Finite Group Databasebeta

← #30649091

Subgroups — #30649091

Order 16256.

372 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550504C211yesmaximal, characteristic, complemented
2#22550450C211yesmaximal, characteristic
3#22550243C212yesmaximal, complemented
4#22550237C212yesmaximal, complemented
5#22550232C212yesmaximal, complemented
6#22550225C212yesmaximal, complemented
7#22549878C211yesmaximal, characteristic
8#22549870C211yesmaximal, characteristic
9C2×#16861133C211yesmaximal, characteristic, complemented
10#22549834C211yesmaximal, characteristic, complemented
11#419×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861209—21no—
15#16861208C2212yes—
16#16861207C2212yes—
17#16861201—21no—
18#16861200C2212yes—
19#16861198C2212yes—
20#16861192—22no—
21#16861186—22no—
22#16861185—21no—
23#16861185—21no—
24#16861168C2211yescharacteristic
25C2×C4⋊Dic127C2211yescharacteristic
26#16861134C2212yes—
27#16861133C2212yes—
28#16861133C2212yes—
29#16861132C2211yescharacteristic
30#16861132C2211yescharacteristic
31#16861131C2212yes—
32#16861129C2211yescharacteristic
33#16861129C2211yescharacteristic
34#16861119C2212yes—
35C8×Dic127C2212yes—
36C22×Q8×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38#117×C127C2211yescharacteristic
39#103×C127C2212yes—
40#103×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—22no—
44C2×Dic254C2311yescharacteristic
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—21no—
47C22×Dic127—21no—
48C2×C4×D127—21no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127—21no—
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2312yes—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59Dic127⋊C4—22no—
60Dic127⋊C4—22no—
61C22⋊Dic127D412yes—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64C4×Dic127C2312yes—
65C2×Q8×C127—21no—
66C2×Q8×C127—22no—
67C2×Q8×C127—22no—
68C2×Q8×C127C2312yes—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71M16×C127—22no—
72C2×C8×C127C2312yes—
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C42×C127—21no—
77C22×D127—21no—
78Dic254—21no—
79Dic254—21no—
80Dic254—82no—
81D508—21no—
82D508—21no—
83C22⋊D127—42no—
84C2×Dic127—21no—
85C2×Dic127—22no—
86C2×Dic127—81no—
87C2×Dic127C2×D412yes—
88C2×Dic127C2×D412yes—
89C4×D127—42no—
90C127⋊C8—21no—
91C127⋊C8—21no—
92C23×C127C2×D411yescharacteristic
93Q8×C127—22no—
94Q8×C127—22no—
95Q8×C127—42no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127—42no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4112yes—
102C2×C4×C127#4112yes—
103C2×C4×C127C2411yescharacteristic
104C8×C127—22no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127#12412yes—
108C22×C127C22×D411yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127#12412yes—
116C4×C127C2×#4111yescharacteristic
117C4×C127C2×#4111yescharacteristic
118Dic127—22no—
119Dic127—22no—
120Dic127—41no—
121Dic127—42no—
122C2×C127—22no—
123C2×C127C2×#12411yescharacteristic
124C2×C127#57711yescharacteristic
125C2×C127#57811yescharacteristic
126D127—81no—
127#3137—1271nomaximal, Sylow
128C127#313711yescharacteristic, complemented, Sylow
129#544—1271no—
130#536—1271no—
131#491—1272no—
132#489—1272no—
133#488—1272no—
134#487—1272no—
135#419D12711yescharacteristic, complemented
136#417—1271no—
137#416—1271no—
138C2×#103—1271no—
139#405—1271no—
140C2×#41—1271no—
141C22×Q8C2×D12711yescharacteristic
142C2×M16C2×D12711yescharacteristic
143C22×C8—1271no—
144C4⋊Q8—1272no—
145C4⋊Q8—1272no—
146C4⋊Q8—2541no—
147#125—1272no—
148#125—2541no—
149#124—1272no—
150#123—2542no—
151C22⋊Q8—2542no—
152C22⋊Q8—2541no—
153C22⋊Q8—2541no—
154#117—1271no—
155#117C2×D12711yescharacteristic
156C2×C4⋊C4—1271no—
157#103—1272no—
158#103—1272no—
159#103—1272no—
160#103—1271no—
161#103—1271no—
162#103C2×D12712yes—
163#103C2×D12712yes—
164#102—1272no—
165#102—1271no—
166#102—1271no—
167#97—1272no—
168C4×C8—1272no—
169#41—2542no—
170#41—2542no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—1271no—
175C2×Q8—2542no—
176C2×Q8C22×D12712yes—
177C2×D4—1271no—
178C2×D4—2541no—
179C22×C4—21no—
180C22×C4—2541no—
181C22×C4—2541no—
182C22×C4C22×D12711yescharacteristic
183M16—22no—
184C2×C8—1271no—
185C2×C8—1271no—
186C2×C8—2542no—
187C2×C8C22×D12712yes—
188C4⋊C4—1271no—
189C4⋊C4—1271no—
190C4⋊C4—1272no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4—2542no—
194C4⋊C4—2542no—
195C4⋊C4—2542no—
196C4⋊C4—2541no—
197C4⋊C4C22×D12711yescharacteristic
198C4⋊C4C22×D12711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—1272no—
201C22⋊C4—2542no—
202C22⋊C4—2542no—
203C42—21no—
204C42—1272no—
205C23—2541no—
206C23D4×D12711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—10162no—
211Q8—2541no—
212Q8—2541no—
213D4—5082no—
214D4—2541no—
215D4—2541no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—1272no—
221C2×C4—1272no—
222C2×C4—10161no—
223C2×C4—5082no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4D4×D12711yescharacteristic
227C2×C4#1452324712yes—
228C2×C4#1452324812yes—
229C2×C4C23×D12711yescharacteristic
230C8—22no—
231C8—2541no—
232C8—2541no—
233C22—21no—
234C22—21no—
235C22—5081no—
236C22—5081no—
237C22—5081no—
238C22#1686119512yes—
239C22C2×D4×D12711yescharacteristic
240C4—42no—
241C4—41no—
242C4—41no—
243C4—5081no—
244C4—5082no—
245C4—2542no—
246C4—2542no—
247C4#1686119512yes—
248C4C2×#1452324711yescharacteristic
249C4C2×#1452324811yescharacteristic
250C2—22no—
251C2—10161no—
252C2C2×#1686119511yescharacteristic
253C2#2255068511yescharacteristic
254C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#419D127
hypocenterC127#3137
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #22550504C2semidirect
3. #22550243C2semidirect
4. #22550237C2semidirect
5. #22550232C2semidirect
6. #22550225C2semidirect
9. C2×#16861133C2semidirect
10. #22549834C2semidirect
11. #419×C127C2semidirect
128. C127#3137semidirect
135. #419D127semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225504502
#225502432
#225502372
#225502322
#225502252
#225498782
#225498702
C2×#168611332
#225498342
#419×C1272
#3137127