Finite Group Databasebeta

← #30638749

Subgroups — #30638749

Order 16256.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#22550344C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic
4C2×#16861135C211yesmaximal, characteristic
5C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861174C411yescharacteristic, complemented
11#16861174C411yescharacteristic, complemented
12#16861171C411yescharacteristic, complemented
13#16861171C411yescharacteristic, complemented
14(C22⋊D127)⋊C4—21no—
15(C22⋊D127)⋊C4—21no—
16#16861166—21no—
17C2×Dic127⋊C4—21no—
18C2×C4×Dic127C2211yescharacteristic
19C2×C4×Dic127C2211yescharacteristic
20C42×D127—21no—
21Dic127⋊C8—21no—
22Dic127⋊C8—21no—
23#16861137—22no—
24#16861135—22no—
25C127⋊#95—21no—
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29#14523248—21no—
30#14523248—21no—
31#14523248D412yes—
32#14523248Q812yes—
33C2×Dic254C2×C411yescharacteristic
34C2×D508C2×C411yescharacteristic
35C2×C22⋊D127—21no—
36C22×Dic127—21no—
37C22×Dic127D411yescharacteristic
38C22×Dic127D411yescharacteristic
39C2×C4×D127—21no—
40C2×C4×D127—21no—
41C2×C4×D127—21no—
42#14523232—21no—
43#14523232—21no—
44C2×C127⋊C8—21no—
45Dic127⋊C4—24no—
46Dic127⋊C4—21no—
47Dic127⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50C4×Dic127—21no—
51C4×Dic127—21no—
52C4×Dic127—21no—
53C4×Dic127—21no—
54C4×Dic127C2×C411yescharacteristic
55C4×Dic127C2×C411yescharacteristic
56C4×Dic127D411yescharacteristic
57C4×Dic127D411yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—21no—
61M16×C127—21no—
62C2×C8×C127—21no—
63C4⋊C4×C127C2×C411yescharacteristic
64C4⋊C4×C127C2×C411yescharacteristic
65C22⋊C4×C127—21no—
66C42×C127—21no—
67C42×C127—21no—
68C42×C127—21no—
69C22×D127—21no—
70Dic254—21no—
71Dic254C4⋊C412yescomplemented
72D508—21no—
73D508C4⋊C412yescomplemented
74C22⋊D127—22no—
75C22⋊D127—22no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—41no—
80C2×Dic127—41no—
81C2×Dic127—41no—
82C2×Dic127C22⋊C411yescharacteristic
83C2×Dic127C22⋊C411yescharacteristic
84C2×Dic127C22⋊C411yescharacteristic
85C2×Dic127C22⋊C411yescharacteristic
86C4×D127—22no—
87C4×D127—22no—
88C4×D127—42no—
89C4×D127—42no—
90C127⋊C8—41no—
91C23×C127C2×D411yescharacteristic
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127C22×C411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×Q811yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C8×C127—41no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127C2×C22⋊C411yescharacteristic
107C22×C127#12111yescharacteristic
108C22×C127C22⋊Q811yescharacteristic
109C2×D127—22no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127C2×C4⋊C411yescharacteristic
116C4×C127#11711yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127C22⋊Q811yescharacteristic
119Dic127—22no—
120Dic127—21no—
121Dic127—21no—
122Dic127—21no—
123Dic127—42no—
124Dic127#10412yescomplemented
125C2×C127—21no—
126C2×C127—21no—
127C2×C127#38011yescharacteristic
128C2×C127C2×#10411yescharacteristic
129C2×C127#42111yescharacteristic
130D127—42no—
131#1806—1271nomaximal, Sylow
132C127#180611yescharacteristic, complemented, Sylow
133C4×#41—1271no—
134#423—1271no—
135C2×#104—1271no—
136C2×#104—1271no—
137C4×C4⋊C4—1271no—
138#339—1271no—
139#339D12711yescharacteristic, complemented
140C2×#41—1271no—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143C4×Q8—1271no—
144C4×Q8—1271no—
145C4×D4—1271no—
146C4×D4—1271no—
147C4×D4—2541no—
148C4×D4—2541no—
149#117—2541no—
150#117C2×D12711yescharacteristic
151C2×C4⋊C4—2541no—
152C2×C42—1271no—
153C2×C42—1271no—
154C2×C42—2541no—
155C2×C42C2×D12711yescharacteristic
156C4⋊C8—2541no—
157C4⋊C8—2541no—
158#104—2542no—
159#104—2542no—
160#95—2541no—
161#41—1272no—
162#41—1272no—
163#41—2541no—
164#41—2541no—
165C2×Q8—1271no—
166C2×D4—1271no—
167C2×D4—2541no—
168C22×C4—21no—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176M16—21no—
177M16—21no—
178M16—2541no—
179M16—2541no—
180C2×C8—21no—
181C2×C8—2541no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4—2544no—
185C4⋊C4C4×D12711yescharacteristic
186C4⋊C4C4×D12711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—2541no—
189C22⋊C4—2541no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—1271no—
194C42—1271no—
195C42—1271no—
196C42—1271no—
197C42—2541no—
198C42—2541no—
199C42—2541no—
200C42—2541no—
201C23—2541no—
202C23D4×D12711yescharacteristic
203Q8—1272no—
204Q8—2541no—
205D4—1272no—
206D4—2542no—
207D4—2542no—
208D4—2541no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1271no—
215C2×C4—1271no—
216C2×C4—1271no—
217C2×C4—1271no—
218C2×C4—5082no—
219C2×C4—5082no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—5081no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2541no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4C2×C4×D12711yescharacteristic
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4Q8×D12711yescharacteristic
232C2×C4#1452324811yescharacteristic
233C2×C4#1452324811yescharacteristic
234C8—41no—
235C8—5081no—
236C22—21no—
237C22—21no—
238C22—5081no—
239C22—5081no—
240C22—2542no—
241C22C22⋊C4×D12711yescharacteristic
242C22#1686118211yescharacteristic
243C22#1686118711yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—1272no—
248C4—5082no—
249C4—2541no—
250C4—2541no—
251C4—2541no—
252C4—2542no—
253C4C4⋊C4×D12711yescharacteristic
254C4#1686116611yescharacteristic
255C4#1686118211yescharacteristic
256C4#1686118711yescharacteristic
257C2—21no—
258C2—21no—
259C2—5082no—
260C2#2254973511yescharacteristic
261C2#104×D12711yescharacteristic
262C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1806
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
5. C4×Dic127⋊C4C2semidirect
6. #22549635C2semidirect
7. #339×C127C2semidirect
10. #16861174C4semidirect
11. #16861174C4semidirect
12. #16861171C4semidirect
13. #16861171C4semidirect
71. Dic254C4⋊C4semidirect
73. D508C4⋊C4semidirect
124. Dic127#104semidirect
132. C127#1806semidirect
139. #339D127semidirect

Maximal subgroups

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

subgroupindex
#225503442
#225498992
C2×#168611372
C2×#168611352
C4×Dic127⋊C42
#225496352
#339×C1272
#1806127