Finite Group Databasebeta

← #30639759

Subgroups — #30639759

Order 16256.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861240C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549885C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#124×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11C2×#14523238—21no—
12C2×#14523235—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861205—21no—
16#16861196—21no—
17#16861180—22no—
18#16861178—22no—
19#16861168C2211yescharacteristic
20#16861137—22no—
21#16861133—22no—
22#16861126—22no—
23C22×Q8×C127C2211yescharacteristic
24C22×D4×C127C2211yescharacteristic
25#124×C127—22no—
26#124×C127—22no—
27C2×C22⋊C4×C127—22no—
28C2×C42×C127C2211yescharacteristic
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254D411yescharacteristic
34C2×D508D411yescharacteristic
35C2×C22⋊D127—24no—
36C2×C22⋊D127—21no—
37C2×C4×D127—21no—
38#14523238—24no—
39#14523235—24no—
40#14523232—21no—
41#14523232—21no—
42#14523232—21no—
43#14523232D412yescomplemented
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C4⋊Dic127D412yes—
47C22⋊Dic127—24no—
48C22⋊Dic127—21no—
49C4×Dic127—21no—
50C24×C127D412yes—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127D412yescomplemented
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66C22⋊C4×C127—28no—
67C42×C127—21no—
68C42×C127D412yescomplemented
69C22×D127—21no—
70Dic254—21no—
71Dic254—21no—
72D508—21no—
73D508—21no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C22⋊D127—84no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C4×D127—41no—
80C4×D127—41no—
81C127⋊C8—22no—
82C127⋊C8—41no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—44no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—22no—
89Q8×C127—22no—
90Q8×C127—41no—
91Q8×C127—41no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127#4111yescharacteristic
107C22×C127—22no—
108C22×C127—44no—
109C22×C127—42no—
110C22×C127—44no—
111C22×C127—82no—
112C22×C127#12011yescharacteristic
113C22×C127#12011yescharacteristic
114C22×C127#12111yescharacteristic
115C2×D127—41no—
116C2×D127—42no—
117C4×C127—42no—
118C4×C127—42no—
119C4×C127#12011yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12711yescharacteristic
123Dic127—41no—
124Dic127—42no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—82no—
128C2×C127#39211yescharacteristic
129C2×C127#45512yes—
130D127—81no—
131#2004—1271nomaximal, Sylow
132C127#200411yescharacteristic, complemented, Sylow
133C2×#137—1271no—
134#535—1271no—
135C2×#124D12711yescharacteristic, complemented
136C2×#104—1271no—
137#419—1271no—
138C2×#100—1271no—
139#339—1271no—
140C2×#41—1271no—
141C22×Q8C2×D12711yescharacteristic
142C22×D4C2×D12711yescharacteristic
143#137—2542no—
144#137—2542no—
145C2×Q16—2541no—
146C2×SD16—2541no—
147C2×M16—1271no—
148C2×M16—1271no—
149#127—2541no—
150#124—22no—
151#124—22no—
152#124—2541no—
153#121—2542no—
154#120—2542no—
155#117—1271no—
156C2×C22⋊C4—22no—
157C2×C42C2×D12711yescharacteristic
158#104—2542no—
159#103—2542no—
160#100—2542no—
161C24C22⋊D12712yes—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165#41—2541no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1271no—
172C2×Q8C22⋊D12712yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—1271no—
180C2×D4—2544no—
181C2×D4—2541no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—2541no—
185C22×C4C22×D12711yescharacteristic
186Q16—2544no—
187SD16—2544no—
188M16—1272no—
189M16—2541no—
190M16—2541no—
191M16—2541no—
192C2×C8—2541no—
193C2×C8—2541no—
194C4⋊C4—1272no—
195C22⋊C4—28no—
196C22⋊C4—2544no—
197C22⋊C4—2541no—
198C42—21no—
199C42—2541no—
200C42C22⋊D12712yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—2541no—
206C23C2×C22⋊D12711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—2541no—
212Q8—2541no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—10164no—
218D4—5081no—
219D4—5081no—
220D4—2541no—
221D4—2541no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—5081no—
228C2×C4—5081no—
229C2×C4—2542no—
230C2×C4—2541no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4#1452324711yescharacteristic
237C8—5081no—
238C8—2542no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—5081no—
245C22—5082no—
246C22#1686117711yescharacteristic
247C22#1686117811yescharacteristic
248C22#1686118111yescharacteristic
249C4—42no—
250C4—42no—
251C4—5081no—
252C4—5082no—
253C4#1686117711yescharacteristic
254C4#1686118011yescharacteristic
255C4#1686118111yescharacteristic
256C4#1686120511yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—10161no—
261C2#2254978411yescharacteristic
262C2#2255005312yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC22×C127#120
prime radical#22550443C2
hypercenterC2×#124D127
hypocenterC127#2004
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861137C2semidirect
4. #22549885C2semidirect
5. C2×#16861126C2semidirect
6. #22549636C2semidirect
7. C2×#124×C127C2semidirect
43. #14523232D4semidirect
56. C2×Q8×C127D4semidirect
68. C42×C127D4semidirect
132. C127#2004semidirect
135. C2×#124D127semidirect
172. C2×Q8C22⋊D127semidirect
200. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504432
C2×#168611372
#225498852
C2×#168611262
#225496362
C2×#124×C1272
#2004127