Finite Group Databasebeta

← #30651384

Subgroups — #30651384

Order 16256.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550600C211yesmaximal, characteristic, complemented
2#22550587C211yesmaximal, characteristic, complemented
3#22550232C212yesmaximal, complemented
4#22550185C212yesmaximal, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550094C212yesmaximal
7#22550078C211yesmaximal, characteristic, complemented
8C8×Dic254C211yesmaximal, characteristic, complemented
9#22549961C212yesmaximal, complemented
10C4×D1016C211yesmaximal, characteristic
11#487×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2212yes—
13C2×D1016C2211yescharacteristic
14#16861204—22no—
15#16861200C2212yes—
16#16861198C2212yes—
17#16861189—21no—
18#16861189—21no—
19#16861185—22no—
20Q8×Dic127C2212yes—
21C4×Dic254C2211yescharacteristic
22C4×D508C2211yescharacteristic
23C4×D508C2211yescharacteristic
24#16861171C2212yes—
25#16861167—21no—
26C4⋊C4×D127—21no—
27#16861143C2211yescharacteristic
28#16861140C2212yes—
29C4⋊(C127⋊C8)C2211yescharacteristic
30Dic127⋊C8C2212yes—
31#16861133C2212yes—
32#16861131C2211yescharacteristic
33#16861131C2211yescharacteristic
34#16861129C2212yes—
35C8×Dic127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37C4⋊Q8×C127C2211yescharacteristic
38#103×C127C2212yes—
39#103×C127C2212yes—
40C4×C8×C127C2211yescharacteristic
41C2×Dic254D411yescharacteristic
42C2×D508C2311yescharacteristic
43C2×D508C2311yescharacteristic
44C2×C4×D127—22no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47#14523235—24no—
48D1016—22no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127—22no—
51C4⋊Dic127—22no—
52C4⋊Dic127D411yescharacteristic
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4D412yes—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—22no—
59(C2×D127)⋊C4—22no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—22no—
63C4×Dic127C2312yes—
64C2×Q8×C127—21no—
65C2×Q8×C127C2312yes—
66C2×C8×C127C2311yescharacteristic
67C2×C8×C127C2311yescharacteristic
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74C22×D127—21no—
75Dic254—22no—
76D508—21no—
77D508—21no—
78D508—21no—
79D508#4112yes—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C4×D127—42no—
85C4×D127—42no—
86C4×D127—82no—
87C127⋊C8—22no—
88Q8×C127—22no—
89Q8×C127—22no—
90Q8×C127—81no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C8×C127—21no—
97C8×C127—21no—
98C22×C127C22×D411yescharacteristic
99C2×D127—22no—
100C2×D127—41no—
101C2×D127—41no—
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—41no—
105C4×C127—21no—
106C4×C127—42no—
107C4×C127—42no—
108C4×C127#13512yes—
109C4×C127C2×#4111yescharacteristic
110C4×C1272−1+411yescharacteristic
111Dic127—22no—
112Dic127—22no—
113Dic127—41no—
114Dic127—41no—
115C2×C127#54811yescharacteristic
116C2×C127C2×#13511yescharacteristic
117C2×C127#57711yescharacteristic
118D127—42no—
119D127—81no—
120#3360—1271nomaximal, Sylow
121C127#336011yescharacteristic, complemented, Sylow
122#556—1271no—
123#554—1271no—
124#488—1272no—
125#487D12711yescharacteristic, complemented
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⋊Q8C2×D12711yescharacteristic
136C4⋊Q8C2×D12711yescharacteristic
137#126—2542no—
138#125—1272no—
139#124—1272no—
140C22⋊Q8—2541no—
141C22⋊Q8—2541no—
142C22⋊Q8—2542no—
143C4×Q8—1271no—
144C4×Q8—1272no—
145C4×D4—1272no—
146C4×D4—1271no—
147C4×D4—1271no—
148#117—2541no—
149C2×C4⋊C4—2541no—
150#107—1271no—
151#106—1272no—
152C4⋊C8—1272no—
153C4⋊C8—1271no—
154#103—1272no—
155#103C2×D12712yes—
156#103C2×D12712yes—
157#102—1271no—
158#102—1271no—
159#102—1272no—
160C4×C8—1272no—
161C4×C8C2×D12711yescharacteristic
162C2×Q8—21no—
163C2×Q8—1271no—
164C2×Q8C22×D12712yes—
165C2×D4—1271no—
166C2×D4—1271no—
167C22×C4—2542no—
168C22×C4—2541no—
169C22×C4—2541no—
170SD16—2544no—
171D8—2542no—
172C2×C8—1272no—
173C2×C8C22×D12711yescharacteristic
174C2×C8C22×D12711yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—1272no—
179C4⋊C4—1271no—
180C4⋊C4—1271no—
181C4⋊C4—1271no—
182C4⋊C4—2542no—
183C4⋊C4—2542no—
184C4⋊C4—2542no—
185C4⋊C4C22×D12712yes—
186C22⋊C4—2542no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C22⋊C4—2541no—
190C22⋊C4—2541no—
191C42—1272no—
192C42—2542no—
193C42C22×D12711yescharacteristic
194C23—2541no—
195C23—2541no—
196Q8—22no—
197Q8—22no—
198Q8—81no—
199Q8—2542no—
200D4—1272no—
201D4—2541no—
202D4—2541no—
203D4—2541no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—1272no—
207C2×C4—1272no—
208C2×C4—10162no—
209C2×C4—5082no—
210C2×C4—5082no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C2×C4D4×D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—21no—
217C8—21no—
218C8—2542no—
219C22—5081no—
220C22—5081no—
221C22—5081no—
222C22—5081no—
223C22—5081no—
224C22—2542no—
225C22C2×D4×D12711yescharacteristic
226C4—21no—
227C4—42no—
228C4—42no—
229C4—5081no—
230C4—5081no—
231C4—2542no—
232C4—2542no—
233C4#1686123212yes—
234C4#41×D12711yescharacteristic
235C4#1686125611yescharacteristic
236C2—10161no—
237C2—5082no—
238C2#2255053411yescharacteristic
239C2C2×#1686123211yescharacteristic
240C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#487×C127C2
socleC22×C127C22×D4
prime radicalC2×D1016C22
hypercenter#487D127
hypocenterC127#3360
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22550600C2semidirect
2. #22550587C2semidirect
3. #22550232C2semidirect
4. #22550185C2semidirect
5. #22550163C2semidirect
7. #22550078C2semidirect
8. C8×Dic254C2semidirect
9. #22549961C2semidirect
11. #487×C127C2semidirect
121. C127#3360semidirect
125. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225506002
#225505872
#225502322
#225501852
#225501632
#225500942
#225500782
C8×Dic2542
#225499612
C4×D10162
#487×C1272
#3360127