Finite Group Databasebeta

← #30640382

Subgroups — #30640382

Order 16256.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550448C211yesmaximal, characteristic
2#22549857C211yesmaximal, characteristic
3#22549848C211yesmaximal, characteristic
4#22549667C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549665C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8C2×#14523247C2211yescharacteristic
9C22×Dic254C411yescharacteristic, complemented
10M16×D127—21no—
11#16861208—21no—
12#16861195—21no—
13#16861193C411yescharacteristic, complemented
14#16861192C411yescharacteristic, complemented
15#16861190C2211yescharacteristic
16#16861187—21no—
17#16861186—21no—
18#16861165—21no—
19#16861165C411yescharacteristic, complemented
20#16861127—21no—
21#16861126C2211yescharacteristic
22#16861124C2211yescharacteristic
23#16861123—21no—
24#123×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27#14523247—41no—
28#14523247—41no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—41no—
32C2×Dic254—41no—
33C2×Dic254D411yescharacteristic
34C2×C22⋊D127C2×C411yescharacteristic
35C2×C22⋊D127D411yescharacteristic
36C22×Dic127C2×C411yescharacteristic
37C22×Dic127C2×C411yescharacteristic
38C22×Dic127D411yescharacteristic
39C2×C4×D127D411yescharacteristic
40#14523232—21no—
41#14523231—41no—
42C8×D127—41no—
43C4⋊Dic127—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46Dic127⋊C4—21no—
47Dic127⋊C4—41no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127C2×C411yescharacteristic
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C4×Dic127—21no—
54C4×Dic127—41no—
55C2×D4×C127C2311yescharacteristic
56C22×C4×C127C2×C411yescharacteristic
57M16×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127C2×C411yescharacteristic
62C22×D127C22⋊C411yescharacteristic
63Dic254—41no—
64Dic254—42no—
65Dic254—41no—
66Dic254—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127C22⋊C411yescharacteristic
78C2×Dic127C22⋊C411yescharacteristic
79C2×Dic127C22⋊C411yescharacteristic
80C4×D127—21no—
81C4×D127—21no—
82C127⋊C8—41no—
83C23×C127C22×C411yescharacteristic
84C23×C127C2×D411yescharacteristic
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C8×C127—41no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127C2×C22⋊C411yescharacteristic
97C2×D127—21no—
98C2×D127—41no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—41no—
102C4×C127—81no—
103Dic127—21no—
104Dic127—21no—
105Dic127—41no—
106Dic127—41no—
107Dic127—81no—
108Dic127#9911yescharacteristic
109Dic127#9911yescharacteristic
110C2×C127—21no—
111C2×C127—41no—
112C2×C127—41no—
113C2×C127C2×#9911yescharacteristic
114D127—41no—
115#2123—1271nomaximal, Sylow
116C127#212311yescharacteristic, complemented, Sylow
117#536—1271no—
118#413—1271no—
119#410—1271no—
120#352—1271no—
121#352—1271no—
122#352—1271no—
123#352D12711yescharacteristic, complemented
124C2×#41—1271no—
125C22×Q8—1271no—
126C2×M16—2541no—
127C4⋊Q8—2541no—
128#124—2541no—
129#123—1271no—
130#123—1271no—
131#123—1271no—
132#123C2×D12711yescharacteristic
133C22⋊Q8—2541no—
134C22⋊Q8—2541no—
135#117—1271no—
136#117—2541no—
137#101—2541no—
138#100—1271no—
139#100C2×D12711yescharacteristic
140#99—1271no—
141#99—2541no—
142#99C2×D12711yescharacteristic
143#41—5081no—
144#41—5081no—
145C2×Q8—1271no—
146C2×Q8—5081no—
147C2×Q8—5081no—
148C2×Q8—2541no—
149C2×Q8—2541no—
150C2×D4—1271no—
151C2×D4—1271no—
152C2×D4C22×D12711yescharacteristic
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4C4×D12711yescharacteristic
158M16—21no—
159M16—5081no—
160M16—2541no—
161C2×C8—5081no—
162C4⋊C4—21no—
163C4⋊C4—5081no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—1271no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4C4×D12711yescharacteristic
176C42—5081no—
177C42—2541no—
178C23—1271no—
179C23C2×C4×D12711yescharacteristic
180C23D4×D12711yescharacteristic
181Q8—5081no—
182Q8—5081no—
183Q8—5081no—
184Q8—5082no—
185D4—41no—
186D4—5081no—
187D4—5081no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—1271no—
193C2×C4—1271no—
194C2×C4—1271no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4D4×D12711yescharacteristic
206C8—41no—
207C8—5081no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—5081no—
213C22—2541no—
214C22C22⋊C4×D12711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—1271no—
220C4—1271no—
221C4—10161no—
222C4—5081no—
223C4—5081no—
224C4—2541no—
225C4—2541no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—5081no—
230C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127C2×#99
prime radicalC2×#14523247C22
cosocleC2×D4×C127C23
hypercenter#352D127
hypocenterC127#2123
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #22549667C2semidirect
5. #22549666C2semidirect
6. #22549665C2semidirect
7. #352×C127C2semidirect
9. C22×Dic254C4semidirect
13. #16861193C4semidirect
14. #16861192C4semidirect
19. #16861165C4semidirect
116. C127#2123semidirect
123. #352D127semidirect

Maximal subgroups

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

subgroupindex
#225504482
#225498572
#225498482
#225496672
#225496662
#225496652
#352×C1272
#2123127