Finite Group Databasebeta

← #30648707

Subgroups — #30648707

Order 16256.

398 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550454C211yesmaximal, characteristic
2#22550357C211yesmaximal, characteristic, complemented
3#22550215C212yesmaximal, complemented
4#22550210C212yesmaximal, complemented
5#22550199C212yesmaximal, complemented
6#22550188C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8C2×#16861142C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861188—21no—
15#16861185C2212yes—
16#16861183—21no—
17#16861181—22no—
18#16861180C2212yes—
19#16861174—21no—
20#16861171—21no—
21C4×C22⋊D127—22no—
22#16861168C2211yescharacteristic
23#16861167C2212yes—
24C2×C4⋊Dic127C2211yescharacteristic
25C4⋊C4×D127C2212yes—
26#16861143C2212yes—
27#16861142C2212yes—
28#16861142C2212yes—
29#16861141C2212yes—
30#16861133C2211yescharacteristic
31#16861133C2211yescharacteristic
32#16861130C2211yescharacteristic
33#16861130C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35(C2×D127)⋊C8C2212yes—
36C2×#41×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#103×C127C2212yes—
40#102×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254D411yescharacteristic
44C2×D508D411yescharacteristic
45C2×C22⋊D127—22no—
46C2×C22⋊D127D412yes—
47C22×Dic127—21no—
48C2×C4×D127—22no—
49C2×C4×D127C2312yes—
50C2×C127⋊C8—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C4⋊Dic127—21no—
54C4⋊Dic127C2312yes—
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60C22⋊Dic127—21no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—22no—
64C4×Dic127—21no—
65#41×C127—22no—
66#41×C127—22no—
67C2×Q8×C127C2311yescharacteristic
68C2×D4×C127—21no—
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127C2311yescharacteristic
73M16×C127—22no—
74C2×C8×C127C2312yes—
75C4⋊C4×C127—21no—
76C4⋊C4×C127C2312yes—
77C22×D127C2×D412yes—
78Dic254—22no—
79D508—22no—
80C22⋊D127—24no—
81C22⋊D127—82no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—81no—
86C2×Dic127C2×D412yes—
87C4×D127—22no—
88C4×D127—22no—
89C4×D127—82no—
90C127⋊C8—21no—
91C127⋊C8—21no—
92C23×C127—21no—
93C23×C127C2×D411yescharacteristic
94Q8×C127—21no—
95Q8×C127—21no—
96D4×C127—21no—
97D4×C127—21no—
98D4×C127—42no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—42no—
102C2×C4×C127—81no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4112yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127#12312yes—
114C22×C127C22×D411yescharacteristic
115C2×D127—22no—
116C2×D127—22no—
117C2×D127—22no—
118C4×C127—41no—
119C4×C127—42no—
120C4×C127#12312yes—
121C4×C127C2×#4111yescharacteristic
122C4×C127C2×#4111yescharacteristic
123Dic127—22no—
124Dic127—41no—
125Dic127—42no—
126Dic127—41no—
127C2×C127—22no—
128C2×C127—81no—
129C2×C127C2×#12311yescharacteristic
130C2×C127#57611yescharacteristic
131C2×C127#57811yescharacteristic
132D127—42no—
133#3098—1271nomaximal, Sylow
134C127#309811yescharacteristic, complemented, Sylow
135#537—1271no—
136#520—1271no—
137#485—1272no—
138#484—1272no—
139#482—1272no—
140#480—1272no—
141#428—1271no—
142C2×#107—1271no—
143#417D12711yescharacteristic, complemented
144#416—1271no—
145#408—1271no—
146C2×#41—1271no—
147C2×#41C2×D12711yescharacteristic
148C2×M16C2×D12711yescharacteristic
149C22×C8—1271no—
150C22⋊Q8—1272no—
151C22⋊Q8—2541no—
152#121—1272no—
153#121—2541no—
154#121—2542no—
155C4×Q8—2541no—
156C4×D4—2541no—
157C4×D4—2542no—
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#103C2×D12712yes—
170#102—1271no—
171#102—1271no—
172#102C2×D12712yes—
173C22⋊C8—1272no—
174C22⋊C8—1272no—
175#41—22no—
176#41—22no—
177#41—2542no—
178#41—2542no—
179C2×Q8—1271no—
180C2×Q8C22×D12711yescharacteristic
181C2×D4—21no—
182C2×D4—1272no—
183C2×D4—1271no—
184C2×D4—2542no—
185C2×D4C22×D12711yescharacteristic
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—1272no—
189C22×C4—2542no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192M16—22no—
193C2×C8—1271no—
194C2×C8—1271no—
195C2×C8—2542no—
196C2×C8C22×D12712yes—
197C4⋊C4—21no—
198C4⋊C4—1271no—
199C4⋊C4—1271no—
200C4⋊C4—1272no—
201C4⋊C4—2542no—
202C4⋊C4—2542no—
203C4⋊C4—2541no—
204C4⋊C4C22×D12712yes—
205C22⋊C4—2542no—
206C22⋊C4—2542no—
207C22⋊C4—2542no—
208C22⋊C4—2541no—
209C42—2542no—
210C42—2541no—
211C23—21no—
212C23—1272no—
213C23D4×D12711yescharacteristic
214Q8—21no—
215Q8—21no—
216Q8—2542no—
217D4—21no—
218D4—21no—
219D4—42no—
220D4—10162no—
221D4—2544no—
222D4—2542no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—1272no—
228C2×C4—10162no—
229C2×C4—10161no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4—2542no—
235C2×C4D4×D12711yescharacteristic
236C2×C4#1452324712yes—
237C2×C4#1452324812yes—
238C2×C4C23×D12711yescharacteristic
239C8—22no—
240C8—2541no—
241C8—2541no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—2542no—
248C22—2542no—
249C22—2542no—
250C22#1686119112yes—
251C22C2×D4×D12711yescharacteristic
252C4—41no—
253C4—42no—
254C4—5081no—
255C4—5082no—
256C4—5081no—
257C4—2542no—
258C4#1686119112yes—
259C4C2×#1452324711yescharacteristic
260C4C2×#1452324811yescharacteristic
261C2—22no—
262C2—81no—
263C2—5082no—
264C2C2×#1686119111yescharacteristic
265C2#2255068011yescharacteristic
266C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550454C2
hypercenter#417D127
hypocenterC127#3098
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #22550357C2semidirect
3. #22550215C2semidirect
4. #22550210C2semidirect
5. #22550199C2semidirect
6. #22550188C2semidirect
7. #22549918C2semidirect
8. C2×#16861142C2semidirect
9. #22549872C2semidirect
10. #22549844C2semidirect
11. #417×C127C2semidirect
134. C127#3098semidirect
143. #417D127semidirect

Maximal subgroups

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

subgroupindex
#225504542
#225503572
#225502152
#225502102
#225501992
#225501882
#225499182
C2×#168611422
#225498722
#225498442
#417×C1272
#3098127