Finite Group Databasebeta

← #30639233

Subgroups — #30639233

Order 16256.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#41×Dic127C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7#575×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—21no—
10D4×Dic127—22no—
11D4×Dic127—21no—
12#16861168—21no—
13#16861168C2211yescharacteristic
14#16861168C2211yescharacteristic
15C2×C4×Dic127—21no—
16C2×C4×Dic127C2211yescharacteristic
17#16861136—21no—
18#16861136—21no—
19#16861136—22no—
20#16861124—22no—
21C2×#41×C127C2211yescharacteristic
22C2×#41×C127C2211yescharacteristic
23#137×C127—21no—
24#137×C127C412yescomplemented
25AGL(1,ℤ/8)×C127—21no—
26AGL(1,ℤ/8)×C127C412yescomplemented
27#135×C127—22no—
28C2×M16×C127C2211yescharacteristic
29C22×Dic127—22no—
30C22×Dic127—21no—
31#14523232—21no—
32#14523232—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—22no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C22⋊Dic127—22no—
38C22⋊Dic127—21no—
39C22⋊Dic127D412yes—
40C4×Dic127—22no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C4×Dic127D411yescharacteristic
45C4×Dic127D411yescharacteristic
46#41×C127—21no—
47#41×C127—21no—
48#41×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127C2×C412yes—
53#41×C127D412yes—
54C2×Q8×C127C2×C411yescharacteristic, complemented
55C2×Q8×C127D411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127C2×C411yescharacteristic, complemented
59C2×D4×C127D411yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63Q16×C127—22no—
64SD16×C127—22no—
65SD16×C127—22no—
66D8×C127—22no—
67M16×C127—21no—
68M16×C127C2×C412yescomplemented
69C2×C8×C127—21no—
70C2×Dic127—21no—
71C2×Dic127—22no—
72C2×Dic127—21no—
73C2×Dic127—42no—
74C2×Dic127—44no—
75C2×Dic127—41no—
76C127⋊C8—41no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127C2×D411yescharacteristic
80Q8×C127—21no—
81Q8×C127—21no—
82Q8×C127—21no—
83Q8×C127C22⋊C412yes—
84D4×C127—22no—
85D4×C127—22no—
86D4×C127—21no—
87D4×C127—21no—
88D4×C127—21no—
89D4×C127—41no—
90D4×C127—41no—
91D4×C127C22⋊C412yes—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C22×C411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4111yescharacteristic
103C2×C4×C127#4111yescharacteristic
104C8×C127—22no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—41no—
109C22×C127—81no—
110C22×C127C4×D411yescharacteristic
111C22×C127#12011yescharacteristic
112C22×C127#12111yescharacteristic
113C4×C127—22no—
114C4×C127—41no—
115C4×C127C2×C22⋊C411yescharacteristic
116C4×C127C4×D411yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127#12311yescharacteristic
119Dic127—22no—
120Dic127—42no—
121Dic127—41no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—42no—
127C2×C127—81no—
128C2×C127#38611yescharacteristic
129#1906—1271nomaximal, Sylow
130C127#190611yescharacteristic, complemented, Sylow
131#575D12711yescharacteristic
132C4×#41—1271no—
133#421—1271no—
134C2×#104—1271no—
135#410—1271no—
136#344—1271no—
137#339—1271no—
138C2×#41C2×D12711yescharacteristic
139C2×#41C2×D12711yescharacteristic
140#137—21no—
141#137Dic12712yescomplemented
142AGL(1,ℤ/8)—21no—
143AGL(1,ℤ/8)Dic12712yescomplemented
144#135—22no—
145C2×M16—1271no—
146C2×M16C2×D12711yescharacteristic
147C4×Q8—2541no—
148C4×D4—2542no—
149C4×D4—2541no—
150#117—1271no—
151#117—1271no—
152#117—2541no—
153C2×C42—1271no—
154C2×C42—2541no—
155#104—2541no—
156#104—2541no—
157#104—2542no—
158#99—2542no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41C2×Dic12712yes—
166#41C22⋊D12712yes—
167C2×Q8C2×Dic12711yescharacteristic, complemented
168C2×Q8C22⋊D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×Dic12711yescharacteristic, complemented
172C2×D4C22⋊D12711yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—2542no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178Q16—22no—
179SD16—22no—
180SD16—22no—
181D8—22no—
182M16—21no—
183M16—2541no—
184M16—2541no—
185M16C2×Dic12712yescomplemented
186C2×C8—21no—
187C2×C8—2541no—
188C4⋊C4—2542no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C22⋊C4—1272no—
192C22⋊C4—2541no—
193C22⋊C4—2542no—
194C42—1271no—
195C42—1271no—
196C42—2542no—
197C42—2541no—
198C42—2541no—
199C42—2541no—
200C23—21no—
201C23—21no—
202C23D4×D12711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—21no—
206Q8C22⋊Dic12712yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—41no—
213D4—41no—
214D4C22⋊Dic12712yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—5081no—
222C2×C4—5082no—
223C2×C4—5084no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4—2541no—
227C2×C4C22×Dic12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4C2×C22⋊D12711yescharacteristic
231C2×C4#1452324711yescharacteristic
232C2×C4#1452324711yescharacteristic
233C8—22no—
234C8—5081no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—81no—
240C22D4×Dic12711yescharacteristic
241C22#1686117711yescharacteristic
242C22#1686118111yescharacteristic
243C4—22no—
244C4—41no—
245C4—10161no—
246C4—5081no—
247C4—5082no—
248C4—2542no—
249C4C2×C22⋊Dic12711yescharacteristic
250C4D4×Dic12711yescharacteristic
251C4#1686118111yescharacteristic
252C4#1686119011yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#2254975611yescharacteristic
subgroupG/H
centerC4#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#575D127
hypocenterC127#1906
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #41×Dic127C2semidirect
2. #22549893C2semidirect
3. C2×#16861136C2semidirect
4. #22549850C2semidirect
5. #22549648C2semidirect
6. #22549637C2semidirect
24. #137×C127C4semidirect
26. AGL(1,ℤ/8)×C127C4semidirect
54. C2×Q8×C127C2×C4semidirect
58. C2×D4×C127C2×C4semidirect
68. M16×C127C2×C4semidirect
130. C127#1906semidirect
141. #137Dic127semidirect
143. AGL(1,ℤ/8)Dic127semidirect
167. C2×Q8C2×Dic127semidirect
171. C2×D4C2×Dic127semidirect
185. M16C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1272
#225498932
C2×#168611362
#225498502
#225496482
#225496372
#575×C1272
#1906127