Finite Group Databasebeta

← #30650376

Subgroups — #30650376

Order 16256.

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

normal subgroups only

This group has 472 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
1C2×#16861240C212yesmaximal, complemented
2#22550531C211yesmaximal, characteristic
3#22550255C211yesmaximal, characteristic, complemented
4#22550132C212yesmaximal, complemented
5#22550086C211yesmaximal, characteristic, complemented
6#22550078C211yesmaximal, characteristic, complemented
7#22550036C212yesmaximal, complemented
8#22550032C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10#22549954C211yesmaximal, characteristic, complemented
11D4×Q8×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13#16861240—28no—
14C2×#14523238C2212yes—
15C2×#14523238C2212yes—
16C2×#14523238C2212yes—
17C2×#14523235C2212yes—
18C2×#14523235C2211yescharacteristic
19C2×#14523232C2212yes—
20#16861209C2211yescharacteristic
21#16861191—22no—
22#16861188C2212yes—
23#16861180—21no—
24C4×D508C2211yescharacteristic
25C4⋊C4×D127—21no—
26#16861140C2211yescharacteristic
27C4⋊(C127⋊C8)C2211yescharacteristic
28#16861133C2211yescharacteristic
29#16861133C2212yes—
30#16861132C2212yes—
31#16861129C2211yescharacteristic
32C22⋊(C127⋊C8)C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34C22×Q8×C127C2212yes—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127C2211yescharacteristic
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127C2212yes—
39C4×Q8×C127C2211yescharacteristic
40C4×D4×C127—21no—
41C4×D4×C127C2211yescharacteristic
42#14523248—24no—
43#14523248—42no—
44C2×Dic254C2312yes—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49#14523238—24no—
50#14523238—24no—
51#14523238—42no—
52#14523235—24no—
53#14523235D414yescomplemented
54#14523232—24no—
55C2×C127⋊C8C2312yes—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63(C2×D127)⋊C4—21no—
64C2×Q8×C127—22no—
65C2×Q8×C127—24no—
66C2×Q8×C127—42no—
67C2×Q8×C127C2312yes—
68C2×Q8×C127C2311yescharacteristic
69C2×D4×C127D411yescharacteristic
70C22×C4×C127—22no—
71C22×C4×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127—22no—
76C4⋊C4×C127D411yescharacteristic
77C4⋊C4×C127C2311yescharacteristic
78C22⋊C4×C127—22no—
79C22⋊C4×C127D412yes—
80C42×C127—21no—
81C42×C127C2311yescharacteristic
82C22×D127—21no—
83Dic254—22no—
84Dic254—22no—
85D508—21no—
86D508C2×D412yes—
87C22⋊D127—44no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C4×D127—44no—
91C4×D127—42no—
92C127⋊C8—22no—
93C127⋊C8—21no—
94C127⋊C8C2×D412yes—
95C23×C127C2×D412yes—
96Q8×C127—21no—
97Q8×C127—22no—
98Q8×C127—22no—
99Q8×C127—44no—
100Q8×C127C2×D412yes—
101D4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—44no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D412yes—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2411yescharacteristic
112C22×C127—22no—
113C22×C127—22no—
114C22×C127—22no—
115C22×C127C22×D411yescharacteristic
116C2×D127—22no—
117C2×D127—41no—
118C2×D127—41no—
119C4×C127—22no—
120C4×C127—22no—
121C4×C127—21no—
122C4×C127—42no—
123C4×C127—41no—
124C4×C127#13712yescomplemented
125C4×C127C22×D411yescharacteristic
126C4×C127H(5,𝔽2)11yescharacteristic
127Dic127—42no—
128Dic127—41no—
129C2×C127—42no—
130C2×C127D4211yescharacteristic
131C2×C127C2×#13711yescharacteristic
132C2×C127#57811yescharacteristic
