Finite Group Databasebeta

← #30637756

Subgroups — #30637756

Order 16256.

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

normal subgroups only

This group has 236 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#22550212C211yesmaximal, characteristic, complemented
3#22550209C211yesmaximal, characteristic, complemented
4#22549813C211yesmaximal, characteristic
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic
7#328×C127C211yesmaximal, characteristic, complemented
8#16861189—21no—
9#16861188C2211yescharacteristic
10#16861187—21no—
11#16861187—21no—
12#16861186—21no—
13#16861185—21no—
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4—21no—
17C2×(C2×D127)⋊C4C2211yescharacteristic
18C22⋊C4×D127—21no—
19#16861143—21no—
20#16861142—21no—
21#16861134—21no—
22#16861133—21no—
23C22⋊(C127⋊C8)C2211yescharacteristic
24C22⋊Q8×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254D411yescharacteristic
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—42no—
39C4⋊Dic127D411yescharacteristic
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—42no—
49C2×Q8×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127D411yescharacteristic
55C22⋊C4×C127—21no—
56C22×D127—21no—
57C22×D127—21no—
58C22×D127—41no—
59C22×D127—41no—
60C22×D127Q1612yes—
61Dic254—41no—
62Dic254—81no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C4×D127—81no—
71C4×D127—81no—
72C127⋊C8—41no—
73C23×C127C2×D411yescharacteristic
74Q8×C127—41no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C8×C127—41no—
84C22×C127—21no—
85C22×C127—21no—
86C22×C127#12011yescharacteristic
87C22×C127C2×Q1611yescharacteristic
88C22×C127#13711yescharacteristic
89C2×D127—22no—
90C2×D127—41no—
91C2×D127—41no—
92C2×D127—42no—
93C2×D127—81no—
94C4×C127—21no—
95C4×C127—21no—
96C4×C127—42no—
97C4×C127—41no—
98C4×C127—41no—
99Dic127—41no—
100Dic127—41no—
101Dic127—41no—
102Dic127—81no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#45111yescharacteristic
106C2×C127#45311yescharacteristic
107C2×C127#45811yescharacteristic
108D127—42no—
109#1614—1271nomaximal, Sylow
110C127#161411yescharacteristic, complemented, Sylow
111#543—1271no—
112#484—1271no—
113#484—1271no—
114#399—1271no—
115#328—1271no—
116#328D12711yescharacteristic, complemented
117#323—1271no—
118C22⋊Q8—1271no—
119C22⋊Q8—2541no—
120C22⋊Q8—2541no—
121C22⋊Q8—2541no—
122C22⋊Q8—2541no—
123C22⋊Q8—2541no—
124C22⋊Q8C2×D12711yescharacteristic
125C2×C4⋊C4—1271no—
126C2×C4⋊C4—2541no—
127C2×C22⋊C4—1271no—
128C2×C22⋊C4—2541no—
129C2×C22⋊C4—2541no—
130#107—2541no—
131#107—2541no—
132#103—2541no—
133#103—2541no—
134C22⋊C8—1271no—
135C22⋊C8C2×D12711yescharacteristic
136#95C2×D12711yescharacteristic
137C24—1271no—
138C2×Q8—1271no—
139C2×Q8—2541no—
140C2×Q8D50811yescharacteristic
141C22×C4—21no—
142C22×C4—1271no—
143C22×C4—2541no—
144C22×C4—2541no—
145C22×C4—2541no—
146C22×C4C22×D12711yescharacteristic
147C2×C8—21no—
148C2×C8—2541no—
149C4⋊C4—21no—
150C4⋊C4—1271no—
151C4⋊C4—5082no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C4⋊C4—2541no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4D50811yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—5082no—
161C22⋊C4—2541no—
162C22⋊C4—2541no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C23—1272no—
167C23—5081no—
168C23—5081no—
169C23—2541no—
170C23—2541no—
171C23D4×D12711yescharacteristic
172Q8—41no—
173Q8—10161no—
174Q8—5081no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—10161no—
182C2×C4—10161no—
183C2×C4—5081no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—5081no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4D4×D12711yescharacteristic
191C2×C4C2×D50811yescharacteristic
192C8—41no—
193C8—5081no—
194C22—21no—
195C22—21no—
196C22—10161no—
197C22—5082no—
198C22—5081no—
199C22—5081no—
200C22—2542no—
201C22#1686117911yescharacteristic
202C22Q16×D12711yescharacteristic
203C22#1686123811yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—10161no—
210C4—5081no—
211C4—5081no—
212C4—5081no—
213C2—21no—
214C2—21no—
215C2—5082no—
216C2#2255003311yescharacteristic
217C2#2255004511yescharacteristic
218C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#328×C127C2
socleC22×C127#120
prime radicalC23×D127D4
hypercenter#328D127
hypocenterC127#1614
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. #22550212C2semidirect
3. #22550209C2semidirect
5. #22549612C2semidirect
7. #328×C127C2semidirect
110. C127#1614semidirect
116. #328D127semidirect

Maximal subgroups

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

subgroupindex
#225505012
#225502122
#225502092
#225498132
#225496122
#225495992
#328×C1272
#1614127