Finite Group Databasebeta

← #30648697

Subgroups — #30648697

Order 16256.

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

normal subgroups only

This group has 386 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#22550450C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic
3#22550216C212yesmaximal, complemented
4#22550215C212yesmaximal, complemented
5#22550210C212yesmaximal, complemented
6#22550205C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549912C211yesmaximal, characteristic, complemented
9C2×#16861133C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic
11#419×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861207—21no—
15#16861198—21no—
16#16861192—22no—
17#16861185C2212yes—
18#16861185C2212yes—
19C4×Dic254—21no—
20C4×D508—21no—
21(C22⋊D127)⋊C4—22no—
22#16861168C2211yescharacteristic
23#16861167C2212yes—
24C2×C4⋊Dic127C2211yescharacteristic
25C4⋊C4×D127C2212yes—
26#16861143C2212yes—
27#16861142C2211yescharacteristic
28#16861142C2211yescharacteristic
29#16861141C2212yes—
30#16861140C2211yescharacteristic
31#16861140C2211yescharacteristic
32#16861133C2212yes—
33#16861133C2212yes—
34(C2×D127)⋊C8C2212yes—
35(C2×D127)⋊C8C2212yes—
36C22×Q8×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38#117×C127C2211yescharacteristic
39#103×C127C2212yes—
40#103×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254D411yescharacteristic
44C2×D508D411yescharacteristic
45C2×C22⋊D127D412yes—
46C22×Dic127—21no—
47C2×C4×D127—22no—
48C2×C4×D127C2312yes—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—21no—
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—21no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—22no—
64C4×Dic127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—22no—
67C2×Q8×C127—22no—
68C2×Q8×C127C2312yes—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71M16×C127—22no—
72C2×C8×C127C2312yes—
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2311yescharacteristic
75C22⋊C4×C127—21no—
76C42×C127—21no—
77C22×D127C2×D412yes—
78Dic254—22no—
79D508—22no—
80C22⋊D127—24no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C2×Dic127—81no—
85C2×Dic127C2×D412yes—
86C4×D127—22no—
87C4×D127—22no—
88C4×D127—82no—
89C127⋊C8—21no—
90C127⋊C8—21no—
91C23×C127C2×D411yescharacteristic
92Q8×C127—22no—
93Q8×C127—22no—
94Q8×C127—42no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—42no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4112yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127#12312yes—
107C22×C127C22×D411yescharacteristic
108C2×D127—22no—
109C2×D127—22no—
110C2×D127—22no—
111C4×C127—42no—
112C4×C127—41no—
113C4×C127—41no—
114C4×C127#12312yes—
115C4×C127C2×#4111yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—22no—
118Dic127—42no—
119Dic127—41no—
120Dic127—41no—
121C2×C127—22no—
122C2×C127C2×#12311yescharacteristic
123C2×C127#57711yescharacteristic
124C2×C127#57811yescharacteristic
125D127—42no—
126#3097—1271nomaximal, Sylow
127C127#309711yescharacteristic, complemented, Sylow
128#536—1271no—
129#520—1271no—
130#485—1272no—
131#485—1272no—
132#484—1272no—
133#483—1272no—
134#428—1271no—
135#427—1271no—
136#419D12711yescharacteristic, complemented
137C2×#103—1271no—
138#408—1271no—
139C2×#41—1271no—
140C22×Q8C2×D12711yescharacteristic
141C2×M16C2×D12711yescharacteristic
142C22×C8—1271no—
143C4⋊Q8—2541no—
144#124—2541no—
145#123—2542no—
146C22⋊Q8—1272no—
147C22⋊Q8—1272no—
148C4×Q8—2541no—
149C4×D4—2541no—
150C4×D4—2542no—
151#117—1271no—
152#117—1272no—
153#117C2×D12711yescharacteristic
154C2×C4⋊C4—1271no—
155C2×C4⋊C4—1272no—
156#107—1272no—
157#107—1271no—
158#107—1271no—
159#106—1272no—
160#106—1271no—
161#106—1271no—
162#103—1272no—
163#103—1272no—
164#103C2×D12712yes—
165#103C2×D12712yes—
166C22⋊C8—1272no—
167C22⋊C8—1272no—
168#41—2542no—
169#41—2542no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—1271no—
174C2×Q8C22×D12712yes—
175C2×D4—1272no—
176C2×D4—1271no—
177C22×C4—21no—
178C22×C4—1272no—
179C22×C4—2542no—
180C22×C4—2541no—
181C22×C4C22×D12711yescharacteristic
182M16—22no—
183C2×C8—1271no—
184C2×C8—1271no—
185C2×C8—2542no—
186C2×C8C22×D12712yes—
187C4⋊C4—1272no—
188C4⋊C4—1271no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4—2541no—
194C4⋊C4C22×D12711yescharacteristic
195C4⋊C4C22×D12711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4—2542no—
200C22⋊C4—2541no—
201C42—21no—
202C42—2542no—
203C42—2541no—
204C23—1272no—
205C23D4×D12711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—2542no—
210D4—2544no—
211D4—2542no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—1272no—
217C2×C4—10162no—
218C2×C4—10161no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4#1452324712yes—
226C2×C4#1452324812yes—
227C2×C4C23×D12711yescharacteristic
228C8—22no—
229C8—2541no—
230C8—2541no—
231C22—21no—
232C22—21no—
233C22—2542no—
234C22—2542no—
235C22—2542no—
236C22#1686119112yes—
237C22C2×D4×D12711yescharacteristic
238C4—42no—
239C4—41no—
240C4—41no—
241C4—5082no—
242C4—5081no—
243C4—5081no—
244C4—2542no—
245C4#1686119112yes—
246C4C2×#1452324711yescharacteristic
247C4C2×#1452324811yescharacteristic
248C2—22no—
249C2—5082no—
250C2C2×#1686119111yescharacteristic
251C2#2255068511yescharacteristic
252C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#419D127
hypocenterC127#3097
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22550216C2semidirect
4. #22550215C2semidirect
5. #22550210C2semidirect
6. #22550205C2semidirect
7. #22549918C2semidirect
8. #22549912C2semidirect
9. C2×#16861133C2semidirect
11. #419×C127C2semidirect
127. C127#3097semidirect
136. #419D127semidirect

Maximal subgroups

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

subgroupindex
#225504502
#225503562
#225502162
#225502152
#225502102
#225502052
#225499182
#225499122
C2×#168611332
#225498442
#419×C1272
#3097127