteristic
258C22#1686117911yescharacteristic
259C22#1686118211yescharacteristic
260C22#1686118411yescharacteristic
261C22#1686119411yescharacteristic
262C4—42no—
263C4—42no—
264C4—81no—
265C4—10161no—
266C4—5082no—
267C4—5081no—
268C4—2542no—
269C4—2542no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—10161no—
277C2#2254975411yescharacteristic
278C2#2254984812yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C22
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1899
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
5. #22549737C2semidirect
6. #22549643C2semidirect
7. #342×C127C2semidirect
15. C2×C4⋊Dic127C4semidirect
19. C2×(C2×D127)⋊C4C4semidirect
140. C127#1899semidirect
147. #342D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611932
C2×#168611652
C2×#168611232
#225497372
#225496432
#342×C1272
#1899127