Finite Group Databasebeta

← #30647990

Subgroups — #30647990

Order 16256.

754 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16861239C212yesmaximal, complemented
2C2×#16861188C211yesmaximal, characteristic, complemented
3#22550038C214yesmaximal, complemented
4#22550032C212yesmaximal, complemented
5#22550027C212yesmaximal, complemented
6#22549877C212yesmaximal, complemented
7#22549840C211yesmaximal, characteristic, complemented
8C22×#41×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic, complemented
10#16861239—28no—
11C2×#14523238C2214yescomplemented
12C2×#14523236C2214yescomplemented
13C2×#14523232C2212yescomplemented
14#16861188—21no—
15#16861188—21no—
16#16861188C2212yes—
17#16861188C2212yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic, complemented
19C2×Dic127⋊C4—21no—
20C2×C22⋊Dic127—21no—
21#16861133C2214yescomplemented
22#16861130C2214yescomplemented
23C22⋊(C127⋊C8)C2212yes—
24C22⋊(C127⋊C8)C2212yescomplemented
25C2×#41×C127—28no—
26C2×#41×C127—21no—
27C2×#41×C127—21no—
28C2×#41×C127C2212yes—
29C2×#41×C127C2212yes—
30C22×Q8×C127C2211yescharacteristic
31C22×D4×C127—21no—
32C22×D4×C127C2211yescharacteristic
33C23×C4×C127—21no—
34C23×C4×C127C2211yescharacteristic
35C2×Dic254—21no—
36C2×Dic254—24no—
37C2×Dic254C2312yes—
38C22×Dic127—21no—
39C22×Dic127—21no—
40#14523238—28no—
41#14523236—28no—
42#14523232—24no—
43C2×C127⋊C8C2314yes—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—22no—
49C22⋊Dic127—22no—
50C24×C127—21no—
51C24×C127D411yescharacteristic
52#41×C127—28no—
53#41×C127—28no—
54#41×C127—24no—
55#41×C127—24no—
56#41×C127—24no—
57#41×C127D418yes—
58C2×Q8×C127—24no—
59C2×Q8×C127—21no—
60C2×Q8×C127C2312yes—
61C2×D4×C127—28no—
62C2×D4×C127—22no—
63C2×D4×C127—22no—
64C2×D4×C127—24no—
65C2×D4×C127—21no—
66C2×D4×C127C2312yes—
67C22×C4×C127—28no—
68C22×C4×C127—21no—
69C22×C4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—24no—
72C22×C4×C127D411yescharacteristic
73C22×C4×C127D411yescharacteristic
74C22×C4×C127D411yescharacteristic
75C22×C4×C127C2311yescharacteristic
76C22×C4×C127C2311yescharacteristic
77C22×C4×C127C2311yescharacteristic
78Dic254—24no—
79Dic254—42no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—42no—
83C2×Dic127—42no—
84C127⋊C8—24no—
85C23×C127—24no—
86C23×C127—22no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127—41no—
90C23×C127—42no—
91C23×C127—42no—
92C23×C127C2×D411yescharacteristic
93C23×C127C2×D411yescharacteristic
94C23×C127C2×D411yescharacteristic
95Q8×C127—24no—
96Q8×C127—22no—
97Q8×C127C2×D414yes—
98D4×C127—28no—
99D4×C127—24no—
100D4×C127—28no—
101D4×C127—22no—
102D4×C127C2×D414yes—
103C2×C4×C127—24no—
104C2×C4×C127—28no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—28no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D414yes—
116C2×C4×C127C2411yescharacteristic
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127—21no—
122C22×C127—24no—
123C22×C127—21no—
124C22×C127—21no—
125C22×C127—44no—
126C22×C127—48no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127#12012yes—
130C22×C127#13712yescomplemented
131C22×C127C22×D411yescharacteristic
132C4×C127—21no—
133C4×C127—24no—
134C4×C127—21no—
135C4×C127#12012yes—
136C4×C127C22×D412yes—
137Dic127—42no—
138Dic127—42no—
139C2×C127—22no—
140C2×C127—22no—
141C2×C127—41no—
142C2×C127—44no—
143C2×C127C2×#12011yescharacteristic
144C2×C127C2×#13711yescharacteristic
145C2×C127#57511yescharacteristic
