Finite Group Databasebeta

← #30650652

Subgroups — #30650652

Order 16256.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550531C211yesmaximal, characteristic, complemented
2#22550228C211yesmaximal, characteristic, complemented
3#22550217C212yesmaximal, complemented
4#22550163C211yesmaximal, characteristic, complemented
5#22550103C212yesmaximal, complemented
6#22550081C211yesmaximal, characteristic, complemented
7#22550019C212yesmaximal, complemented
8C4×#14523233C211yesmaximal, characteristic, complemented
9D4×C127⋊C8C211yesmaximal, characteristic, complemented
10#22549870C212yesmaximal, complemented
11#555×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13C2×#14523236C2212yes—
14C2×#14523233C2211yescharacteristic
15C22×C127⋊C8C2212yes—
16#16861209C2211yescharacteristic
17#16861191—22no—
18#16861188C2212yes—
19#16861180—21no—
20C4×D508C2211yescharacteristic
21C4⋊C4×D127—21no—
22#16861142C2211yescharacteristic
23#16861140C2212yes—
24C4⋊(C127⋊C8)C2211yescharacteristic
25#16861132C2212yes—
26#16861132C2212yes—
27#16861132C2212yes—
28#16861130C2211yescharacteristic
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#124×C127—22no—
35#121×C127—21no—
36#121×C127C2212yes—
37C4×D4×C127C2211yescharacteristic
38C4×D4×C127C2211yescharacteristic
39#117×C127—21no—
40#117×C127C2212yes—
41#14523248—24no—
42#14523248—42no—
43C2×Dic254C2312yes—
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—22no—
46C2×C4×D127—22no—
47C2×C4×D127—21no—
48#14523236—42no—
49#14523233—22no—
50C2×C127⋊C8—42no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2312yes—
53C2×C127⋊C8C2311yescharacteristic
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—21no—
60C2×Q8×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127D411yescharacteristic
63C2×D4×C127C2311yescharacteristic
64C22×C4×C127—21no—
65C22×C4×C127C2312yes—
66C4⋊C4×C127—21no—
67C4⋊C4×C127D411yescharacteristic
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C22⋊C4×C127—22no—
71C22⋊C4×C127—22no—
72C22⋊C4×C127—21no—
73C22⋊C4×C127D412yes—
74C42×C127—22no—
75C42×C127C2311yescharacteristic
76C22×D127—21no—
77Dic254—22no—
78Dic254—22no—
79D508—21no—
80D508#4112yes—
81C22⋊D127—44no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C4×D127—44no—
85C4×D127—42no—
86C127⋊C8—21no—
87C127⋊C8—21no—
88C127⋊C8—22no—
89C23×C127—21no—
90C23×C127C2×D412yes—
91Q8×C127—81no—
92D4×C127—22no—
93D4×C127—21no—
94D4×C127—21no—
95D4×C127—82no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—42no—
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2411yescharacteristic
105C22×C127—22no—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127C22×D411yescharacteristic
112C2×D127—22no—
113C2×D127—41no—
114C2×D127—41no—
115C4×C127—22no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—42no—
119C4×C127#13512yes—
120C4×C127C2×#4111yescharacteristic
121C4×C127H(5,𝔽2)11yescharacteristic
122Dic127—42no—
123Dic127—41no—
124C2×C127—42no—
125C2×C127—81no—
126C2×C127#54611yescharacteristic
127C2×C127C2×#13511yescharacteristic
128C2×C127#57711yescharacteristic
129D127—42no—
130#3303—1271nomaximal, Sylow
131C127#330311yescharacteristic, complemented, Sylow
132#555D12711yescharacteristic, complemented
133#547—1271no—
134#487—1271no—
135#485—1272no—
136#476—1271no—
137#465—1272no—
138#461—1271no—
139#449—1272no—
140C4×D8—1271no—
141C8×D4—1271no—
142#416—1272no—
143C2×#41—1272no—
144C2×SD16—1272no—
145C2×D8—1271no—
146C22×C8—1272no—
147C4⋊Q8—1271no—
148C4⋊Q8C2×D12711yescharacteristic
149#124—22no—
150#123—2542no—
151C22⋊Q8—1272no—
152#121—21no—
153#121—2541no—
154#121C2×D12712yes—
155C4×D4—1271no—
156C4×D4C2×D12711yescharacteristic
157C4×D4C2×D12711yescharacteristic
158#117—21no—
159#117C2×D12712yes—
160C2×C4⋊C4—2541no—
161#107—1271no—
162#106—1272no—
163C4⋊C8—1271no—
164#103—1272no—
165#103—1272no—
166#103—1272no—
167#102—1271no—
168#102—1271no—
169#102—1272no—
170C22⋊C8—1272no—
171C4×C8—1271no—
172#41—5082no—
173#41—2544no—
174C2×Q8—21no—
175C2×Q8—1272no—
176C2×D4—22no—
177C2×D4—1271no—
178C2×D4—2542no—
179C2×D4C22⋊D12711yescharacteristic
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—2541no—
183C22×C4—2542no—
184C22×C4C22×D12712yes—
185SD16—5082no—
186D8—2542no—
187C2×C8—1271no—
188C2×C8—1271no—
189C2×C8—1272no—
190C2×C8—5082no—
191C4⋊C4—21no—
192C4⋊C4—1271no—
193C4⋊C4—2542no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C4⋊C4C22⋊D12711yescharacteristic
197C4⋊C4C22×D12712yes—
198C4⋊C4C22×D12711yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—21no—
202C22⋊C4—2542no—
203C22⋊C4—2541no—
204C22⋊C4C22⋊D12712yes—
205C42—22no—
206C42C22×D12711yescharacteristic
207C23—21no—
208C23—2541no—
209C23C2×C22⋊D12712yes—
210Q8—81no—
211Q8—2542no—
212Q8—2542no—
213D4—22no—
214D4—21no—
215D4—21no—
216D4—82no—
217D4—1272no—
218D4—5084no—
219D4—2541no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—5084no—
226C2×C4—5082no—
227C2×C4—2541no—
228C2×C4—2542no—
229C2×C4C2×C22⋊D12712yes—
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4C2×C22⋊D12711yescharacteristic
232C2×C4C23×D12711yescharacteristic
233C8—2541no—
234C8—2541no—
235C8—2542no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—5081no—
243C22—5081no—
244C22—2542no—
245C22C22×C22⋊D12711yescharacteristic
246C4—21no—
247C4—22no—
248C4—41no—
249C4—42no—
250C4—5081no—
251C4—5082no—
252C4#1686122912yes—
253C4#41×D12711yescharacteristic
254C4#1686125511yescharacteristic
255C2—42no—
256C2—81no—
257C2—5082no—
258C2#2255051411yescharacteristic
259C2C2×#1686122911yescharacteristic
260C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#555×C127C2
socleC22×C127C22×D4
hypercenter#555D127
hypocenterC127#3303
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550531C2semidirect
2. #22550228C2semidirect
3. #22550217C2semidirect
4. #22550163C2semidirect
5. #22550103C2semidirect
6. #22550081C2semidirect
7. #22550019C2semidirect
8. C4×#14523233C2semidirect
9. D4×C127⋊C8C2semidirect
10. #22549870C2semidirect
11. #555×C127C2semidirect
131. C127#3303semidirect
132. #555D127semidirect

Maximal subgroups

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

subgroupindex
#225505312
#225502282
#225502172
#225501632
#225501032
#225500812
#225500192
C4×#145232332
D4×C127⋊C82
#225498702
#555×C1272
#3303127