Finite Group Databasebeta

← #30651470

Subgroups — #30651470

Order 16256.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550599C211yesmaximal, characteristic
2#22550238C212yesmaximal
3#22550228C211yesmaximal, characteristic, complemented
4#22550163C211yesmaximal, characteristic
5#22550158C212yesmaximal, complemented
6#22550098C212yesmaximal
7#22550086C211yesmaximal, characteristic, complemented
8#22550007C211yesmaximal, characteristic, complemented
9#22549981C212yesmaximal, complemented
10C4×#14523235C211yesmaximal, characteristic
11Q82×C127C211yesmaximal, characteristic, complemented
12C2×#14523238C2212yes—
13C2×#14523235C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861204—22no—
16#16861197C2212yes—
17#16861185—21no—
18C4×Dic254C2212yes—
19C4×D508C2211yescharacteristic
20#16861167—21no—
21#16861140C2211yescharacteristic
22#16861140C2212yes—
23C4⋊(C127⋊C8)C2211yescharacteristic
24C4⋊(C127⋊C8)C2212yes—
25#16861133C2211yescharacteristic
26#16861132C2212yes—
27#16861132C2212yes—
28#16861132C2212yes—
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31#16861120C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127—22no—
34C4⋊Q8×C127—21no—
35C4⋊Q8×C127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37C4×Q8×C127—21no—
38C4×Q8×C127C2211yescharacteristic
39C4×Q8×C127C2212yes—
40C4×Q8×C127C2211yescharacteristic
41C2×Dic254C2312yes—
42C2×D508C2311yescharacteristic
43C2×C4×D127—21no—
44#14523238—24no—
45#14523235—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127C2311yescharacteristic
52Dic127⋊C4—22no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—22no—
55(C2×D127)⋊C4—21no—
56C4×Dic127—22no—
57C2×Q8×C127—22no—
58C2×Q8×C127—21no—
59C2×Q8×C127D411yescharacteristic
60C2×Q8×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127D412yes—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127C2311yescharacteristic
68C4⋊C4×C127C2312yes—
69C42×C127—22no—
70C42×C127—21no—
71C42×C127C2312yes—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—22no—
75Dic254—22no—
76D508—21no—
77D508—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C4×D127—42no—
81C127⋊C8—21no—
82C127⋊C8—22no—
83C127⋊C8—21no—
84Q8×C127—22no—
85Q8×C127—21no—
86Q8×C127—82no—
87Q8×C127—81no—
88Q8×C127#4112yes—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C22×C127C22×D411yescharacteristic
98C2×D127—41no—
99C2×D127—41no—
100C2×D127—41no—
101C4×C127—22no—
102C4×C127—22no—
103C4×C127—22no—
104C4×C127—21no—
105C4×C127—42no—
106C4×C127—41no—
107C4×C127#13712yes—
108C4×C127C2×#4111yescharacteristic
109C4×C1272−1+411yescharacteristic
110Dic127—42no—
111Dic127—41no—
112C2×C127#54811yescharacteristic
113C2×C127C2×#13711yescharacteristic
114C2×C127#57711yescharacteristic
115D127—81no—
116#3369—1271nomaximal, Sylow
117C127#336911yescharacteristic, complemented, Sylow
118Q82D12711yescharacteristic, complemented
119#556—1271no—
120#489—1272no—
121#487—1271no—
122#476—1271no—
123#475—1272no—
124#464—1272no—
125#462—1271no—
126#446—1271no—
127#441—1272no—
128C4×SD16—1271no—
129C2×Q16—1272no—
130C2×SD16—1271no—
131C4⋊Q8—21no—
132C4⋊Q8—22no—
133C4⋊Q8—1271no—
134C4⋊Q8C2×D12712yes—
135C4⋊Q8C2×D12711yescharacteristic
136#126—2542no—
137#124—1272no—
138C22⋊Q8—2541no—
139C4×Q8—21no—
140C4×Q8—1272no—
141C4×Q8C2×D12711yescharacteristic
142C4×Q8C2×D12712yes—
143C4×Q8C2×D12711yescharacteristic
144C4×D4—1271no—
145#117—2541no—
146#106—1271no—
147#106—1272no—
148C4⋊C8—1271no—
149C4⋊C8—1272no—
150#103—1271no—
151#103—1272no—
152#103—1272no—
153#103—1272no—
154#102—1271no—
155#102—1272no—
156#97—1272no—
157C4×C8—1271no—
158C2×Q8—22no—
159C2×Q8—21no—
160C2×Q8—1272no—
161C2×Q8C22⋊D12711yescharacteristic
162C2×Q8C22×D12711yescharacteristic
163C2×D4—1271no—
164C22×C4—2541no—
165Q16—2544no—
166SD16—2542no—
167C2×C8—1271no—
168C2×C8—1271no—
169C2×C8—1272no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—1271no—
175C4⋊C4—2542no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4C22⋊D12711yescharacteristic
179C4⋊C4C22⋊D12712yes—
180C4⋊C4C22×D12711yescharacteristic
181C4⋊C4C22×D12712yes—
182C22⋊C4—2542no—
183C22⋊C4—2541no—
184C22⋊C4—2542no—
185C42—22no—
186C42—21no—
187C42—2542no—
188C42C22×D12711yescharacteristic
189C42C22×D12712yes—
190C23—2541no—
191Q8—22no—
192Q8—21no—
193Q8—82no—
194Q8—81no—
195Q8—2542no—
196Q8—2542no—
197Q8#1452324912yes—
198D4—2541no—
199D4—2541no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—5082no—
204C2×C4—2541no—
205C2×C4—2542no—
206C2×C4C2×C22⋊D12712yes—
207C2×C4C2×C22⋊D12712yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C23×D12711yescharacteristic
211C8—2541no—
212C8—2542no—
213C8—2541no—
214C22—5081no—
215C22—5081no—
216C22—5081no—
217C22C22×C22⋊D12711yescharacteristic
218C4—22no—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—42no—
223C4—41no—
224C4—5082no—
225C4—5081no—
226C4#1686124012yes—
227C4C2×#1452324911yescharacteristic
228C4#1686125711yescharacteristic
229C2—10161no—
230C2#2255053711yescharacteristic
231C2C2×#1686124011yescharacteristic
232C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingQ82×C127C2
socleC22×C127C22×D4
prime radicalC2×D508C23
hypercenterQ82D127
hypocenterC127#3369
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
3. #22550228C2semidirect
5. #22550158C2semidirect
7. #22550086C2semidirect
8. #22550007C2semidirect
9. #22549981C2semidirect
11. Q82×C127C2semidirect
117. C127#3369semidirect
118. Q82D127semidirect

Maximal subgroups

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

subgroupindex
#225505992
#225502382
#225502282
#225501632
#225501582
#225500982
#225500862
#225500072
#225499812
C4×#145232352
Q82×C1272
#3369127