Finite Group Databasebeta

← #30639884

Subgroups — #30639884

Order 16256.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550504C211yesmaximal, characteristic, complemented
2#22549881C211yesmaximal, characteristic
3#22549779C211yesmaximal, characteristic
4#22549644C211yesmaximal, characteristic, complemented
5#22549637C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861209—21no—
10#16861208—21no—
11#16861201—21no—
12#16861200—21no—
13#16861186—22no—
14#16861168C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861130—21no—
19#16861130—21no—
20C22×D4×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#102×C127—21no—
24#102×C127—21no—
25C2×Dic254—22no—
26C22×Dic127—21no—
27C22×Dic127D412yes—
28#14523232—21no—
29#14523232—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D411yescharacteristic
35C4⋊Dic127D411yescharacteristic
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38C22⋊Dic127—21no—
39C22⋊Dic127—44no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C24×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127Q812yescomplemented
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127D411yescharacteristic
55C4⋊C4×C127D411yescharacteristic
56C22⋊C4×C127—21no—
57C42×C127—21no—
58Dic254—82no—
59C2×Dic127—22no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—22no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—42no—
66C127⋊C8—41no—
67C23×C127—22no—
68C23×C127—42no—
69C23×C127—42no—
70C23×C127—41no—
71C23×C127—41no—
72C23×C127C2×D411yescharacteristic
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×Q811yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—41no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—44no—
89C22×C127—42no—
90C22×C127—82no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127#12011yescharacteristic
94C22×C127C22⋊Q811yescharacteristic
95C22×C127#12411yescharacteristic
96C4×C127—41no—
97C4×C127—41no—
98C4×C127C22⋊Q811yescharacteristic
99C4×C127C22⋊Q811yescharacteristic
100C4×C127#12411yescharacteristic
101C4×C127#12411yescharacteristic
102Dic127—42no—
103Dic127—42no—
104Dic127—41no—
105Dic127—41no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—82no—
109C2×C127#39311yescharacteristic
110C2×C127#45311yescharacteristic
111C2×C127#45411yescharacteristic
112#2027—1271nomaximal, Sylow
113C127#202711yescharacteristic, complemented, Sylow
114#544—1271no—
115#418—1271no—
116#418D12711yescharacteristic
117#390—1271no—
118#343—1271no—
119#339—1271no—
120#339—1271no—
121C22×D4C2×D12711yescharacteristic
122C2×M16—1271no—
123C2×M16C2×D12711yescharacteristic
124C4⋊Q8—2541no—
125C4⋊Q8—2541no—
126#125—2541no—
127#125—2541no—
128C22⋊Q8—2542no—
129#117—1271no—
130#117C2×D12711yescharacteristic
131C2×C4⋊C4—1271no—
132C2×C22⋊C4—2542no—
133C2×C42—1271no—
134#102—21no—
135#102—21no—
136#102—2541no—
137#102—2541no—
138C24—21no—
139C2×Q8—2542no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4Dic25412yescomplemented
146C22×C4—1272no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149M16—21no—
150M16—21no—
151M16—2541no—
152M16—2541no—
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—1271no—
156C4⋊C4—1271no—
157C4⋊C4—2542no—
158C4⋊C4—2542no—
159C4⋊C4—2542no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4C22⋊D12711yescharacteristic
163C4⋊C4C22⋊D12711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—5084no—
166C22⋊C4—2541no—
167C42—21no—
168C42—2541no—
169C42—2541no—
170C42—2541no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—41no—
175C23—41no—
176C23D4×D12711yescharacteristic
177Q8—10162no—
178D4—22no—
179D4—22no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—5082no—
187C2×C4—2542no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2542no—
191C2×C4D4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C2×C4C2×Dic25411yescharacteristic
194C2×C4#1452324711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—41no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22#1686117711yescharacteristic
207C22#1686118611yescharacteristic
208C22#1686119511yescharacteristic
209C4—41no—
210C4—41no—
211C4—5082no—
212C4—5082no—
213C4—5081no—
214C4—5081no—
215C4#1686118611yescharacteristic
216C4#1686118811yescharacteristic
217C4#1686119511yescharacteristic
218C4#1686119611yescharacteristic
219C2—21no—
220C2—21no—
221C2—82no—
222C2#2254978811yescharacteristic
223C2#2255004211yescharacteristic
224C2#2255004711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
hypercenter#418D127
hypocenterC127#2027
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22550504C2semidirect
4. #22549644C2semidirect
5. #22549637C2semidirect
6. #22549635C2semidirect
49. C2×D4×C127Q8semidirect
113. C127#2027semidirect
145. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225498812
#225497792
#225496442
#225496372
#225496352
#418×C1272
#2027127