Finite Group Databasebeta

← #30639875

Subgroups — #30639875

Order 16256.

294 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#16861200C211yesmaximal, characteristic, complemented
2C2×#16861131C211yesmaximal, characteristic
3C2×#16861129C211yesmaximal, characteristic
4#22549777C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861200—22no—
11#16861200—22no—
12C2×C4⋊Dic127—21no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17C2×C4×Dic127C2211yescharacteristic
18#16861131—22no—
19#16861129—22no—
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C23×D127—21no—
24C2×D508—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508Q812yes—
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C22×Dic127D411yescharacteristic
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—24no—
37C4⋊Dic127—21no—
38C4⋊Dic127D412yes—
39Dic127⋊C4—24no—
40(C2×D127)⋊C4—42no—
41(C2×D127)⋊C4—42no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127D412yes—
54C22×D127—22no—
55C22×D127—41no—
56C22×D127—41no—
57C22×D127—41no—
58C22×D127—41no—
59C22×D127—41no—
60C22×D127—41no—
61D508—22no—
62D508—22no—
63D508—41no—
64D508—41no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—22no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C127⋊C8—41no—
75C23×C127C2×D411yescharacteristic
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—41no—
89C22×C127#12011yescharacteristic
90C22×C127C22⋊Q811yescharacteristic
91C22×C127#12411yescharacteristic
92C22×C127#13511yescharacteristic
93C22×C127#13511yescharacteristic
94C22×C127AGL(1,ℤ/8)11yescharacteristic
95C22×C127AGL(1,ℤ/8)11yescharacteristic
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—81no—
100C2×D127—81no—
101C2×D127—81no—
102C2×D127—81no—
103C4×C127—42no—
104C4×C127—81no—
105C4×C127C22⋊Q811yescharacteristic
106C4×C127C22⋊Q811yescharacteristic
107C4×C127#12411yescharacteristic
108C4×C127#12411yescharacteristic
109Dic127—41no—
110Dic127—41no—
111Dic127—42no—
112C2×C127#39311yescharacteristic
113C2×C127#44911yescharacteristic
114C2×C127#44911yescharacteristic
115C2×C127#47811yescharacteristic
116C2×C127#47811yescharacteristic
117C2×C127#48811yescharacteristic
118C2×C127#49011yescharacteristic
119D127—82no—
120#2025—1271nomaximal, Sylow
121C127#202511yescharacteristic, complemented, Sylow
122C2×#125—1271no—
123C2×#102—1271no—
124C2×#102—1271no—
125#390—1271no—
126#340—1271no—
127#340D12711yescharacteristic, complemented
128#336—1271no—
129C22×D4—1271no—
130C22×C8—1271no—
131C22×C8C2×D12711yescharacteristic
132#125—2542no—
133#125—2542no—
134C2×C4⋊C4—1271no—
135C2×C4⋊C4—2541no—
136C2×C4⋊C4—2541no—
137C2×C4⋊C4C2×D12711yescharacteristic
138C2×C4⋊C4C2×D12711yescharacteristic
139C2×C22⋊C4—2541no—
140C2×C22⋊C4—2541no—
141C2×C42—1271no—
142#102—2542no—
143#102—2542no—
144C24—2541no—
145C2×D4—1272no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C2×D4—2541no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—2541no—
156C22×C4C22×D12711yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—2541no—
161C2×C8—2541no—
162C2×C8—2541no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—1272no—
167C4⋊C4—2544no—
168C4⋊C4—2544no—
169C4⋊C4—2541no—
170C4⋊C4D50812yes—
171C22⋊C4—5082no—
172C22⋊C4—5082no—
173C42—2541no—
174C42—2541no—
175C23—5081no—
176C23—5081no—
177C23—5081no—
178C23—5081no—
179C23—5081no—
180C23—5081no—
181C23—2542no—
182C23D4×D12711yescharacteristic
183D4—5081no—
184D4—5081no—
185D4—2542no—
186D4—2542no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4—2542no—
202C2×C4D4×D12711yescharacteristic
203C2×C4C2×D50811yescharacteristic
204C2×C4Q8×D12711yescharacteristic
205C2×C4#1452324711yescharacteristic
206C2×C4#1452324811yescharacteristic
207C2×C4#1452324811yescharacteristic
208C8—41no—
209C8—5081no—
210C22—10161no—
211C22—10161no—
212C22—10161no—
213C22—10161no—
214C22—5082no—
215C22—5082no—
216C22—5082no—
217C22#1686117911yescharacteristic
218C22#1686118711yescharacteristic
219C22#1686119511yescharacteristic
220C22#1686123011yescharacteristic
221C22#1686123211yescharacteristic
222C22#1686123311yescharacteristic
223C22#1686123611yescharacteristic
224C4—42no—
225C4—81no—
226C4—5081no—
227C4—5082no—
228C4—5081no—
229C4#1686118711yescharacteristic
230C4#1686118911yescharacteristic
231C4#1686119511yescharacteristic
232C4#1686119711yescharacteristic
233C2—10162no—
234C2#2254978911yescharacteristic
235C2#2255002211yescharacteristic
236C2#2255002411yescharacteristic
237C2#2255017511yescharacteristic
238C2#2255017911yescharacteristic
239C2#2255023211yescharacteristic
240C2#2255023911yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#340D127
hypocenterC127#2025
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#16861200C2semidirect
5. #22549638C2semidirect
6. #22549630C2semidirect
7. #340×C127C2semidirect
121. C127#2025semidirect
127. #340D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612002
C2×#168611312
C2×#168611292
#225497772
#225496382
#225496302
#340×C1272
#2025127