Finite Group Databasebeta

← #30648896

Subgroups — #30648896

Order 16256.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550407C211yesmaximal, characteristic
2C2×Q8×Dic127C211yesmaximal, characteristic, complemented
3#22550212C211yesmaximal, characteristic, complemented
4#22550192C211yesmaximal, characteristic
5#22550185C212yesmaximal, complemented
6#22550096C212yesmaximal, complemented
7#22550032C211yesmaximal, characteristic, complemented
8#22549980C212yesmaximal, complemented
9#22549977C212yesmaximal, complemented
10#22549930C211yesmaximal, characteristic, complemented
11#448×C127C211yesmaximal, characteristic
12C2×#14523238C2212yes—
13#16861202—21no—
14#16861200C2211yescharacteristic
15#16861196C2211yescharacteristic
16#16861190—21no—
17#16861188C2211yescharacteristic
18Q8×Dic127—21no—
19Q8×Dic127—21no—
20Q8×Dic127C2212yes—
21Q8×Dic127C2212yes—
22#16861174C2211yescharacteristic
23D4×Dic127—21no—
24D4×Dic127C2211yescharacteristic
25#16861165—21no—
26C2×C4⋊Dic127—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×C4×Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yes—
31#16861140C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861133C2212yes—
35#16861130C2212yes—
36C22⋊(C127⋊C8)C2211yescharacteristic
37#16861119C2212yes—
38C22×Q8×C127C2211yescharacteristic
39C2×SD16×C127C2212yes—
40#121×C127C2211yescharacteristic
41#103×C127C2212yes—
42C22⋊C8×C127C2211yescharacteristic
43C2×Dic254C2311yescharacteristic
44C22×Dic127—21no—
45C22×Dic127—21no—
46C22×Dic127D411yescharacteristic
47C22×Dic127D411yescharacteristic
48#14523238—24no—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127—21no—
51C4⋊Dic127—22no—
52C4⋊Dic127—22no—
53C4⋊Dic127—21no—
54C4⋊Dic127C2311yescharacteristic
55C4⋊Dic127C2312yes—
56Dic127⋊C4—21no—
57Dic127⋊C4—21no—
58Dic127⋊C4—21no—
59C22⋊Dic127—21no—
60C22⋊Dic127—21no—
61C22⋊Dic127—21no—
62C22⋊Dic127—21no—
63C4×Dic127—22no—
64C4×Dic127—22no—
65C4×Dic127—21no—
66C4×Dic127D411yescharacteristic
67C4×Dic127D411yescharacteristic
68C4×Dic127C2311yescharacteristic
69C4×Dic127C2311yescharacteristic
70C2×Q8×C127—24no—
71C2×Q8×C127—21no—
72C2×Q8×C127C2312yes—
73C2×D4×C127—21no—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2311yescharacteristic
77SD16×C127—24no—
78C2×C8×C127C2312yes—
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81Dic254—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—41no—
87C2×Dic127—42no—
88C2×Dic127—42no—
89C2×Dic127C2×D411yescharacteristic
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C127⋊C8—22no—
94C23×C127—21no—
95C23×C127C2×D411yescharacteristic
96Q8×C127—24no—
97Q8×C127—22no—
98Q8×C127#4114yes—
99D4×C127—22no—
100D4×C127—42no—
101C2×C4×C127—24no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127#13712yes—
114C22×C127C22×D411yescharacteristic
115C4×C127—24no—
116C4×C127—21no—
117C4×C127—41no—
118C4×C127C2×#4112yes—
119Dic127—21no—
120Dic127—21no—
121Dic127—21no—
122Dic127—21no—
123Dic127—41no—
124Dic127—42no—
125Dic127—41no—
126C2×C127—22no—
127C2×C127—81no—
128C2×C127#52511yescharacteristic
129C2×C127C2×#13711yescharacteristic
130C2×C127#57511yescharacteristic
131#3117—1271nomaximal, Sylow
132C127#311711yescharacteristic, complemented, Sylow
133#529—1271no—
134C2×C4×Q8—1271no—
135#484—1271no—
136#481—1271no—
137#479—1272no—
138#464—1272no—
139#451—1271no—
140#448D12711yescharacteristic
141#441—1272no—
142#440—1272no—
143#432—1271no—
144C22×Q8C2×D12711yescharacteristic
145C2×Q16—1272no—
146C2×SD16C2×D12712yes—
147#126—2541no—
148#125—1271no—
149#124—1271no—
150#123—2541no—
151C22⋊Q8—1271no—
152#121C2×D12711yescharacteristic
153C4×Q8—1272no—
154C4×Q8—1272no—
155C4×Q8—1271no—
156C4×Q8—2541no—
157C4×Q8—2541no—
158C4×D4—1271no—
159C4×D4—2541no—
160#117—2541no—
161C2×C4⋊C4—1271no—
162C2×C4⋊C4—2541no—
163C2×C42—1271no—
164C2×C42—2541no—
165#107—1272no—
166#106—1272no—
167C4⋊C8—1272no—
168#103—1272no—
169#103—1272no—
170#103C2×D12712yes—
171#102—1272no—
172C22⋊C8—1271no—
173C22⋊C8C2×D12711yescharacteristic
174#97—1272no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1271no—
178C2×Q8C22×D12712yes—
179C2×D4—21no—
180C2×D4C22×D12711yescharacteristic
181C22×C4—21no—
182C22×C4—1271no—
183C22×C4—1271no—
184C22×C4—2541no—
185C22×C4—2541no—
186C22×C4C22×D12711yescharacteristic
187Q16—2544no—
188SD16—24no—
189C2×C8—1272no—
190C2×C8C22×D12712yes—
191C4⋊C4—1272no—
192C4⋊C4—1271no—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C4⋊C4—2542no—
196C4⋊C4—2542no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4C22×D12711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C42—1271no—
207C42—1271no—
208C42—1271no—
209C42—1271no—
210C42—2542no—
211C42—2542no—
212C42—2541no—
213C23—21no—
214C23D4×D12711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2542no—
218Q8#1452324714yes—
219D4—22no—
220D4—42no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—1271no—
226C2×C4—1271no—
227C2×C4—1271no—
228C2×C4—1271no—
229C2×C4—5082no—
230C2×C4—5082no—
231C2×C4—5081no—
232C2×C4—2542no—
233C2×C4—2541no—
234C2×C4—2541no—
235C2×C4—2542no—
236C2×C4D4×D12711yescharacteristic
237C2×C4C23×D12711yescharacteristic
238C8—22no—
239C8—2542no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1686123812yes—
246C22C2×D4×D12711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—5081no—
251C4—5082no—
252C4—5081no—
253C4—2541no—
254C4—2541no—
255C4—2541no—
256C4—2541no—
257C4C2×#1452324712yes—
258C2—22no—
259C2—81no—
260C2#2255038211yescharacteristic
261C2C2×#1686123811yescharacteristic
262C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#121×C127C22
hypercenter#448D127
hypocenterC127#3117
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic127C2semidirect
3. #22550212C2semidirect
5. #22550185C2semidirect
6. #22550096C2semidirect
7. #22550032C2semidirect
8. #22549980C2semidirect
9. #22549977C2semidirect
10. #22549930C2semidirect
132. C127#3117semidirect

Maximal subgroups

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

subgroupindex
#225504072
C2×Q8×Dic1272
#225502122
#225501922
#225501852
#225500962
#225500322
#225499802
#225499772
#225499302
#448×C1272
#3117127