Finite Group Databasebeta

← #30639885

Subgroups — #30639885

Order 16256.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#22549882C211yesmaximal, characteristic
3#22549880C211yesmaximal, characteristic
4#22549778C211yesmaximal, characteristic
5#22549644C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861209—21no—
11#16861208—21no—
12#16861201—21no—
13#16861200—21no—
14#16861186—22no—
15#16861168C2211yescharacteristic
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—22no—
18#16861131—21no—
19#16861131—21no—
20#16861129—21no—
21#16861129—21no—
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24C2×C42×C127C2211yescharacteristic
25C23×D127—21no—
26C2×Dic254—22no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508Q812yescomplemented
33C22×Dic127D412yes—
34#14523232—21no—
35#14523232—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—22no—
38C4⋊Dic127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127D411yescharacteristic
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—44no—
46C4×Dic127—21no—
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49M16×C127—21no—
50M16×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127D411yescharacteristic
53C4⋊C4×C127D411yescharacteristic
54C22⋊C4×C127—21no—
55C42×C127—21no—
56C42×C127—21no—
57C42×C127—21no—
58C22×D127—22no—
59C22×D127—42no—
60C22×D127—42no—
61C22×D127—41no—
62C22×D127—41no—
63Dic254—82no—
64D508—22no—
65D508—22no—
66D508—41no—
67D508—41no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—22no—
71C2×Dic127—42no—
72C127⋊C8—41no—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—22no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
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#12011yescharacteristic
89C22×C127C22⋊Q811yescharacteristic
90C22×C127#12411yescharacteristic
91C2×D127—44no—
92C2×D127—42no—
93C2×D127—82no—
94C2×D127—81no—
95C2×D127—81no—
96C4×C127—42no—
97C4×C127—41no—
98C4×C127—41no—
99C4×C127C22⋊Q811yescharacteristic
100C4×C127C22⋊Q811yescharacteristic
101C4×C127#12411yescharacteristic
102C4×C127#12411yescharacteristic
103Dic127—42no—
104Dic127—41no—
105Dic127—41no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127#39311yescharacteristic
109C2×C127#45311yescharacteristic
110C2×C127#45411yescharacteristic
111D127—82no—
112#2027—1271nomaximal, Sylow
113C127#202711yescharacteristic, complemented, Sylow
114#544—1271no—
115#418—1271no—
116#418—1271no—
117#390—1271no—
118#343—1271no—
119#339—1271no—
120#339D12711yescharacteristic, complemented
121C22×D4—1271no—
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×C42C2×D12711yescharacteristic
134#102—2541no—
135#102—2541no—
136#102—2541no—
137#102—2541no—
138C24—2541no—
139C2×Q8—2542no—
140C2×D4—1272no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C22×C4—21no—
147C22×C4—1272no—
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⋊C4D50811yescharacteristic
163C4⋊C4D50811yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—5084no—
166C22⋊C4—2541no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—2541no—
171C23—5082no—
172C23—5082no—
173C23—5081no—
174C23—5081no—
175C23—2542no—
176C23D4×D12711yescharacteristic
177Q8—10162no—
178D4—5081no—
179D4—5081no—
180D4—2542no—
181D4—2542no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5082no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2542no—
191C2×C4D4×D12711yescharacteristic
192C2×C4C2×D50811yescharacteristic
193C2×C4Q8×D12711yescharacteristic
194C2×C4#1452324711yescharacteristic
195C2×C4#1452324811yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—41no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—10162no—
202C22—10161no—
203C22—10161no—
204C22—5084no—
205C22—5082no—
206C22#1686117911yescharacteristic
207C22#1686118711yescharacteristic
208C22#1686119511yescharacteristic
209C4—42no—
210C4—41no—
211C4—41no—
212C4—5082no—
213C4—5081no—
214C4—5081no—
215C4#1686118711yescharacteristic
216C4#1686118911yescharacteristic
217C4#1686119511yescharacteristic
218C4#1686119711yescharacteristic
219C2—21no—
220C2—21no—
221C2—10162no—
222C2#2254978911yescharacteristic
223C2#2255004511yescharacteristic
224C2#2255005011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127#120
prime radicalC22×D508C22
hypercenter#339D127
hypocenterC127#2027
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #22550504C2semidirect
5. #22549644C2semidirect
6. #22549636C2semidirect
7. #339×C127C2semidirect
32. C2×D508Q8semidirect
113. C127#2027semidirect
120. #339D127semidirect

Maximal subgroups

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

subgroupindex
#225505042
#225498822
#225498802
#225497782
#225496442
#225496362
#339×C1272
#2027127