Finite Group Databasebeta

← #30649100

Subgroups — #30649100

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#22550452C211yesmaximal, characteristic
2#22550246C212yesmaximal, complemented
3#22550238C212yesmaximal, complemented
4#22550233C212yesmaximal, complemented
5#22550228C212yesmaximal, complemented
6#22549884C211yesmaximal, characteristic, complemented
7#22549875C211yesmaximal, characteristic
8#22549870C211yesmaximal, characteristic
9C2×#16861132C211yesmaximal, characteristic, complemented
10#22549832C211yesmaximal, characteristic, complemented
11#544×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861209C2212yes—
17#16861197C2212yes—
18#16861193—22no—
19#16861189—21no—
20#16861189—21no—
21#16861132C2212yes—
22#16861132C2212yes—
23#16861132C2211yescharacteristic
24#16861132C2211yescharacteristic
25#16861132C2212yes—
26#16861132C2212yes—
27#16861132C2212yes—
28#16861129C2211yescharacteristic
29#16861129C2211yescharacteristic
30#16861129C2212yes—
31#16861120C2212yes—
32C4×C127⋊C8C2212yes—
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127C2212yes—
35#125×C127—21no—
36#125×C127C2212yes—
37C22⋊Q8×C127—22no—
38#117×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—21no—
44C2×Dic254—22no—
45C2×Dic254—22no—
46C2×Dic254C2312yes—
47C2×Dic254C2311yescharacteristic
48C2×D508C2311yescharacteristic
49C2×C22⋊D127—21no—
50C22×Dic127—21no—
51C2×C4×D127—21no—
52#14523232—22no—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2311yescharacteristic
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2312yes—
57C4⋊Dic127—22no—
58C4⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—22no—
61C2×Q8×C127—22no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64C4⋊C4×C127—21no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—22no—
68C4⋊C4×C127D411yescharacteristic
69C4⋊C4×C127D411yescharacteristic
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—21no—
74C22⋊C4×C127D412yes—
75C42×C127—21no—
76C42×C127C2312yes—
77C22×D127—21no—
78Dic254—22no—
79Dic254—22no—
80Dic254—21no—
81Dic254—21no—
82Dic254—42no—
83D508—21no—
84D508—21no—
85C22⋊D127—42no—
86C2×Dic127—22no—
87C2×Dic127—21no—
88C2×Dic127—42no—
89C4×D127—42no—
90C127⋊C8—21no—
91C127⋊C8—21no—
92C127⋊C8—22no—
93C23×C127C2×D411yescharacteristic
94Q8×C127—82no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—81no—
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#12412yes—
108C22×C127C22×D411yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—42no—
115C4×C127—41no—
116C4×C127—41no—
117C4×C127#12412yes—
118C4×C127C2×#4111yescharacteristic
119C4×C127C2×#4111yescharacteristic
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#544D12711yescharacteristic, complemented
130#536—1271no—
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×#41—1271no—
141C22×Q8—1271no—
142C2×M16—1271no—
143C22×C8—1271no—
144C4⋊Q8—21no—
145C4⋊Q8—1272no—
146C4⋊Q8C2×D12712yes—
147#125—21no—
148#125C2×D12712yes—
149#124—1272no—
150#123—2542no—
151C22⋊Q8—22no—
152C22⋊Q8—2541no—
153C22⋊Q8—2541no—
154#117C2×D12711yescharacteristic
155#117C2×D12711yescharacteristic
156C2×C4⋊C4C2×D12711yescharacteristic
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—2542no—
170#41—2542no—
171C2×Q8—22no—
172C2×Q8—1272no—
173C2×Q8—1271no—
174C2×Q8—2542no—
175C2×Q8—2542no—
176C2×Q8—2541no—
177C2×D4—1271no—
178C2×D4—2541no—
179C22×C4—21no—
180C22×C4—2541no—
181C22×C4—2541no—
182C22×C4C22×D12711yescharacteristic
183M16—2542no—
184C2×C8—1271no—
185C2×C8—1272no—
186C2×C8—1271no—
187C2×C8—2542no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—2542no—
193C4⋊C4—2542no—
194C4⋊C4C22⋊D12711yescharacteristic
195C4⋊C4C22⋊D12711yescharacteristic
196C4⋊C4C22×D12712yes—
197C4⋊C4C22×D12711yescharacteristic
198C4⋊C4C22×D12711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—2542no—
201C22⋊C4—2542no—
202C22⋊C4C22⋊D12712yes—
203C42—21no—
204C42C22×D12712yes—
205C23—2541no—
206C23C2×C22⋊D12711yescharacteristic
207Q8—82no—
208Q8—5082no—
209Q8—2542no—
210Q8—2542no—
211Q8—2541no—
212Q8—2541no—
213D4—5082no—
214D4—2541no—
215D4—2541no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—81no—
220C2×C4—5082no—
221C2×C4—5082no—
222C2×C4—2542no—
223C2×C4—2541no—
224C2×C4C2×C22⋊D12712yes—
225C2×C4C2×C22⋊D12712yes—
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4#1452324912yes—
228C2×C4#1452324912yes—
229C2×C4C23×D12711yescharacteristic
230C8—2541no—
231C8—2541no—
232C8—2542no—
233C22—21no—
234C22—21no—
235C22—5081no—
236C22—5081no—
237C22—5081no—
238C22#1686119812yes—
239C22C22×C22⋊D12711yescharacteristic
240C4—22no—
241C4—22no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—5081no—
246C4—5082no—
247C4#1686119812yes—
248C4C2×#1452324911yescharacteristic
249C4C2×#1452324911yescharacteristic
250C2—22no—
251C2—10161no—
252C2C2×#1686119811yescharacteristic
253C2#2255068711yescharacteristic
254C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#544×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#544D127
hypocenterC127#3137
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #22550246C2semidirect
3. #22550238C2semidirect
4. #22550233C2semidirect
5. #22550228C2semidirect
6. #22549884C2semidirect
9. C2×#16861132C2semidirect
10. #22549832C2semidirect
11. #544×C127C2semidirect
128. C127#3137semidirect
129. #544D127semidirect

Maximal subgroups

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

subgroupindex
#225504522
#225502462
#225502382
#225502332
#225502282
#225498842
#225498752
#225498702
C2×#168611322
#225498322
#544×C1272
#3137127