Finite Group Databasebeta

← #30639195

Subgroups — #30639195

Order 16256.

338 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D127C211yesmaximal, characteristic
2C2×#16861185C211yesmaximal, characteristic
3C2×#16861128C211yesmaximal, characteristic, complemented
4#22549839C211yesmaximal, characteristic
5#22549644C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8C22×C4×D127C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11C2×#14523231—21no—
12M16×D127—22no—
13M16×D127—22no—
14C2×C8×D127—21no—
15#16861185—21no—
16#16861185—21no—
17#16861185—21no—
18#16861185—21no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C412yes—
21C2×(C2×D127)⋊C4C412yes—
22#16861128—22no—
23(C2×D127)⋊C8—21no—
24(C2×D127)⋊C8—21no—
25C22×Q8×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#101×C127—22no—
29C23×D127C2×C411yescharacteristic
30C22×Dic127C2×C411yescharacteristic
31C22×Dic127C2×C412yes—
32C2×C4×D127—21no—
33C2×C4×D127—21no—
34C2×C4×D127—21no—
35C2×C4×D127—21no—
36C2×C4×D127D411yescharacteristic
37C2×C4×D127D411yescharacteristic
38C2×C4×D127D411yescharacteristic
39C2×C4×D127D411yescharacteristic
40#14523232—21no—
41#14523232—21no—
42#14523232—21no—
43#14523232D412yes—
44#14523231—24no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C8×D127—24no—
48C4⋊Dic127—21no—
49C4⋊Dic127—21no—
50Dic127⋊C4—22no—
51Dic127⋊C4—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C22×C4×C127C2×C412yes—
61C22×C4×C127C2311yescharacteristic
62M16×C127—21no—
63M16×C127—21no—
64M16×C127—21no—
65M16×C127D412yes—
66C2×C8×C127—21no—
67C2×C8×C127—21no—
68C22×D127—21no—
69C22×D127—21no—
70C22×D127—22no—
71C22×D127—41no—
72C22×D127C22⋊C411yescharacteristic
73C22×D127C22⋊C411yescharacteristic
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—22no—
77C2×Dic127—42no—
78C2×Dic127C22⋊C411yescharacteristic
79C2×Dic127C22⋊C411yescharacteristic
80C4×D127—21no—
81C4×D127—21no—
82C4×D127—21no—
83C4×D127—21no—
84C4×D127—21no—
85C4×D127—21no—
86C4×D127—21no—
87C4×D127—21no—
88C127⋊C8—22no—
89C127⋊C8—41no—
90C23×C127C22×C411yescharacteristic
91Q8×C127—41no—
92Q8×C127—41no—
93Q8×C127—41no—
94Q8×C127—41no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C8×C127—22no—
105C8×C127—41no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127C2×C22⋊C411yescharacteristic
109C22×C127C4×D411yescharacteristic
110C22×C127C4×D411yescharacteristic
111C2×D127—21no—
112C2×D127—21no—
113C2×D127—21no—
114C2×D127—21no—
115C2×D127—21no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127#10112yescomplemented
119C4×C127—42no—
120C4×C127#12011yescharacteristic
121C4×C127#12111yescharacteristic
122C4×C127#12111yescharacteristic
123C4×C127#12311yescharacteristic
124Dic127—21no—
125Dic127—21no—
126Dic127—42no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127#38611yescharacteristic
130C2×C127C2×#10111yescharacteristic
131C2×C127#41311yescharacteristic
132D127—22no—
133D127—41no—
134#1901—1271nomaximal, Sylow
135C127#190111yescharacteristic, complemented, Sylow
136C22×M16—1271no—
137C2×C22⋊Q8—1271no—
138C2×#101—1271no—
139C2×#101D12711yescharacteristic, complemented
140#407—1271no—
141#343—1271no—
142#343—1271no—
143C22×Q8C2×D12711yescharacteristic
144C23×C4—1271no—
145C2×M16—1271no—
146C2×M16—1271no—
147C2×M16—2541no—
148C2×M16—2542no—
149C2×M16—2542no—
150C2×M16C2×D12711yescharacteristic
151C2×M16C2×D12711yescharacteristic
152C22×C8—2541no—
153C22⋊Q8—2541no—
154C22⋊Q8—2541no—
155C22⋊Q8—2541no—
156C22⋊Q8—2541no—
157C2×C4⋊C4—1272no—
158C2×C4⋊C4—1271no—
159C2×C22⋊C4—1272no—
160#101—22no—
161#101—2542no—
162C22⋊C8—2541no—
163C22⋊C8—2541no—
164C24—1271no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C22×C4—1271no—
172C22×C4—1271no—
173C22×C4—1271no—
174C22×C4—1271no—
175C22×C4—1271no—
176C22×C4—1272no—
177C22×C4—2541no—
178C22×C4—2541no—
179C22×C4—2541no—
180C22×C4—2541no—
181C22×C4C4×D12712yes—
182C22×C4C22×D12711yescharacteristic
183M16—21no—
184M16—21no—
185M16—21no—
186M16—1272no—
187M16—2544no—
188M16—2541no—
189M16—2541no—
190M16—2541no—
191M16C22⋊D12712yes—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—2544no—
195C2×C8—2541no—
196C2×C8—2541no—
197C4⋊C4—2542no—
198C4⋊C4—2542no—
199C4⋊C4—2541no—
200C4⋊C4—2541no—
201C22⋊C4—2542no—
202C22⋊C4—2542no—
203C23—1271no—
204C23—1271no—
205C23—5081no—
206C23—2542no—
207C23—2541no—
208C23—2541no—
209C23C2×C4×D12711yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—1271no—
218C2×C4—1271no—
219C2×C4—5082no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2541no—
226C2×C4—2541no—
227C2×C4—2541no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×C22⋊D12711yescharacteristic
235C2×C4#1452324711yescharacteristic
236C2×C4#1452324811yescharacteristic
237C8—22no—
238C8—41no—
239C8—5081no—
240C8—2542no—
241C22—21no—
242C22—21no—
243C22—1272no—
244C22—5082no—
245C22—5082no—
246C22—2541no—
247C22—2541no—
248C22—2541no—
249C22—2541no—
250C22—2541no—
251C22C22⋊C4×D12711yescharacteristic
252C22(C22⋊D127)⋊C411yescharacteristic
253C22C4×C22⋊D12711yescharacteristic
254C4—42no—
255C4—5082no—
256C4—2541no—
257C4—2541no—
258C4#1686117711yescharacteristic
259C4#1686118011yescharacteristic
260C4#1686118211yescharacteristic
261C4#1686119111yescharacteristic
262C2—21no—
263C2—21no—
264C2—5081no—
265C2—2542no—
266C2#2254975211yescharacteristic
267C2#101×D12711yescharacteristic
268C2#2254985911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1901
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
3. C2×#16861128C2semidirect
5. #22549644C2semidirect
6. #22549644C2semidirect
7. C2×#101×C127C2semidirect
118. C2×D127#101semidirect
135. C127#1901semidirect
139. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1272
C2×#168611852
C2×#168611282
#225498392
#225496442
C2×#101×C1272
#1901127