Finite Group Databasebeta

← #30639752

Subgroups — #30639752

Order 16256.

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

normal subgroups only

This group has 382 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×#16861235C211yesmaximal, characteristic
2#22550450C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5C2×#16861128C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7C2×#124×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861207—21no—
16#16861198—21no—
17#16861192—22no—
18#16861185—22no—
19#16861168C2211yescharacteristic
20#16861137—22no—
21#16861130—22no—
22#16861128—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—21no—
36C2×C4×D127—21no—
37#14523236—24no—
38#14523233—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232D412yes—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127D412yes—
46Dic127⋊C4—24no—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—24no—
49C4×Dic127—21no—
50C24×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127D412yes—
63C22×C4×C127—21no—
64C22×C4×C127D412yes—
65C22×C4×C127C2311yescharacteristic
66C22⋊C4×C127—28no—
67C42×C127—21no—
68C42×C127D412yes—
69C22×D127—21no—
70Dic254—21no—
71Dic254—21no—
72D508—21no—
73D508—21no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C2×Dic127—22no—
77C2×Dic127—21no—
78C4×D127—41no—
79C4×D127—41no—
80C127⋊C8—22no—
81C127⋊C8—41no—
82C23×C127—22no—
83C23×C127—44no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127C2×D411yescharacteristic
87Q8×C127—41no—
88Q8×C127—41no—
89Q8×C127—41no—
90Q8×C127—41no—
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—41no—
94D4×C127—41no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—42no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4111yescharacteristic
106C22×C127—22no—
107C22×C127—44no—
108C22×C127—42no—
109C22×C127—82no—
110C22×C127—82no—
111C22×C127#12011yescharacteristic
112C22×C127#12011yescharacteristic
113C22×C127#12111yescharacteristic
114C2×D127—41no—
115C2×D127—42no—
116C4×C127—42no—
117C4×C127—42no—
118C4×C127#12011yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12111yescharacteristic
121C4×C127#12711yescharacteristic
122Dic127—41no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—82no—
127C2×C127#39211yescharacteristic
128C2×C127#45412yes—
129D127—81no—
130#2003—1271nomaximal, Sylow
131C127#200311yescharacteristic, complemented, Sylow
132C2×AGL(1,ℤ/8)—1271no—
133#536—1271no—
134C2×#124D12711yescharacteristic, complemented
135C2×#104—1271no—
136#418—1271no—
137C2×#101—1271no—
138#339—1271no—
139C2×#41—1271no—
140C22×Q8C2×D12711yescharacteristic
141C22×D4C2×D12711yescharacteristic
142AGL(1,ℤ/8)—2542no—
143AGL(1,ℤ/8)—2542no—
144C2×SD16—2541no—
145C2×D8—2541no—
146C2×M16—1271no—
147C2×M16—1271no—
148C4⋊Q8—2541no—
149#124—22no—
150#124—22no—
151#124—2541no—
152#123—2542no—
153C22⋊Q8—2542no—
154#117—1271no—
155C2×C22⋊C4—22no—
156C2×C42C2×D12711yescharacteristic
157#104—2542no—
158#102—2542no—
159#101—2542no—
160C24—21no—
161#41—2541no—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1271no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—1271no—
178C2×D4—2541no—
179C2×D4C22⋊D12712yes—
180C22×C4—21no—
181C22×C4—2541no—
182C22×C4C22⋊D12712yes—
183C22×C4C22×D12711yescharacteristic
184SD16—2544no—
185D8—2544no—
186M16—1272no—
187M16—2541no—
188M16—2541no—
189M16—2541no—
190C2×C8—2541no—
191C2×C8—2541no—
192C4⋊C4—1272no—
193C4⋊C4—2544no—
194C22⋊C4—28no—
195C22⋊C4—2544no—
196C22⋊C4—2541no—
197C42—21no—
198C42—2541no—
199C42C22⋊D12712yes—
200C23—22no—
201C23—44no—
202C23—41no—
203C23—41no—
204C23—2541no—
205C23C2×C22⋊D12711yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—2541no—
211Q8—2541no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—5081no—
217D4—5081no—
218D4—2541no—
219D4—2541no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—5081no—
226C2×C4—5081no—
227C2×C4—2542no—
228C2×C4—2541no—
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4#1452324711yescharacteristic
235C8—5081no—
236C8—2542no—
237C22—22no—
238C22—44no—
239C22—42no—
240C22—82no—
241C22—82no—
242C22—5081no—
243C22—5082no—
244C22#1686117711yescharacteristic
245C22#1686117811yescharacteristic
246C22#1686118111yescharacteristic
247C4—42no—
248C4—42no—
249C4—5081no—
250C4—5082no—
251C4#1686117711yescharacteristic
252C4#1686118011yescharacteristic
253C4#1686118111yescharacteristic
254C4#1686120511yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—10161no—
259C2#2254978411yescharacteristic
260C2#2255004812yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#124×C127C2
socleC22×C127#120
prime radicalC2×#16861235C2
hypercenterC2×#124D127
hypocenterC127#2003
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
2. #22550450C2semidirect
3. C2×#16861137C2semidirect
4. #22549881C2semidirect
5. C2×#16861128C2semidirect
6. #22549636C2semidirect
7. C2×#124×C127C2semidirect
131. C127#2003semidirect
134. C2×#124D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225504502
C2×#168611372
#225498812
C2×#168611282
#225496362
C2×#124×C1272
#2003127