Finite Group Databasebeta

← #30639229

Subgroups — #30639229

Order 16256.

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

normal subgroups only

This group has 332 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
1#22550674C211yesmaximal, characteristic
2#41×Dic127C211yesmaximal, characteristic, complemented
3C2×#16861135C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239C412yescomplemented
11#16861235—21no—
12#16861234C412yescomplemented
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15Q8×Dic127—21no—
16D4×Dic127—22no—
17D4×Dic127—21no—
18#16861168—21no—
19#16861168C2211yescharacteristic
20C2×C4×Dic127—21no—
21C2×C4×Dic127C2211yescharacteristic
22#16861135—21no—
23#16861135—21no—
24#16861123—22no—
25C2×#41×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28#104×C127—22no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254C2×C411yescharacteristic, complemented
34C2×D508C2×C411yescharacteristic, complemented
35C2×C22⋊D127—21no—
36C22×Dic127—22no—
37C22×Dic127—21no—
38C2×C4×D127—21no—
39#14523238—22no—
40#14523236—22no—
41#14523235—22no—
42#14523233—22no—
43#14523232—21no—
44#14523232C2×C412yescomplemented
45C2×C127⋊C8—21no—
46C4⋊Dic127—22no—
47C4⋊Dic127—21no—
48C22⋊Dic127—22no—
49C22⋊Dic127D412yes—
50C4×Dic127—22no—
51C4×Dic127—21no—
52C4×Dic127—21no—
53C4×Dic127D411yescharacteristic
54C4×Dic127D411yescharacteristic
55#41×C127—21no—
56#41×C127—21no—
57#41×C127C2×C412yes—
58#41×C127D412yes—
59C2×Q8×C127D411yescharacteristic
60C2×D4×C127—21no—
61C2×D4×C127D411yescharacteristic
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64M16×C127—21no—
65M16×C127—21no—
66C2×C8×C127—21no—
67C4⋊C4×C127—21no—
68C22⋊C4×C127—21no—
69C42×C127—21no—
70C22×D127—21no—
71Dic254—21no—
72Dic254—21no—
73D508—21no—
74D508—21no—
75C22⋊D127—41no—
76C22⋊D127—41no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127—42no—
82C2×Dic127—44no—
83C4×D127—41no—
84C4×D127—41no—
85C127⋊C8—22no—
86C23×C127—21no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—21no—
89Q8×C127C22⋊C412yes—
90D4×C127—22no—
91D4×C127—22no—
92D4×C127—21no—
93D4×C127C22⋊C412yes—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
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—41no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127C4×D411yescharacteristic
109C22×C127#12011yescharacteristic
110C22×C127#12111yescharacteristic
111C2×D127—41no—
112C2×D127—81no—
113C4×C127—22no—
114C4×C127—81no—
115C4×C127C2×C22⋊C411yescharacteristic
116C4×C127C4×D411yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127#12311yescharacteristic
119Dic127—22no—
120Dic127—42no—
121Dic127—41no—
122Dic127—41no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—42no—
127C2×C127#38611yescharacteristic
128D127—81no—
129#1906—1271nomaximal, Sylow
130C127#190611yescharacteristic, complemented, Sylow
131#575—1271no—
132C4×#41—1271no—
133#421D12711yescharacteristic, complemented
134C2×#104—1271no—
135#410—1271no—
136#344—1271no—
137#339—1271no—
138C2×#41—1271no—
139C2×#41C2×D12711yescharacteristic
140#137—1272no—
141#137—2541no—
142AGL(1,ℤ/8)—1272no—
143AGL(1,ℤ/8)—2541no—
144#135—2542no—
145C2×M16—1271no—
146C2×M16C2×D12711yescharacteristic
147C4×Q8—2541no—
148C4×D4—2542no—
149C4×D4—2541no—
150#117—1271no—
151#117—2541no—
152#117C2×D12711yescharacteristic
153C2×C42—1271no—
154C2×C42—2541no—
155#104—22no—
156#104—2541no—
157#104—2541no—
158#99—2542no—
159#41—21no—
160#41—21no—
161#41—2541no—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165#41C4×D12712yes—
166#41C22⋊D12712yes—
167C2×Q8—1271no—
168C2×Q8D50811yescharacteristic
169C2×D4—21no—
170C2×D4—1271no—
171C2×D4—2541no—
172C2×D4D50811yescharacteristic
173C22×C4—21no—
174C22×C4—2542no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178Q16—2542no—
179SD16—2542no—
180SD16—2542no—
181D8—2542no—
182M16—21no—
183M16—21no—
184M16—1272no—
185M16—2541no—
186C2×C8—21no—
187C2×C8—2541no—
188C4⋊C4—21no—
189C4⋊C4—2542no—
190C4⋊C4—2541no—
191C22⋊C4—21no—
192C22⋊C4—1272no—
193C22⋊C4—2542no—
194C42—21no—
195C42—1271no—
196C42—1271no—
197C42—2542no—
198C42—2541no—
199C42—2541no—
200C23—21no—
201C23—2541no—
202C23D4×D12711yescharacteristic
203Q8—21no—
204Q8—2541no—
205Q8—2541no—
206Q8(C2×D127)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—5081no—
211D4—5081no—
212D4—2541no—
213D4—2541no—
214D4(C2×D127)⋊C412yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—5082no—
220C2×C4—5084no—
221C2×C4—5081no—
222C2×C4—5081no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4—2541no—
227C2×C4C2×C4×D12711yescharacteristic
228C2×C4D4×D12711yescharacteristic
229C2×C4C2×C22⋊D12711yescharacteristic
230C2×C4C2×D50811yescharacteristic
231C2×C4#1452324711yescharacteristic
232C2×C4#1452324711yescharacteristic
233C8—41no—
234C8—2542no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—10161no—
239C22—5081no—
240C22(C22⋊D127)⋊C411yescharacteristic
241C22#1686117911yescharacteristic
242C22#1686118111yescharacteristic
243C4—22no—
244C4—81no—
245C4—5081no—
246C4—5082no—
247C4—5081no—
248C4—2542no—
249C4C2×(C2×D127)⋊C411yescharacteristic
250C4(C22⋊D127)⋊C411yescharacteristic
251C4#1686118111yescharacteristic
252C4#1686119311yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—10161no—
258C2#2254975311yescharacteristic
subgroupG/H
centerC4#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radical#22550674C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1906
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #41×Dic127C2semidirect
3. C2×#16861135C2semidirect
4. #22549849C2semidirect
5. #22549647C2semidirect
6. #22549635C2semidirect
7. #421×C127C2semidirect
10. #16861239C4semidirect
12. #16861234C4semidirect
33. C2×Dic254C2×C4semidirect
34. C2×D508C2×C4semidirect
44. #14523232C2×C4semidirect
130. C127#1906semidirect
133. #421D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#41×Dic1272
C2×#168611352
#225498492
#225496472
#225496352
#421×C1272
#1906127