Finite Group Databasebeta

← #30648689

Subgroups — #30648689

Order 16256.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550440C211yesmaximal, characteristic
2#22550359C211yesmaximal, characteristic, complemented
3#22550201C212yesmaximal, complemented
4#22550201C212yesmaximal, complemented
5#22550195C212yesmaximal, complemented
6#22550190C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549914C211yesmaximal, characteristic, complemented
9#22549882C211yesmaximal, characteristic, complemented
10C2×#16861131C211yesmaximal, characteristic, complemented
11#408×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13#16861205—21no—
14#16861198—21no—
15#16861183C2212yes—
16#16861183C2212yes—
17#16861177—22no—
18Q8×Dic127—21no—
19D4×Dic127—21no—
20D4×Dic127—22no—
21#16861168C2211yescharacteristic
22#16861168C2211yescharacteristic
23#16861168C2212yescomplemented
24C2×C4⋊Dic127C2211yescharacteristic
25C2×C4⋊Dic127C2212yescomplemented
26#16861143C2211yescharacteristic, complemented
27#16861143C2211yescharacteristic, complemented
28#16861143C2212yescomplemented
29#16861141C2211yescharacteristic, complemented
30#16861141C2211yescharacteristic, complemented
31#16861141C2212yescomplemented
32#16861131C2212yes—
33#16861131C2212yes—
34#16861131C2212yes—
35#16861131C2212yes—
36C2×#41×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C22×C8×C127C2211yescharacteristic
39C22⋊C8×C127C2212yescomplemented
40C22⋊C8×C127C2212yescomplemented
41C23×D127—21no—
42C2×D508—21no—
43C2×D508—22no—
44C2×D508—22no—
45C2×D508C2312yes—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C22×Dic127—21no—
49C22×Dic127—22no—
50C4⋊Dic127—21no—
51C4⋊Dic127—22no—
52C4⋊Dic127C2312yes—
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57C22⋊Dic127—21no—
58C22⋊Dic127—21no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62C4×Dic127—21no—
63C4×Dic127—21no—
64C4×Dic127—22no—
65#41×C127—22no—
66#41×C127—22no—
67C2×Q8×C127D411yescharacteristic
68C2×D4×C127D411yescharacteristic
69C2×D4×C127D412yes—
70C22×C4×C127C2312yes—
71C22×C4×C127C2311yescharacteristic
72M16×C127—22no—
73C2×C8×C127—22no—
74C2×C8×C127C2311yescharacteristic
75C2×C8×C127C2311yescharacteristic
76C2×C8×C127C2312yes—
77C22×D127—22no—
78C22×D127—42no—
79C22×D127—81no—
80C22×D127—81no—
81D508—22no—
82D508—22no—
83D508—42no—
84C22⋊D127—82no—
85C22⋊D127—82no—
86C2×Dic127—21no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C2×Dic127—81no—
92C2×Dic127—82no—
93C23×C127C2×D412yes—
94C23×C127C2×D411yescharacteristic
95Q8×C127—22no—
96D4×C127—24no—
97D4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127#4112yes—
103C2×C4×C127#4112yes—
104C2×C4×C127C2411yescharacteristic
105C8×C127—21no—
106C8×C127—21no—
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127#12312yes—
114C22×C127C22×D411yescharacteristic
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—84no—
119C4×C127—22no—
120C4×C127#12312yes—
121C4×C127C2×#4111yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—41no—
124Dic127—42no—
125Dic127—41no—
126Dic127—41no—
127Dic127—41no—
128C2×C127—22no—
129C2×C127—42no—
130C2×C127C2×#12311yescharacteristic
131C2×C127#57611yescharacteristic
132C2×C127#57711yescharacteristic
133D127—82no—
134#3096—1271nomaximal, Sylow
135C127#309611yescharacteristic, complemented, Sylow
136#535—1271no—
137#520—1271no—
138#482—1272no—
139#482—1272no—
