Finite Group Databasebeta

← #30651375

Subgroups — #30651375

Order 16256.

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

normal subgroups only

This group has 350 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#22550586C211yesmaximal, characteristic, complemented
2#22550230C212yesmaximal, complemented
3#22550226C211yesmaximal, characteristic, complemented
4#22550183C212yesmaximal, complemented
5#22550166C211yesmaximal, characteristic, complemented
6#22550088C212yesmaximal
7#22550078C211yesmaximal, characteristic, complemented
8Q8×C127⋊C8C211yesmaximal, characteristic, complemented
9C4×#14523235C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic
11#556×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2212yes—
13C2×#14523233C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861201C2212yes—
16#16861187—22no—
17#16861185—21no—
18C4×D508C2211yescharacteristic
19C4×D508C2212yes—
20C4⋊C4×D127—21no—
21#16861142C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2212yes—
26#16861133C2212yes—
27#16861133C2212yes—
28#16861130C2212yes—
29#16861130C2211yescharacteristic
30#16861129C2211yescharacteristic
31C4×C127⋊C8C2211yescharacteristic
32C4×C127⋊C8C2212yes—
33C4⋊Q8×C127C2211yescharacteristic
34#126×C127—22no—
35#124×C127C2212yes—
36C22⋊Q8×C127—21no—
37C4×Q8×C127C2212yes—
38C4×Q8×C127C2211yescharacteristic
39C4×D4×C127C2211yescharacteristic
40#117×C127—21no—
41C2×Dic254—21no—
42C2×D508C2311yescharacteristic
43C2×C4×D127—21no—
44C2×C4×D127—22no—
45#14523235—24no—
46#14523233—22no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—21no—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55(C2×D127)⋊C4—22no—
56(C2×D127)⋊C4—21no—
57C2×Q8×C127D411yescharacteristic
58C2×Q8×C127C2312yes—
59C2×D4×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127—22no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127D412yes—
65C4⋊C4×C127D411yescharacteristic
66C4⋊C4×C127C2311yescharacteristic
67C22⋊C4×C127—22no—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—22no—
70C42×C127—22no—
71C42×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—81no—
75D508—21no—
76D508#4112yes—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C4×D127—42no—
80C4×D127—82no—
81C127⋊C8—21no—
82C127⋊C8—22no—
83C127⋊C8—21no—
84C23×C127—21no—
85Q8×C127—22no—
86Q8×C127—22no—
87Q8×C127—22no—
88D4×C127—21no—
89D4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2411yescharacteristic
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127C22×D411yescharacteristic
102C2×D127—22no—
103C2×D127—41no—
104C2×D127—41no—
105C4×C127—22no—
106C4×C127—22no—
107C4×C127—21no—
108C4×C127—42no—
109C4×C127—41no—
110C4×C127#13512yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—41no—
114Dic127—42no—
115C2×C127—81no—
116C2×C127#54811yescharacteristic
117C2×C127C2×#13511yescharacteristic
118C2×C127#57711yescharacteristic
119D127—42no—
120#3360—1271nomaximal, Sylow
121C127#336011yescharacteristic, complemented, Sylow
122#556D12711yescharacteristic, complemented
123#554—1271no—
124#488—1272no—
125#487—1271no—
126#479—1272no—
127#476—1271no—
128#463—1272no—
129#461—1271no—
130C8×Q8—1271no—
131C4×SD16—1272no—
132C4×D8—1271no—
133C2×SD16—1272no—
134C2×D8—1271no—
135C4⋊Q8—1271no—
136C4⋊Q8C2×D12711yescharacteristic
137#126—22no—
138#125—1272no—
139#124C2×D12712yes—
140C22⋊Q8—21no—
141C22⋊Q8—2541no—
142C22⋊Q8—2542no—
143C4×Q8C2×D12712yes—
144C4×Q8C2×D12711yescharacteristic
145C4×D4—1271no—
146C4×D4—1272no—
147C4×D4C2×D12711yescharacteristic
148#117—21no—
149C2×C4⋊C4—2541no—
150#107—1271no—
151#106—1272no—
152C4⋊C8—1271no—
153C4⋊C8—1272no—
154#103—1272no—
155#103—1272no—
156#103—1272no—
157#102—1271no—
158#102—1272no—
159#102—1271no—
160C4×C8—1271no—
161C4×C8—1272no—
162C2×Q8—2541no—
163C2×Q8C22⋊D12711yescharacteristic
164C2×Q8C22×D12712yes—
165C2×D4—1271no—
166C2×D4C22×D12711yescharacteristic
167C22×C4—21no—
168C22×C4—2542no—
169C22×C4—2541no—
170SD16—2544no—
171D8—2542no—
172C2×C8—1271no—
173C2×C8—1272no—
174C2×C8—1271no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—1272no—
179C4⋊C4—1271no—
180C4⋊C4—2542no—
181C4⋊C4—2542no—
182C4⋊C4—2541no—
183C4⋊C4C22⋊D12711yescharacteristic
184C4⋊C4C22⋊D12712yes—
185C4⋊C4C22×D12711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—2541no—
190C22⋊C4—2542no—
191C42—22no—
192C42C22×D12711yescharacteristic
193C42C22×D12712yes—
194C23—21no—
195C23—2541no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—10161no—
200D4—21no—
201D4—21no—
202D4—1272no—
203D4—2541no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—10162no—
208C2×C4—5082no—
209C2×C4—2541no—
210C2×C4—2542no—
211C2×C4C2×C22⋊D12712yes—
212C2×C4C2×C22⋊D12712yes—
213C2×C4C2×C22⋊D12711yescharacteristic
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—2541no—
217C8—2542no—
218C8—2541no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—5081no—
223C22—5081no—
224C22—2542no—
225C22C22×C22⋊D12711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—5081no—
232C4—5082no—
233C4#1686122912yes—
234C4#41×D12711yescharacteristic
235C4#1686125611yescharacteristic
236C2—81no—
237C2—5082no—
238C2#2255053311yescharacteristic
239C2C2×#1686122911yescharacteristic
240C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#556×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523233C22
hypercenter#556D127
hypocenterC127#3360
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #22550586C2semidirect
2. #22550230C2semidirect
3. #22550226C2semidirect
4. #22550183C2semidirect
5. #22550166C2semidirect
7. #22550078C2semidirect
8. Q8×C127⋊C8C2semidirect
9. C4×#14523235C2semidirect
11. #556×C127C2semidirect
121. C127#3360semidirect
122. #556D127semidirect

Maximal subgroups

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

subgroupindex
#225505862
#225502302
#225502262
#225501832
#225501662
#225500882
#225500782
Q8×C127⋊C82
C4×#145232352
C4×#145232332
#556×C1272
#3360127