133D127—42no—
134#3276—1271nomaximal, Sylow
135C127#327611yescharacteristic, complemented, Sylow
136C2×#137—1272no—
137D4×Q8D12711yescharacteristic, complemented
138#547—1271no—
139#494—1271no—
140#470—1272no—
141#462—1271no—
142#461—1271no—
143#452—1272no—
144#451—1272no—
145C4×SD16—1271no—
146#436—1271no—
147C2×#41—1272no—
148C22×Q8C2×D12712yes—
149#137—2548no—
150C2×Q16—1272no—
151C2×Q16—1272no—
152C2×Q16—1272no—
153C2×SD16—1271no—
154C2×SD16—1272no—
155C2×M16—1272no—
156C4⋊Q8—21no—
157C4⋊Q8—1271no—
158C4⋊Q8C2×D12711yescharacteristic
159#123—2542no—
160C22⋊Q8—22no—
161C22⋊Q8—1272no—
162C22⋊Q8C2×D12712yes—
163#121—2541no—
164C4×Q8C2×D12711yescharacteristic
165C4×D4—21no—
166C4×D4—1271no—
167C4×D4C2×D12711yescharacteristic
168C2×C4⋊C4—2541no—
169#106—1271no—
170C4⋊C8—1271no—
171#103—1271no—
172#103—1272no—
173#103—1272no—
174#102—1271no—
175C22⋊C8—1272no—
176C4×C8—1271no—
177#41—5082no—
178#41—2544no—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—1272no—
183C2×Q8C22×D12712yes—
184C2×Q8C22×D12711yescharacteristic
185C2×D4—1271no—
186C2×D4—2542no—
187C2×D4C22⋊D12711yescharacteristic
188C22×C4—22no—
189C22×C4—2541no—
190C22×C4—2542no—
191C22×C4C22×D12712yes—
192Q16—5082no—
193Q16—2544no—
194Q16—2544no—
195SD16—1274no—
196SD16—2544no—
197M16—2544no—
198C2×C8—1271no—
199C2×C8—1271no—
200C2×C8—1272no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—22no—
205C4⋊C4—1271no—
206C4⋊C4—2542no—
207C4⋊C4—2541no—
208C4⋊C4—2541no—
209C4⋊C4C22⋊D12711yescharacteristic
210C4⋊C4C22×D12711yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—2542no—
213C22⋊C4—2541no—
214C22⋊C4C22⋊D12712yes—
215C42—21no—
216C42C22×D12711yescharacteristic
217C23—2541no—
218C23C2×C22⋊D12712yes—
219Q8—21no—
220Q8—22no—
221Q8—22no—
222Q8—44no—
223Q8—2542no—
224Q8—2542no—
225Q8D4×D12712yes—
226D4—22no—
227D4—1272no—
228D4—5084no—
229D4—2541no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—5084no—
237C2×C4—5082no—
238C2×C4—2541no—
239C2×C4—2542no—
240C2×C4C2×C22⋊D12711yescharacteristic
241C2×C4C2×C22⋊D12712yes—
242C2×C4C2×C22⋊D12711yescharacteristic
243C2×C4C23×D12711yescharacteristic
244C8—1272no—
245C8—2542no—
246C8—2541no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—5081no—
251C22—5081no—
252C22—2542no—
253C22C22×C22⋊D12711yescharacteristic
254C4—21no—
255C4—22no—
256C4—22no—
257C4—41no—
258C4—42no—
259C4—5081no—
260C4—5082no—
261C4#1686124012yescomplemented
262C4C2×D4×D12711yescharacteristic
263C4#1686125411yescharacteristic
264C2—42no—
265C2—5082no—
266C2D4×C22⋊D12711yescharacteristic
267C2C2×#1686124011yescharacteristic
268C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD4×Q8×C127C2
socleC22×C127C22×D4
prime radical#22550531C2
hypercenterD4×Q8D127
hypocenterC127#3276
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. #22550255C2semidirect
4. #22550132C2semidirect
5. #22550086C2semidirect
6. #22550078C2semidirect
7. #22550036C2semidirect
8. #22550032C2semidirect
9. C4×#14523235C2semidirect
10. #22549954C2semidirect
11. D4×Q8×C127C2semidirect
53. #14523235D4semidirect
124. C4×C127#137semidirect
135. C127#3276semidirect
137. D4×Q8D127semidirect
261. C4#16861240semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225505312
#225502552
#225501322
#225500862
#225500782
#225500362
#225500322
C4×#145232352
#225499542
D4×Q8×C1272
#3276127