140#481—1272no—
141#480—1272no—
142#428—1271no—
143#427—1271no—
144#418—1271no—
145C2×#102—1271no—
146#408D12711yescharacteristic, complemented
147C2×#41C2×D12711yescharacteristic
148C22×D4—1271no—
149C2×M16C2×D12711yescharacteristic
150C22×C8C2×D12711yescharacteristic
151#127—2541no—
152#124—2541no—
153#121—1272no—
154#121—1272no—
155#120—2542no—
156C4×Q8—2541no—
157C4×D4—2542no—
158C4×D4—2541no—
159#117—1271no—
160#117—1272no—
161#117—1271no—
162C2×C4⋊C4—1271no—
163C2×C4⋊C4—1272no—
164#107—1271no—
165#107—1271no—
166#107—1272no—
167#106—1271no—
168#106—1271no—
169#106—1272no—
170#102—1272no—
171#102—1272no—
172#102—1272no—
173#102—1272no—
174C22⋊C8C2×D12712yescomplemented
175C22⋊C8C2×D12712yescomplemented
176C24—2541no—
177#41—22no—
178#41—22no—
179C2×Q8D50811yescharacteristic
180C2×D4—1272no—
181C2×D4—2542no—
182C2×D4—2542no—
183C2×D4—2541no—
184C2×D4—2542no—
185C2×D4—2542no—
186C2×D4D50812yes—
187C2×D4D50811yescharacteristic
188C22×C4—2542no—
189C22×C4—2541no—
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D12711yescharacteristic
195C2×C8C22×D12712yes—
196C2×C8C22×D12711yescharacteristic
197C4⋊C4—1272no—
198C4⋊C4—1271no—
199C4⋊C4—1271no—
200C4⋊C4—1271no—
201C4⋊C4—1271no—
202C4⋊C4—2542no—
203C4⋊C4—2541no—
204C22⋊C4—2542no—
205C22⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2542no—
208C22⋊C4—2542no—
209C42—2542no—
210C42—2541no—
211C42—2541no—
212C23—10161no—
213C23—10161no—
214C23—5082no—
215C23—2542no—
216C23C2×D50812yes—
217C23C2×D50811yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—10162no—
222D4—10162no—
223D4—5082no—
224D4—2542no—
225D4—2542no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—10161no—
229C2×C4—10162no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2541no—
233C2×C4—2542no—
234C2×C4—2541no—
235C2×C4C2×D50812yes—
236C2×C4C2×D50811yescharacteristic
237C2×C4#1452324712yes—
238C2×C4#1452324712yes—
239C2×C4C23×D12711yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—10164no—
249C22—5082no—
250C22—5082no—
251C22—5082no—
252C22#1686119312yes—
253C22C22×D50811yescharacteristic
254C4—22no—
255C4—5082no—
256C4—5081no—
257C4—5081no—
258C4—5081no—
259C4—5081no—
260C4#1686119312yes—
261C4C2×#1452324711yescharacteristic
262C4C2×#1452324711yescharacteristic
263C2—22no—
264C2—42no—
265C2—10162no—
266C2C2×#1686119311yescharacteristic
267C2#2255068111yescharacteristic
268C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#408×C127C2
socleC22×C127C22×D4
prime radical#22550440C2
hypercenter#408D127
hypocenterC127#3096
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22550359C2semidirect
3. #22550201C2semidirect
4. #22550201C2semidirect
5. #22550195C2semidirect
6. #22550190C2semidirect
7. #22549918C2semidirect
8. #22549914C2semidirect
9. #22549882C2semidirect
10. C2×#16861131C2semidirect
11. #408×C127C2semidirect
23. #16861168C22semidirect
25. C2×C4⋊Dic127C22semidirect
26. #16861143C22semidirect
27. #16861143C22semidirect
28. #16861143C22semidirect
29. #16861141C22semidirect
30. #16861141C22semidirect
31. #16861141C22semidirect
39. C22⋊C8×C127C22semidirect
40. C22⋊C8×C127C22semidirect
135. C127#3096semidirect
146. #408D127semidirect
174. C22⋊C8C2×D127semidirect
175. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225503592
#225502012
#225501952
#225501902
#225499182
#225499142
#225498822
C2×#168611312
#408×C1272
#3096127