146#3007—1271nomaximal, Sylow
147C127#300711yescharacteristic, complemented, Sylow
148C22×#41D12711yescharacteristic
149C2×#137—1272no—
150C2×C22⋊Q8—1271no—
151#452—1274no—
152#451—1272no—
153#450—1272no—
154#417—1272no—
155#407—1271no—
156C2×#41—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41C2×D12712yes—
160C2×#41C2×D12712yes—
161C22×Q8—1271no—
162C22×Q8C2×D12711yescharacteristic
163C22×D4—21no—
164C22×D4C2×D12711yescharacteristic
165C23×C4—21no—
166C23×C4C2×D12711yescharacteristic
167#137—2548no—
168C2×Q16—1274no—
169C2×SD16—1274no—
170C2×M16—1272no—
171C22⋊Q8—1272no—
172C22⋊Q8—1272no—
173C22⋊Q8—2541no—
174C22⋊Q8—2541no—
175C2×C4⋊C4—1271no—
176C2×C4⋊C4—2541no—
177C2×C22⋊C4—2541no—
178#103—1274no—
179#102—1274no—
180C22⋊C8—1272no—
181C22⋊C8—1272no—
182C24—21no—
183C24C22⋊D12711yescharacteristic
184#41—28no—
185#41—28no—
186#41—24no—
187#41—24no—
188#41—24no—
189#41C22⋊D12718yes—
190C2×Q8—24no—
191C2×Q8—21no—
192C2×Q8—1272no—
193C2×Q8—2544no—
194C2×Q8—2541no—
195C2×Q8C22×D12712yes—
196C2×D4—28no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4C22×D12712yes—
202C22×C4—28no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4C22⋊D12711yescharacteristic
210C22×C4C22⋊D12711yescharacteristic
211C22×C4C22⋊D12711yescharacteristic
212C22×C4C22×D12711yescharacteristic
213C22×C4C22×D12711yescharacteristic
214C22×C4C22×D12711yescharacteristic
215Q16—2548no—
216SD16—2548no—
217M16—2544no—
218C2×C8—1274no—
219C4⋊C4—1272no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C23—24no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23C2×C22⋊D12711yescharacteristic
233C23C2×C22⋊D12711yescharacteristic
234C23C2×C22⋊D12711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—5082no—
238Q8—2544no—
239Q8C2×C22⋊D12714yes—
240D4—28no—
241D4—24no—
242D4—28no—
243D4—22no—
244D4C2×C22⋊D12714yes—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—5082no—
255C2×C4—5082no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4C2×C22⋊D12711yescharacteristic
259C2×C4C2×C22⋊D12711yescharacteristic
260C2×C4C2×C22⋊D12714yes—
261C2×C4C2×C22⋊D12711yescharacteristic
262C2×C4C23×D12711yescharacteristic
263C8—2544no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—21no—
271C22—21no—
272C22—44no—
273C22—48no—
274C22—42no—
275C22—42no—
276C22#1686117812yes—
277C22#1686123912yescomplemented
278C22C22×C22⋊D12711yescharacteristic
279C4—21no—
280C4—24no—
281C4—21no—
282C4—5082no—
283C4—5082no—
284C4#1686117812yes—
285C4C22×C22⋊D12712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—44no—
290C2C2×#1686117811yescharacteristic
291C2C2×#1686123911yescharacteristic
292C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC22×#41×C127C2
socleC22×C127C22×D4
hypercenterC22×#41D127
hypocenterC127#3007
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861188C2semidirect
3. #22550038C2semidirect
4. #22550032C2semidirect
5. #22550027C2semidirect
6. #22549877C2semidirect
7. #22549840C2semidirect
9. C22×Dic254C22semidirect
11. C2×#14523238C22semidirect
12. C2×#14523236C22semidirect
13. C2×#14523232C22semidirect
17. #16861188C22semidirect
18. C2×C4⋊Dic127C22semidirect
21. #16861133C22semidirect
22. #16861130C22semidirect
24. C22⋊(C127⋊C8)C22semidirect
130. C22×C127#137semidirect
147. C127#3007semidirect
277. C22#16861239semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168611882
#225500382
#225500322
#225500272
#225498772
#225498402
C22×#41×C1272
#3007127