Finite Group Databasebeta

← #30639185

Subgroups — #30639185

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×#16861190C211yesmaximal, characteristic
2C2×#16861165C211yesmaximal, characteristic, complemented
3C2×#16861124C211yesmaximal, characteristic
4#22549737C211yesmaximal, characteristic, complemented
5#22549642C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7C2×#99×C127C211yesmaximal, characteristic
8C23×Dic127C2211yescharacteristic
9#16861190—22no—
10#16861190—21no—
11#16861190—21no—
12#16861165—22no—
13#16861165—22no—
14C2×Dic127⋊C4—21no—
15C2×Dic127⋊C4C412yescomplemented
16C2×C22⋊Dic127C412yescomplemented
17C2×C22⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127C2211yescharacteristic
20C2×C4×Dic127—21no—
21#16861124—22no—
22C127⋊#95—22no—
23C22×D4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
26#99×C127—22no—
27C22×Dic127—21no—
28C22×Dic127—21no—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127C2×C412yes—
34C22×Dic127C2×C412yes—
35C22×Dic127D412yes—
36C22×Dic127D411yescharacteristic
37C22×Dic127D411yescharacteristic
38Dic127⋊C4—24no—
39Dic127⋊C4—22no—
40Dic127⋊C4—22no—
41C22⋊Dic127—22no—
42C22⋊Dic127—22no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127D412yes—
49C4×Dic127—24no—
50C24×C127C2×C411yescharacteristic
51C24×C127C2311yescharacteristic
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127C2×C411yescharacteristic
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127—21no—
64C22⋊C4×C127D412yes—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—22no—
70C2×Dic127—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—
80C2×Dic127C22⋊C412yes—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—21no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127C22×C411yescharacteristic
87C23×C127C2×D411yescharacteristic
88C23×C127C2×D411yescharacteristic
89C23×C127C2×D411yescharacteristic
90C23×C127C2×D411yescharacteristic
91C23×C127#4111yescharacteristic
92C23×C127#4111yescharacteristic
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—41no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
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×C127—41no—
112C22×C127—42no—
113C22×C127—81no—
114C22×C127—81no—
115C22×C127C2×C22⋊C411yescharacteristic
116C22×C127C4×D411yescharacteristic
117C22×C127C4×D411yescharacteristic
118C22×C127#12011yescharacteristic
119C22×C127#12111yescharacteristic
120C22×C127#12111yescharacteristic
121C22×C127#12311yescharacteristic
122C4×C127—42no—
123C4×C127—41no—
124C4×C127—81no—
125Dic127—22no—
126Dic127—22no—
127Dic127—42no—
128Dic127—42no—
129Dic127—81no—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—21no—
135C2×C127—21no—
136C2×C127—81no—
137C2×C127#38611yescharacteristic
138C2×C127#41012yes—
139#1899—1271nomaximal, Sylow
140C127#189911yescharacteristic, complemented, Sylow
141C2×#123—1271no—
142C2×#117—1271no—
143C2×#99—1271no—
144C2×#99D12711yescharacteristic
145#381—1271no—
146#342—1271no—
147#342—1271no—
148C22×D4C2×D12711yescharacteristic
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⋊C4—1271no—
161C2×C22⋊C4C2×D12711yescharacteristic
162C2×C22⋊C4C2×D12711yescharacteristic
163C2×C42—2541no—
164#99—22no—
165#99—2542no—
166#95—2542no—
167C24C4×D12711yescharacteristic
168C24C22×D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—1272no—
178C22×C4—1272no—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1272no—
182C22×C4—2541no—
183C22×C4—2541no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4—2541no—
187C22×C4—2541no—
188C22×C4C4×D12711yescharacteristic
189C4⋊C4—2542no—
190C4⋊C4—2542no—
191C4⋊C4—2544no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—1272no—
196C22⋊C4—2542no—
197C22⋊C4—2542no—
198C22⋊C4—2541no—
199C22⋊C4—2541no—
200C22⋊C4—2541no—
201C22⋊C4—2541no—
202C22⋊C4—2541no—
203C22⋊C4C22⋊D12712yes—
204C42—2544no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—42no—
209C23—42no—
210C23C2×C4×D12711yescharacteristic
211C23D4×D12711yescharacteristic
212C23D4×D12711yescharacteristic
213C23D4×D12711yescharacteristic
214C23C2×C22⋊D12711yescharacteristic
215C23#1452324711yescharacteristic
216C23#1452324811yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—1272no—
229C2×C4—1272no—
230C2×C4—5082no—
231C2×C4—5081no—
232C2×C4—5082no—
233C2×C4—5082no—
234C2×C4—2542no—
235C2×C4—2542no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4—2541no—
242C2×C4—2542no—
243C2×C4—2542no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—81no—
254C22—81no—
255C22C22⋊C4×D12711yescharacteristic
256C22(C22⋊D127)⋊C411yescharacteristic
257C22C4×C22⋊D12711yescharacteristic
258C22#1686117711yescharacteristic
259C22#1686118011yescharacteristic
260C22#1686118211yescharacteristic
261C22#1686119111yescharacteristic
262C4—42no—
263C4—41no—
264C4—81no—
265C4—10161no—
266C4—5082no—
267C4—5082no—
268C4—2542no—
269C4—2542no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—81no—
277C2#2254975211yescharacteristic
278C2#2254984812yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC24×C127C23
hypercenterC2×#99D127
hypocenterC127#1899
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
4. #22549737C2semidirect
5. #22549642C2semidirect
6. #22549642C2semidirect
15. C2×Dic127⋊C4C4semidirect
16. C2×C22⋊Dic127C4semidirect
140. C127#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#168611902
C2×#168611652
C2×#168611242
#225497372
#225496422
C2×#99×C1272
#1899127