Finite Group Databasebeta

← #30648859

Subgroups — #30648859

Order 16256.

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

normal subgroups only

This group has 380 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#22550431C211yesmaximal, characteristic
2C4×Q8×D127C211yesmaximal, characteristic
3#22550205C212yesmaximal
4#22550162C211yesmaximal, characteristic, complemented
5#22550161C211yesmaximal, characteristic, complemented
6#22550083C211yesmaximal, characteristic, complemented
7#22550030C212yesmaximal
8#22549983C212yesmaximal, complemented
9#22549980C212yesmaximal, complemented
10#22549932C211yesmaximal, characteristic
11#462×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C2×#14523238C2212yes—
14#16861209C2211yescharacteristic
15#16861200—21no—
16#16861189—21no—
17#16861185C2212yes—
18Q8×Dic127C2211yescharacteristic
19Q8×Dic127C2212yes—
20C4×Dic254—21no—
21C4×Dic254C2211yescharacteristic
22#16861174—21no—
23#16861174—21no—
24#16861174C2211yescharacteristic
25#16861167—21no—
26C4⋊C4×D127—21no—
27C4⋊C4×D127C2211yescharacteristic
28C42×D127—21no—
29C42×D127C2211yescharacteristic
30#16861141C2212yes—
31#16861140C2212yes—
32C4⋊(C127⋊C8)C2211yescharacteristic
33#16861134C2212yes—
34#16861133C2212yes—
35#16861133C2212yes—
36(C2×D127)⋊C8C2212yes—
37#16861119C2212yes—
38C2×Q16×C127C2212yes—
39C4⋊Q8×C127C2211yescharacteristic
40C4×Q8×C127C2211yescharacteristic
41C4⋊C8×C127C2211yescharacteristic
42#103×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C2×Dic254C2311yescharacteristic
46Q8×D127—22no—
47Q8×D127—22no—
48C2×C4×D127—21no—
49C2×C4×D127—21no—
50C2×C4×D127D411yescharacteristic
51C2×C4×D127D411yescharacteristic
52C2×C4×D127C2311yescharacteristic
53#14523238—24no—
54C2×C127⋊C8C2312yes—
55C4⋊Dic127—22no—
56C4⋊Dic127—21no—
57C4⋊Dic127—21no—
58C4⋊Dic127C2312yes—
59C4⋊Dic127C2311yescharacteristic
60Dic127⋊C4—22no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64(C2×D127)⋊C4—22no—
65C4×Dic127—21no—
66C4×Dic127—21no—
67C4×Dic127—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2312yes—
73C2×Q8×C127C2311yescharacteristic
74Q16×C127—24no—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127C2×D411yescharacteristic
83Dic254—24no—
84Dic254—21no—
85Dic254—42no—
86Dic254#4112yes—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—24no—
94C4×D127—22no—
95C4×D127—22no—
96C4×D127—21no—
97C4×D127—21no—
98C4×D127—42no—
99C127⋊C8—22no—
100Q8×C127—21no—
101Q8×C127—22no—
102Q8×C127—22no—
103Q8×C127#4112yes—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C8×C127—22no—
111C22×C127C22×D411yescharacteristic
112C2×D127—21no—
113C2×D127—21no—
114C2×D127#13712yes—
115C4×C127—22no—
116C4×C127—21no—
117C4×C127—42no—
118C4×C127—41no—
119C4×C127#13712yes—
120C4×C127C2×#4111yescharacteristic
121C4×C127C2×#4111yescharacteristic
122Dic127—22no—
123Dic127—21no—
124Dic127—21no—
125Dic127—21no—
126Dic127—41no—
127Dic127—42no—
128C2×C127#52511yescharacteristic
129C2×C127C2×#13711yescharacteristic
130C2×C127C2×#13711yescharacteristic
131D127—22no—
132#3113—1271nomaximal, Sylow
