Finite Group Databasebeta

← #30648704

Subgroups — #30648704

Order 16256.

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

normal subgroups only

This group has 398 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#22550359C211yesmaximal, characteristic, complemented
2#22550214C212yesmaximal, complemented
3#22550209C212yesmaximal, complemented
4#22550197C212yesmaximal, complemented
5#22550187C212yesmaximal, complemented
6#22549916C211yesmaximal, characteristic, complemented
7C2×#16861142C211yesmaximal, characteristic, complemented
8#22549877C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10#22549843C211yesmaximal, characteristic, complemented
11#537×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14Q8×Dic127—21no—
15D4×Dic127—22no—
16D4×Dic127—21no—
17#16861168C2211yescharacteristic
18#16861168C2212yes—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2212yes—
21#16861142C2212yes—
22#16861142C2212yes—
23#16861142C2212yes—
24#16861140C2212yes—
25#16861133C2211yescharacteristic
26#16861133C2211yescharacteristic
27#16861133C2212yes—
28#16861130C2211yescharacteristic
29#16861130C2211yescharacteristic
30#16861130C2212yes—
31C22⋊(C127⋊C8)C2212yes—
32C22⋊(C127⋊C8)C2212yes—
33C2×#41×C127C2211yescharacteristic
34C2×#41×C127C2211yescharacteristic
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127C2212yes—
37#121×C127—21no—
38#121×C127—22no—
39#121×C127C2212yes—
40C2×C4⋊C4×C127C2211yescharacteristic
41C22×Dic127—22no—
42C22×Dic127—21no—
43#14523232—22no—
44C2×C127⋊C8—22no—
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2312yes—
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127C2311yescharacteristic
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2312yes—
53C22⋊Dic127—22no—
54C22⋊Dic127—21no—
55C4×Dic127—22no—
56C4×Dic127—21no—
57#41×C127—22no—
58#41×C127—22no—
59#41×C127—22no—
60#41×C127—22no—
61C2×Q8×C127D411yescharacteristic
62C2×Q8×C127C2311yescharacteristic
63C2×D4×C127—21no—
64C2×D4×C127—22no—
65C2×D4×C127D412yes—
66C2×D4×C127D411yescharacteristic
67C2×D4×C127C2311yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127C2312yes—
71C22×C4×C127C2311yescharacteristic
72C4⋊C4×C127—21no—
73C4⋊C4×C127—22no—
74C4⋊C4×C127C2312yes—
75C22⋊C4×C127—22no—
76C22⋊C4×C127—22no—
77C2×Dic127—21no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127—81no—
81C2×Dic127—82no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—21no—
86C23×C127C2×D412yes—
87C23×C127C2×D411yescharacteristic
88Q8×C127—22no—
89Q8×C127—21no—
90Q8×C127—21no—
91D4×C127—24no—
92D4×C127—22no—
93D4×C127—21no—
94D4×C127—21no—
95D4×C127—42no—
96D4×C127—82no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—42no—
102C2×C4×C127—81no—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4112yes—
106C2×C4×C127#4112yes—
107C2×C4×C127C2411yescharacteristic
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127#12312yes—
117C22×C127C22×D411yescharacteristic
118C4×C127—22no—
119C4×C127—41no—
120C4×C127—42no—
121C4×C127#12312yes—
122C4×C127C2×#4111yescharacteristic
123C4×C127C2×#4111yescharacteristic
124Dic127—42no—
125Dic127—41no—
126Dic127—41no—
127C2×C127—22no—
128C2×C127—42no—
129C2×C127—81no—
130C2×C127C2×#12311yescharacteristic
131C2×C127#57611yescharacteristic
132C2×C127#57811yescharacteristic
133#3098—1271nomaximal, Sylow
134C127#309811yescharacteristic, complemented, Sylow
135#537D12711yescharacteristic
136#520—1271no—
137#485—1272no—
138#484—1272no—
139#482—1272no—
140#480—1272no—
141#428—1271no—
142C2×#107—1271no—
143#417—1271no—
144#416—1271no—
145#408—1271no—
146C2×#41C2×D12711yescharacteristic
147C2×#41C2×D12711yescharacteristic
148C2×M16—1271no—
149C22×C8—1271no—
150C22⋊Q8—21no—
151C22⋊Q8C2×D12712yes—
152#121—21no—
153#121—22no—
154#121C2×D12712yes—
155C4×Q8—2541no—
156C4×D4—2542no—
157C4×D4—2541no—
158#117—1271no—
159#117—1272no—
160C2×C4⋊C4—1271no—
161C2×C4⋊C4—1272no—
162C2×C4⋊C4C2×D12711yescharacteristic
163#107—1272no—
164#107—1272no—
165#107—1272no—
166#106—1272no—
167#103—1271no—
168#103—1271no—
169#103—1272no—
170#102—1271no—
171#102—1271no—
172#102—1272no—
173C22⋊C8—1272no—
174C22⋊C8—1272no—
175#41—22no—
176#41—22no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D12711yescharacteristic
180C2×Q8C22×D12711yescharacteristic
181C2×D4—21no—
182C2×D4—22no—
183C2×D4C22⋊D12712yes—
184C2×D4C22⋊D12711yescharacteristic
185C2×D4C22×D12711yescharacteristic
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—2542no—
189C22×C4—2541no—
190C22×C4C22×D12712yes—
191C22×C4C22×D12711yescharacteristic
192M16—2542no—
193C2×C8—1272no—
194C2×C8—1271no—
195C2×C8—1271no—
196C2×C8—2542no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1271no—
200C4⋊C4—1271no—
201C4⋊C4—1272no—
202C4⋊C4—2542no—
203C4⋊C4—2541no—
204C4⋊C4C22×D12712yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2542no—
208C22⋊C4—2541no—
209C42—2542no—
210C42—2541no—
211C23—21no—
212C23C2×C22⋊D12712yes—
213C23C2×C22⋊D12711yescharacteristic
214Q8—22no—
215Q8—21no—
216Q8—21no—
217D4—24no—
218D4—22no—
219D4—21no—
220D4—21no—
221D4—42no—
222D4—82no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—10161no—
230C2×C4—10162no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4—2541no—
234C2×C4C2×C22⋊D12712yes—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4#1452324712yes—
237C2×C4#1452324712yes—
238C2×C4C23×D12711yescharacteristic
239C8—2541no—
240C8—2541no—
241C8—2542no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22#1686119012yes—
251C22C22×C22⋊D12711yescharacteristic
252C4—22no—
253C4—41no—
254C4—42no—
255C4—5082no—
256C4—5081no—
257C4—5081no—
258C4#1686119012yes—
259C4C2×#1452324711yescharacteristic
260C4C2×#1452324711yescharacteristic
261C2—22no—
262C2—42no—
263C2—81no—
264C2C2×#1686119011yescharacteristic
265C2#2255067911yescharacteristic
266C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#537×C127C2
socleC22×C127C22×D4
hypercenter#537D127
hypocenterC127#3098
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22550359C2semidirect
2. #22550214C2semidirect
3. #22550209C2semidirect
4. #22550197C2semidirect
5. #22550187C2semidirect
6. #22549916C2semidirect
7. C2×#16861142C2semidirect
8. #22549877C2semidirect
9. #22549872C2semidirect
10. #22549843C2semidirect
134. C127#3098semidirect

Maximal subgroups

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

subgroupindex
#225503592
#225502142
#225502092
#225501972
#225501872
#225499162
C2×#168611422
#225498772
#225498722
#225498432
#537×C1272
#3098127