Finite Group Databasebeta

← #30637688

Subgroups — #30637688

Order 16256.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550501C211yesmaximal, characteristic
2#22550034C211yesmaximal, characteristic, complemented
3#22550031C211yesmaximal, characteristic, complemented
4#22549789C211yesmaximal, characteristic
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic
7#328×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C2×Dic508—21no—
10C2×#14523238—21no—
11#16861189—21no—
12#16861188C2211yescharacteristic
13#16861187—21no—
14#16861187—21no—
15#16861186—21no—
16#16861185—21no—
17C2×(C2×D127)⋊C4—21no—
18C2×(C2×D127)⋊C4C2211yescharacteristic
19C22⋊C4×D127—21no—
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—42no—
32C2×Dic254D411yescharacteristic
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37Dic508—42no—
38#14523238—42no—
39C2×C127⋊C8—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45C22⋊Dic127—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—21no—
50(C2×D127)⋊C4—42no—
51C2×Q8×C127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127Q1612yescomplemented
63Dic254—22no—
64Dic254—22no—
65Dic254—41no—
66Dic254—41no—
67Dic254—41no—
68Dic254—81no—
69C2×Dic127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C4×D127—81no—
77C4×D127—81no—
78C127⋊C8—41no—
79C23×C127C2×D411yescharacteristic
80Q8×C127—41no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C8×C127—41no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127#12011yescharacteristic
93C22×C127C2×Q1611yescharacteristic
94C22×C127#13711yescharacteristic
95C2×D127—22no—
96C2×D127—41no—
97C2×D127—41no—
98C2×D127—42no—
99C2×D127—81no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—41no—
103C4×C127—41no—
104C4×C127—81no—
105Dic127—41no—
106Dic127—41no—
107Dic127—41no—
108Dic127—42no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127#45111yescharacteristic
112C2×C127#45511yescharacteristic
113C2×C127B(4,𝔽2)11yescharacteristic
114D127—42no—
115#1602—1271nomaximal, Sylow
116C127#160211yescharacteristic, complemented, Sylow
117#543—1271no—
118#451—1271no—
119#451—1271no—
120#393—1271no—
121#328—1271no—
122#328D12711yescharacteristic, complemented
123#323—1271no—
124C22×Q8—1271no—
125C2×Q16—2541no—
126C2×Q16—2541no—
127C22⋊Q8—1271no—
128C22⋊Q8—2541no—
129C22⋊Q8—2541no—
130C22⋊Q8—2541no—
131C22⋊Q8—2541no—
132C22⋊Q8—2541no—
133C22⋊Q8C2×D12711yescharacteristic
134C2×C22⋊C4—1271no—
135C2×C22⋊C4—2541no—
136C2×C22⋊C4—2541no—
137#103—2541no—
138#103—2541no—
139C22⋊C8—1271no—
140C22⋊C8C2×D12711yescharacteristic
141#95—2541no—
142#95C2×D12711yescharacteristic
143C24—1271no—
144C2×Q8—1271no—
145C2×Q8—5082no—
146C2×Q8—2541no—
147C2×Q8—2541no—
148C2×Q8—2541no—
149C2×Q8D50811yescharacteristic
150C22×C4—21no—
151C22×C4—1271no—
152C22×C4—2541no—
153C22×C4—2541no—
154C22×C4—2541no—
155C22×C4C22×D12711yescharacteristic
156Q16—5082no—
157Q16—5082no—
158C2×C8—21no—
159C2×C8—2541no—
160C4⋊C4—21no—
161C4⋊C4—1271no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4D50811yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—5082no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C23—1272no—
175C23—5081no—
176C23—5081no—
177C23—2541no—
178C23—2541no—
179C23D4×D12711yescharacteristic
180Q8—41no—
181Q8—10161no—
182Q8—5081no—
183Q8—5081no—
184Q8—5081no—
185Q8—2542no—
186Q8—2542no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—10161no—
194C2×C4—10161no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—2542no—
199C2×C4—2541no—
200C2×C4—2541no—
201C2×C4—2541no—
202C2×C4D4×D12711yescharacteristic
203C2×C4C2×D50811yescharacteristic
204C8—41no—
205C8—5081no—
206C22—21no—
207C22—21no—
208C22—10161no—
209C22—5082no—
210C22—5081no—
211C22—5081no—
212C22—2542no—
213C22#1686117911yescharacteristic
214C22Q16×D12711yescharacteristic
215C22#1686123811yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—5081no—
222C4—5081no—
223C4—5081no—
224C4—5082no—
225C2—21no—
226C2—21no—
227C2—5082no—
228C2#2255003311yescharacteristic
229C2#2255005511yescharacteristic
230C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radicalC23×D127D4
hypercenter#328D127
hypocenterC127#1602
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
3. #22550031C2semidirect
5. #22549612C2semidirect
7. #328×C127C2semidirect
62. C22×D127Q16semidirect
116. C127#1602semidirect
122. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225505012
#225500342
#225500312
#225497892
#225496122
#225495992
#328×C1272
#1602127