Finite Group Databasebeta

← #30651377

Subgroups — #30651377

Order 16256.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550596C211yesmaximal, characteristic, complemented
2#22550583C211yesmaximal, characteristic, complemented
3#22550229C212yesmaximal, complemented
4#22550227C211yesmaximal, characteristic, complemented
5#22550182C212yesmaximal, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550089C212yesmaximal
8#22550081C211yesmaximal, characteristic, complemented
9C8×Dic254C211yesmaximal, characteristic, complemented
10#22549967C212yesmaximal, complemented
11C4×D8×C127C211yesmaximal, characteristic
12C2×#14523236C2212yes—
13#16861209C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861202—22no—
16#16861200C2212yes—
17#16861196C2212yes—
18#16861188—21no—
19#16861188—21no—
20#16861186—22no—
21C4×Dic254C2211yescharacteristic
22#16861174C2212yes—
23D4×Dic127C2212yes—
24#16861168—21no—
25C2×C4⋊Dic127—21no—
26#16861140C2212yes—
27C4⋊(C127⋊C8)C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861134C2212yes—
31#16861132C2212yes—
32#16861130C2212yes—
33C8×Dic127C2212yes—
34C2×D8×C127C2211yescharacteristic
35C4×D4×C127C2211yescharacteristic
36C4×D4×C127C2211yescharacteristic
37#107×C127C2211yescharacteristic
38#102×C127C2211yescharacteristic
39#102×C127C2211yescharacteristic
40C4×C8×C127C2211yescharacteristic
41C2×Dic254—21no—
42C2×Dic254D411yescharacteristic
43C2×Dic254C2312yes—
44C22×Dic127—22no—
45#14523236—24no—
46C2×C127⋊C8C2312yes—
47C4⋊Dic127—22no—
48C4⋊Dic127—21no—
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2312yes—
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55Dic127⋊C4D412yes—
56C22⋊Dic127—22no—
57C22⋊Dic127—22no—
58C22⋊Dic127—22no—
59C4×Dic127—22no—
60C4×Dic127C2312yes—
61C2×D4×C127C2311yescharacteristic
62C2×D4×C127C2311yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65D8×C127—22no—
66C2×C8×C127C2311yescharacteristic
67C2×C8×C127C2311yescharacteristic
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2311yescharacteristic
70C22⋊C4×C127—21no—
71C22⋊C4×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—22no—
75Dic254—22no—
76Dic254—81no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—82no—
80C2×Dic127C2×D412yes—
81C2×Dic127C2×D412yes—
82C127⋊C8—22no—
83C23×C127—21no—
84C23×C127—21no—
85D4×C127—21no—
86D4×C127—21no—
87D4×C127—21no—
88D4×C127#4112yes—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—42no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C8×C127—21no—
97C8×C127—21no—
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127C22×D411yescharacteristic
105C4×C127—21no—
106C4×C127—41no—
107C4×C127—41no—
108C4×C127#13512yes—
109C4×C127C2×#4111yescharacteristic
110C4×C1272−1+411yescharacteristic
111Dic127—22no—
112Dic127—22no—
113Dic127—42no—
114Dic127—42no—
115C2×C127—42no—
116C2×C127—81no—
117C2×C127#54811yescharacteristic
118C2×C127C2×#13511yescharacteristic
119C2×C127#57711yescharacteristic
120#3360—1271nomaximal, Sylow
121C127#336011yescharacteristic, complemented, Sylow
122#556—1271no—
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×D8D12711yescharacteristic
133C2×SD16—1272no—
134C2×D8C2×D12711yescharacteristic
135C4⋊Q8—1271no—
136C4⋊Q8—1271no—
137#126—2542no—
138#125—1272no—
139#124—1272no—
140C22⋊Q8—2542no—
141C22⋊Q8—2541no—
142C22⋊Q8—2541no—
143C4×Q8—1271no—
144C4×Q8—1272no—
145C4×D4—1272no—
146C4×D4C2×D12711yescharacteristic
147C4×D4C2×D12711yescharacteristic
148#117—2541no—
149C2×C4⋊C4—2541no—
150#107C2×D12711yescharacteristic
151#106—1272no—
152C4⋊C8—1272no—
153C4⋊C8—1271no—
154#103—1272no—
155#103—1272no—
156#103—1272no—
157#102—1272no—
158#102C2×D12711yescharacteristic
159#102C2×D12711yescharacteristic
160C4×C8—1272no—
161C4×C8C2×D12711yescharacteristic
162C2×Q8—1271no—
163C2×Q8—1272no—
164C2×Q8—2541no—
165C2×D4C22×D12711yescharacteristic
166C2×D4C22×D12711yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—2542no—
170SD16—2544no—
171D8—22no—
172C2×C8—1272no—
173C2×C8C22×D12711yescharacteristic
174C2×C8C22×D12711yescharacteristic
175C4⋊C4—1272no—
176C4⋊C4—1271no—
177C4⋊C4—1272no—
178C4⋊C4—2542no—
179C4⋊C4—2542no—
180C4⋊C4—2542no—
181C4⋊C4—2541no—
182C4⋊C4—2541no—
183C4⋊C4—2541no—
184C4⋊C4C22×D12711yescharacteristic
185C4⋊C4C22×D12711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—2542no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C42—1272no—
192C42—2542no—
193C42C22×D12711yescharacteristic
194C23—21no—
195C23—21no—
196Q8—10161no—
197Q8—2542no—
198Q8—2542no—
199Q8—2542no—
200D4—21no—
201D4—21no—
202D4—21no—
203D4#1452324812yes—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—1272no—
209C2×C4—1272no—
210C2×C4—10162no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4D4×D12711yescharacteristic
214C2×C4D4×D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—21no—
217C8—21no—
218C8—2542no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22C2×D4×D12711yescharacteristic
226C4—21no—
227C4—41no—
228C4—41no—
229C4—5082no—
230C4—5082no—
231C4—2542no—
232C4—2542no—
233C4#1686122812yes—
234C4C2×#1452324811yescharacteristic
235C4#1686125611yescharacteristic
236C2—42no—
237C2—81no—
238C2#2255053511yescharacteristic
239C2C2×#1686122811yescharacteristic
240C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
prime radicalC2×D8×C127C22
hypercenterC4×D8D127
hypocenterC127#3360
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550583C2semidirect
3. #22550229C2semidirect
4. #22550227C2semidirect
5. #22550182C2semidirect
6. #22550166C2semidirect
8. #22550081C2semidirect
9. C8×Dic254C2semidirect
10. #22549967C2semidirect
121. C127#3360semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225505832
#225502292
#225502272
#225501822
#225501662
#225500892
#225500812
C8×Dic2542
#225499672
C4×D8×C1272
#3360127