Finite Group Databasebeta

← #30637646

Subgroups — #30637646

Order 16256.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#22550434C211yesmaximal, characteristic, complemented
2#22550010C211yesmaximal, characteristic, complemented
3#22549784C211yesmaximal, characteristic, complemented
4#22549611C211yesmaximal, characteristic, complemented
5#22549610C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#447×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C2×D4×D127—21no—
10C2×#14523233—21no—
11#16861194—21no—
12#16861191—21no—
13#16861183C2211yescharacteristic
14#16861181—21no—
15#16861179—21no—
16#16861177—21no—
17C2×C22⋊Dic127—21no—
18C2×(C2×D127)⋊C4C2211yescharacteristic
19#16861130—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic, complemented
21C127⋊#95C2211yescharacteristic, complemented
22C22×D4×C127C2211yescharacteristic, complemented
23C2×D8×C127—21no—
24#121×C127C2211yescharacteristic
25#102×C127—21no—
26C22⋊C8×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic
29C2×D508—21no—
30C2×D508D411yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—42no—
35C2×C22⋊D127—42no—
36C2×C22⋊D127—42no—
37D4×D127—42no—
38D4×D127—42no—
39C22×Dic127—21no—
40C22×Dic127D411yescharacteristic
41C2×C4×D127—21no—
42#14523233—42no—
43C2×C127⋊C8—21no—
44C4⋊Dic127D411yescharacteristic, complemented
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—42no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—21no—
51(C2×D127)⋊C4—21no—
52C24×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—42no—
57C2×D4×C127D411yescharacteristic, complemented
58C22×C4×C127C2311yescharacteristic
59D8×C127—42no—
60C2×C8×C127—21no—
61C4⋊C4×C127D411yescharacteristic, complemented
62C22⋊C4×C127—21no—
63C22×D127—21no—
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—42no—
67C22×D127—41no—
68C22×D127—41no—
69C22×D127—81no—
70C22×D127—81no—
71C22×D127D812yescomplemented
72D508—41no—
73D508—42no—
74C22⋊D127—42no—
75C22⋊D127—48no—
76C22⋊D127—41no—
77C22⋊D127—41no—
78C22⋊D127—81no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—41no—
83C2×Dic127—41no—
84C2×Dic127—41no—
85C4×D127—42no—
86C127⋊C8—41no—
87C23×C127—22no—
88C23×C127—21no—
89C23×C127—41no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127C2×D411yescharacteristic
96D4×C127—22no—
97D4×C127—22no—
98D4×C127—41no—
99D4×C127—41no—
100D4×C127—41no—
101D4×C127—41no—
102D4×C127—41no—
103C2×C4×C127—21no—
104C2×C4×C127—41no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C8×C127—41no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—42no—
111C22×C127—42no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127—81no—
117C22×C127—81no—
118C22×C127—81no—
119C22×C127—81no—
120C22×C127#12011yescharacteristic
121C22×C127C2×D811yescharacteristic
122C22×C127AGL(1,ℤ/8)11yescharacteristic
123C2×D127—22no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—42no—
127C2×D127—41no—
128C2×D127—41no—
129C2×D127—41no—
130C2×D127—81no—
131C2×D127—84no—
132C4×C127—21no—
133C4×C127—21no—
134C4×C127—41no—
135Dic127—42no—
136Dic127—41no—
137Dic127—41no—
138C2×C127—21no—
139C2×C127—21no—
140C2×C127—81no—
141C2×C127—82no—
142C2×C127#44711yescharacteristic
143C2×C127#45311yescharacteristic
144C2×C127B(4,𝔽2)11yescharacteristic
145D127—42no—
146D127—81no—
147#1595—1271nomaximal, Sylow
148C127#159511yescharacteristic, complemented, Sylow
149#534—1271no—
150#447—1271no—
151#447D12711yescharacteristic, complemented
152#392—1271no—
153#327—1271no—
154#327—1271no—
155#323—1271no—
156C22×D4—2541no—
157C22×D4—2541no—
158C22×D4C2×D12711yescharacteristic, complemented
159C2×D8—21no—
160C2×D8—2541no—
161#123—2541no—
162#123—2541no—
163#121—1271no—
164#121—2541no—
165#121C2×D12711yescharacteristic
166#120—2541no—
167#120—2541no—
168C2×C22⋊C4—1271no—
169C2×C22⋊C4—2541no—
170#102—21no—
171#102—2541no—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic, complemented
174#95—1271no—
175C24—21no—
176C24—1271no—
177C24—2541no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—1271no—
183C2×D4—5082no—
184C2×D4—5082no—
185C2×D4—5082no—
186C2×D4—5082no—
187C2×D4—5082no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4—2541no—
192C2×D4C22⋊D12711yescharacteristic, complemented
193C22×C4—1271no—
194C22×C4—2541no—
195C22×C4—2541no—
196C22×C4C22×D12711yescharacteristic
197D8—42no—
198D8—5082no—
199C2×C8—21no—
200C2×C8—2541no—
201C4⋊C4—1271no—
202C4⋊C4—2541no—
203C4⋊C4C22⋊D12711yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—5082no—
206C22⋊C4—2541no—
207C22⋊C4—2541no—
208C22⋊C4—2541no—
209C22⋊C4—2541no—
210C22⋊C4—2541no—
211C23—22no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—1272no—
220C23—10161no—
221C23—10161no—
222C23—5081no—
223C23—5081no—
224C23—5082no—
225C23—2541no—
226C23—2541no—
227C23—2541no—
228C23D4×D12711yescharacteristic
229D4—22no—
230D4—22no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—10161no—
237D4—5081no—
238D4—5082no—
239D4—5088no—
240D4—5082no—
241D4—5081no—
242D4—5081no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—5082no—
246C2×C4—5081no—
247C2×C4—5081no—
248C2×C4—5081no—
249C2×C4—2542no—
250C2×C4—2541no—
251C2×C4—2541no—
252C2×C4D4×D12711yescharacteristic
253C2×C4C2×C22⋊D12711yescharacteristic
254C8—41no—
255C8—5081no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—10161no—
269C22—10164no—
270C22—5082no—
271C22—5081no—
272C22—5081no—
273C22—5081no—
274C22—5081no—
275C22—5081no—
276C22—2542no—
277C22#1686117711yescharacteristic
278C22D8×D12711yescharacteristic
279C22#1686123311yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—5082no—
284C4—5081no—
285C4—5081no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—81no—
290C2—10161no—
291C2—5082no—
292C2#2255000811yescharacteristic
293C2#2255004211yescharacteristic
294C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#447×C127C2
socleC22×C127#120
hypercenter#447D127
hypocenterC127#1595
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22550434C2semidirect
2. #22550010C2semidirect
3. #22549784C2semidirect
4. #22549611C2semidirect
5. #22549610C2semidirect
6. #22549599C2semidirect
7. #447×C127C2semidirect
20. C22⋊(C127⋊C8)C22semidirect
21. C127⋊#95C22semidirect
22. C22×D4×C127C22semidirect
26. C22⋊C8×C127C22semidirect
30. C2×D508D4semidirect
44. C4⋊Dic127D4semidirect
57. C2×D4×C127D4semidirect
61. C4⋊C4×C127D4semidirect
71. C22×D127D8semidirect
148. C127#1595semidirect
151. #447D127semidirect
158. C22×D4C2×D127semidirect
173. C22⋊C8C2×D127semidirect
192. C2×D4C22⋊D127semidirect
203. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504342
#225500102
#225497842
#225496112
#225496102
#225495992
#447×C1272
#1595127