Finite Group Databasebeta

← #30649095

Subgroups — #30649095

Order 16256.

372 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550244C212yesmaximal, complemented
3#22550235C212yesmaximal, complemented
4#22550230C212yesmaximal, complemented
5#22550226C212yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549877C211yesmaximal, characteristic
8#22549872C211yesmaximal, characteristic
9C2×#16861133C211yesmaximal, characteristic, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#536×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861209C2212yes—
15#16861208—21no—
16#16861201C2212yes—
17#16861200—21no—
18#16861186—22no—
19#16861168C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21#16861133C2212yes—
22#16861133C2212yes—
23#16861133C2211yescharacteristic
24#16861133C2211yescharacteristic
25#16861133C2212yes—
26#16861133C2212yes—
27#16861133C2212yes—
28#16861130C2211yescharacteristic
29#16861130C2211yescharacteristic
30#16861130C2212yes—
31#16861120C2212yes—
32C4×C127⋊C8C2212yes—
33C2×#41×C127C2211yescharacteristic
34C22×Q8×C127C2211yescharacteristic
35C4⋊Q8×C127C2212yes—
36#124×C127C2212yes—
37#123×C127—22no—
38C22⋊Q8×C127—21no—
39C22⋊Q8×C127—21no—
40#117×C127C2211yescharacteristic
41C2×Dic254—22no—
42C22×Dic127—21no—
43#14523232—22no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127C2311yescharacteristic
51C4⋊Dic127C2312yes—
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—21no—
56C4×Dic127—21no—
57#41×C127—22no—
58#41×C127—22no—
59C2×Q8×C127—21no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127C2312yes—
63C2×Q8×C127C2311yescharacteristic
64C2×D4×C127—21no—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—22no—
70C4⋊C4×C127—22no—
71C4⋊C4×C127D411yescharacteristic
72C4⋊C4×C127D411yescharacteristic
73C22⋊C4×C127—22no—
74C22⋊C4×C127—22no—
75C22⋊C4×C127D412yes—
76C42×C127C2312yes—
77Dic254—82no—
78C2×Dic127—21no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—81no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—21no—
86C23×C127C2×D411yescharacteristic
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127—21no—
90Q8×C127—21no—
91Q8×C127—42no—
92D4×C127—21no—
93D4×C127—21no—
94D4×C127—42no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4112yes—
103C2×C4×C127#4112yes—
104C2×C4×C127C2411yescharacteristic
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127#12412yes—
111C22×C127C22×D411yescharacteristic
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—41no—
115C4×C127—42no—
116C4×C127#12412yes—
117C4×C127C2×#4111yescharacteristic
118C4×C127C2×#4111yescharacteristic
119Dic127—42no—
120Dic127—41no—
121Dic127—41no—
122C2×C127—22no—
123C2×C127—81no—
124C2×C127C2×#12411yescharacteristic
125C2×C127#57711yescharacteristic
126C2×C127#57811yescharacteristic
127#3137—1271nomaximal, Sylow
128C127#313711yescharacteristic, complemented, Sylow
129#544—1271no—
130#536D12711yescharacteristic
131#491—1272no—
132#489—1272no—
133#488—1272no—
134#487—1272no—
135#419—1271no—
136#417—1271no—
137#416—1271no—
138C2×#103—1271no—
139#405—1271no—
140C2×#41C2×D12711yescharacteristic
141C22×Q8C2×D12711yescharacteristic
142C2×M16—1271no—
143C22×C8—1271no—
144C4⋊Q8—1272no—
145C4⋊Q8—2541no—
146C4⋊Q8C2×D12712yes—
147#125—1272no—
148#125—2541no—
149#124C2×D12712yes—
150#123—22no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153C22⋊Q8—2542no—
154#117—1271no—
155#117C2×D12711yescharacteristic
156C2×C4⋊C4—1271no—
157#103—1272no—
158#103—1272no—
159#103—1271no—
160#103—1271no—
161#103—1272no—
162#103—1272no—
163#103—1272no—
164#102—1271no—
165#102—1271no—
166#102—1272no—
167#97—1272no—
168C4×C8—1272no—
169#41—22no—
170#41—22no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—2542no—
175C2×Q8C22×D12712yes—
176C2×Q8C22×D12711yescharacteristic
177C2×D4—21no—
178C2×D4C22×D12711yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—2541no—
182C22×C4C22×D12711yescharacteristic
183M16—2542no—
184C2×C8—1271no—
185C2×C8—1272no—
186C2×C8—1271no—
187C2×C8—2542no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—1272no—
191C4⋊C4—1271no—
192C4⋊C4—1271no—
193C4⋊C4—2542no—
194C4⋊C4—2542no—
195C4⋊C4—2541no—
196C4⋊C4—2542no—
197C4⋊C4C22⋊D12711yescharacteristic
198C4⋊C4C22⋊D12711yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—2541no—
202C22⋊C4C22⋊D12712yes—
203C42—2541no—
204C42C22×D12712yes—
205C23—21no—
206C23C2×C22⋊D12711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—10162no—
213D4—21no—
214D4—21no—
215D4—42no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—10161no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2542no—
224C2×C4C2×C22⋊D12712yes—
225C2×C4C2×C22⋊D12712yes—
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4#1452324712yes—
228C2×C4#1452324712yes—
229C2×C4C23×D12711yescharacteristic
230C8—2541no—
231C8—2541no—
232C8—2542no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22#1686119612yes—
239C22C22×C22⋊D12711yescharacteristic
240C4—22no—
241C4—22no—
242C4—41no—
243C4—42no—
244C4—5082no—
245C4—5081no—
246C4—5081no—
247C4#1686119612yes—
248C4C2×#1452324711yescharacteristic
249C4C2×#1452324711yescharacteristic
250C2—22no—
251C2—81no—
252C2C2×#1686119611yescharacteristic
253C2#2255068411yescharacteristic
254C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenter#536D127
hypocenterC127#3137
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #22550504C2semidirect
2. #22550244C2semidirect
3. #22550235C2semidirect
4. #22550230C2semidirect
5. #22550226C2semidirect
6. #22549885C2semidirect
9. C2×#16861133C2semidirect
10. #22549832C2semidirect
128. C127#3137semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225502442
#225502352
#225502302
#225502262
#225498852
#225498772
#225498722
C2×#168611332
#225498322
#536×C1272
#3137127