Finite Group Databasebeta

← #30637711

Subgroups — #30637711

Order 16256.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550463C211yesmaximal, characteristic
2#22550034C211yesmaximal, characteristic, complemented
3#22550014C211yesmaximal, characteristic, complemented
4#22549789C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549599C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C2×Dic508—21no—
10C2×#14523236—21no—
11#16861204—21no—
12#16861195—21no—
13#16861193—21no—
14#16861188C2211yescharacteristic
15#16861181—21no—
16#16861177—21no—
17D4×Dic127—21no—
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C23×D127D411yescharacteristic
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—42no—
31C2×Dic254D411yescharacteristic
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C22×Dic127—21no—
35C22×Dic127—21no—
36C22×Dic127D411yescharacteristic
37Dic508—42no—
38#14523236—42no—
39C2×C127⋊C8—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42Dic127⋊C4—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—42no—
50C4×Dic127—21no—
51C2×D4×C127—21no—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58C22×D127—21no—
59C22×D127—21no—
60C22×D127—21no—
61C22×D127—41no—
62C22×D127—41no—
63Dic254—22no—
64Dic254—22no—
65Dic254—41no—
66Dic254—41no—
67Dic254—41no—
68C22⋊D127—81no—
69C22⋊D127—81no—
70C2×Dic127—22no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—41no—
77C2×Dic127—81no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127C2×D411yescharacteristic
81D4×C127—41no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C8×C127—41no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127#12011yescharacteristic
98C22×C127#13511yescharacteristic
99C22×C127#13711yescharacteristic
100C2×D127—42no—
101C2×D127—41no—
102C2×D127—41no—
103C2×D127—41no—
104C2×D127—81no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—81no—
109Dic127—41no—
110Dic127—41no—
111Dic127—42no—
112Dic127—41no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—81no—
116C2×C127#45211yescharacteristic
117C2×C127#45511yescharacteristic
118C2×C127B(4,𝔽2)11yescharacteristic
119D127—81no—
120#1606—1271nomaximal, Sylow
121C127#160611yescharacteristic, complemented, Sylow
122#538—1271no—
123#451—1271no—
124#448—1271no—
125#393—1271no—
126#328—1271no—
127#327D12711yescharacteristic, complemented
128#323—1271no—
129C22×Q8—1271no—
130C2×Q16—2541no—
131C2×SD16—2541no—
132#126—2541no—
133#124—2541no—
134#123—2541no—
135C22⋊Q8—1271no—
136#121—2541no—
137#121C2×D12711yescharacteristic
138#120—2541no—
139C4×D4—2541no—
140C2×C22⋊C4—1271no—
141C2×C22⋊C4—2541no—
142#103—2541no—
143#103—2541no—
144C22⋊C8—1271no—
145C22⋊C8C2×D12711yescharacteristic
146#95—2541no—
147#95C2×D12711yescharacteristic
148C24—1271no—
149C2×Q8—1271no—
150C2×Q8—5082no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×D4—21no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4D50811yescharacteristic
157C22×C4—21no—
158C22×C4—1271no—
159C22×C4—2541no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—5082no—
163SD16—5082no—
164C2×C8—21no—
165C2×C8—2541no—
166C4⋊C4—1271no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C4⋊C4D50811yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—5082no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C42—2541no—
179C23—21no—
180C23—5081no—
181C23—5081no—
182C23—2541no—
183C23—2541no—
184C23—2541no—
185C23D4×D12711yescharacteristic
186Q8—5081no—
187Q8—5081no—
188Q8—5081no—
189Q8—2542no—
190Q8—2542no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—10161no—
195D4—10161no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—10161no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2542no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4—2541no—
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C8—41no—
212C8—5081no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—10161no—
219C22—5082no—
220C22—5081no—
221C22—5081no—
222C22—5081no—
223C22#1686117911yescharacteristic
224C22#1686122811yescharacteristic
225C22#1686123811yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—81no—
230C4—5081no—
231C4—5081no—
232C4—5082no—
233C4—5081no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—10161no—
238C2#2255003911yescharacteristic
239C2#2255005511yescharacteristic
240C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550463C2
hypercenter#327D127
hypocenterC127#1606
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
3. #22550014C2semidirect
4. #22549789C2semidirect
5. #22549612C2semidirect
6. #22549599C2semidirect
7. #327×C127C2semidirect
121. C127#1606semidirect
127. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504632
#225500342
#225500142
#225497892
#225496122
#225495992
#327×C1272
#1606127