133C127#311311yescharacteristic, complemented, Sylow
134#533—1271no—
135C2×C4×Q8—1271no—
136#483—1272no—
137#475—1271no—
138#475—1271no—
139#462—1271no—
140#462D12711yescharacteristic, complemented
141#451—1272no—
142#441—1272no—
143#441—1272no—
144#432—1271no—
145C22×Q8—1271no—
146C2×Q16—1272no—
147C2×Q16C2×D12712yes—
148C4⋊Q8—1271no—
149C4⋊Q8C2×D12711yescharacteristic
150#125—2541no—
151C22⋊Q8—1272no—
152C22⋊Q8—2541no—
153C4×Q8—1271no—
154C4×Q8—1272no—
155C4×Q8—1271no—
156C4×Q8—1271no—
157C4×Q8—2541no—
158C4×Q8—2541no—
159C4×Q8—2541no—
160C4×Q8C2×D12711yescharacteristic
161#117—2541no—
162C2×C4⋊C4—1271no—
163C2×C4⋊C4—2541no—
164C2×C42—1271no—
165C2×C42—2541no—
166#106—1272no—
167#106—1272no—
168C4⋊C8—1271no—
169C4⋊C8C2×D12711yescharacteristic
170#103—1272no—
171#103—1272no—
172#103—1272no—
173#103C2×D12712yes—
174C22⋊C8—1272no—
175#97—1272no—
176C2×Q8—1271no—
177C2×Q8—2541no—
178C2×Q8—2542no—
179C2×Q8—2542no—
180C2×Q8—2541no—
181C2×Q8C22×D12712yes—
182C2×Q8C22×D12711yescharacteristic
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—1271no—
186C22×C4—2541no—
187C22×C4—2541no—
188Q16—24no—
189Q16—2544no—
190C2×C8—1272no—
191C2×C8C22×D12712yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—1271no—
196C4⋊C4—1272no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4—2542no—
200C4⋊C4—2541no—
201C4⋊C4—2542no—
202C4⋊C4—2541no—
203C4⋊C4—2541no—
204C4⋊C4C22×D12711yescharacteristic
205C22⋊C4—2542no—
206C42—21no—
207C42—1271no—
208C42—1271no—
209C42—1271no—
210C42—2541no—
211C42—2541no—
212C42—2542no—
213C42—2541no—
214C42C22×D12711yescharacteristic
215C23—1271no—
216Q8—21no—
217Q8—22no—
218Q8—22no—
219Q8—1272no—
220Q8—5082no—
221Q8—2544no—
222Q8—2541no—
223Q8#1452324712yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—1271no—
228C2×C4—1271no—
229C2×C4—1271no—
230C2×C4—5082no—
231C2×C4—2542no—
232C2×C4—2544no—
233C2×C4—2541no—
234C2×C4—2542no—
235C2×C4—2541no—
236C2×C4—2542no—
237C2×C4—2541no—
238C2×C4—2541no—
239C2×C4D4×D12711yescharacteristic
240C2×C4D4×D12711yescharacteristic
241C2×C4C23×D12711yescharacteristic
242C8—22no—
243C8—2542no—
244C22—1272no—
245C22—2541no—
246C22—2541no—
247C22C2×D4×D12711yescharacteristic
248C4—22no—
249C4—21no—
250C4—42no—
251C4—41no—
252C4—5081no—
253C4—5082no—
254C4—2541no—
255C4—2542no—
256C4—2541no—
257C4—2541no—
258C4#1686124112yes—
259C4#41×D12711yescharacteristic
260C4C2×#1452324711yescharacteristic
261C2—2542no—
262C2#2255038611yescharacteristic
263C2#137×D12711yescharacteristic
264C2C2×#1686124111yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socleC22×C127C22×D4
prime radicalC22×D127C2×D4
hypercenter#462D127
hypocenterC127#3113
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
4. #22550162C2semidirect
5. #22550161C2semidirect
6. #22550083C2semidirect
8. #22549983C2semidirect
9. #22549980C2semidirect
11. #462×C127C2semidirect
133. C127#3113semidirect
140. #462D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
C4×Q8×D1272
#225502052
#225501622
#225501612
#225500832
#225500302
#225499832
#225499802
#225499322
#462×C1272
#3113127