Finite Group Databasebeta

← #30650460

Subgroups — #30650460

Order 16256.

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

normal subgroups only

This group has 494 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×#16861229C212yesmaximal, complemented
2#22550531C211yesmaximal, characteristic, complemented
3#22550248C211yesmaximal, characteristic, complemented
4#22550103C212yesmaximal, complemented
5#22550086C211yesmaximal, characteristic, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550038C212yesmaximal, complemented
8#22550019C212yesmaximal, complemented
9C4×#14523235C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic, complemented
11#550×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13#16861229—28no—
14C2×#14523238C2212yes—
15C2×#14523236C2212yes—
16C2×#14523236C2212yes—
17C2×#14523235C2211yescharacteristic
18C2×#14523233C2212yes—
19C22×C127⋊C8C2212yes—
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#16861132C2212yes—
30#16861130C2212yes—
31#16861129C2211yescharacteristic
32C22⋊(C127⋊C8)C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34C2×#41×C127C2212yes—
35#127×C127—21no—
36#127×C127C2211yescharacteristic
37#121×C127—22no—
38#121×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#14523236—24no—
51#14523236—42no—
52#14523235D414yescomplemented
53#14523233—24no—
54C2×C127⋊C8—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—
64#41×C127—24no—
65#41×C127—42no—
66C2×Q8×C127C2311yescharacteristic
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127D411yescharacteristic
72C2×D4×C127C2312yes—
73C22×C4×C127—22no—
74C22×C4×C127C2312yes—
75C4⋊C4×C127—21no—
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×C127—22no—
96C23×C127C2×D412yes—
97Q8×C127—21no—
98Q8×C127C2×D412yes—
99D4×C127—22no—
100D4×C127—22no—
101D4×C127—22no—
102D4×C127—44no—
103D4×C127—42no—
104D4×C127—81no—
105D4×C127—82no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—44no—
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2411yescharacteristic
115C22×C127—22no—
116C22×C127—22no—
117C22×C127—22no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127C22×D411yescharacteristic
122C2×D127—22no—
123C2×D127—41no—
124C2×D127—41no—
125C4×C127—22no—
126C4×C127—22no—
127C4×C127—21no—
128C4×C127—41no—
129C4×C127#13512yescomplemented
130C4×C127C22×D411yescharacteristic
131C4×C127H(5,𝔽2)11yescharacteristic
132Dic127—42no—
133Dic127—41no—
134C2×C127—42no—
135C2×C127—82no—
136C2×C127D4211yescharacteristic
137C2×C127C2×#13511yescharacteristic
138C2×C127#57711yescharacteristic
139D127—42no—
140#3284—1271nomaximal, Sylow
141C127#328411yescharacteristic, complemented, Sylow
142C2×#135—1272no—
143#550D12711yescharacteristic, complemented
144#547—1271no—
145#492—1271no—
146#465—1272no—
147#462—1271no—
148#459—1271no—
149#452—1272no—
150#449—1272no—
151C4×SD16—1271no—
152C8×D4—1271no—
153C2×#41—1272no—
154C2×#41C2×D12712yes—
155#135—2548no—
156C2×Q16—1272no—
157C2×SD16—1271no—
158C2×SD16—1272no—
159C2×SD16—1272no—
160C2×D8—1272no—
161C22×C8—1272no—
162C4⋊Q8—1271no—
163#127—21no—
164#127C2×D12711yescharacteristic
165#123—2542no—
166C22⋊Q8—1272no—
167#121—22no—
168#121—2541no—
169#121C2×D12712yes—
170C4×Q8C2×D12711yescharacteristic
171C4×D4—21no—
172C4×D4—1271no—
173C4×D4C2×D12711yescharacteristic
174C2×C4⋊C4—2541no—
175#106—1271no—
176C4⋊C8—1271no—
177#103—1271no—
178#103—1272no—
179#102—1272no—
180#102—1271no—
181C22⋊C8—1272no—
182C4×C8—1271no—
183#41—24no—
184#41—42no—
185#41—5082no—
186#41—2544no—
187C2×Q8—1272no—
188C2×Q8C22×D12711yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—1271no—
194C2×D4—2542no—
195C2×D4C22⋊D12711yescharacteristic
196C2×D4C22×D12712yes—
197C22×C4—22no—
198C22×C4—2541no—
199C22×C4—2542no—
200C22×C4C22×D12712yes—
201Q16—2544no—
202SD16—1274no—
203SD16—5082no—
204SD16—2544no—
205D8—2544no—
206C2×C8—1271no—
207C2×C8—1271no—
208C2×C8—1272no—
209C2×C8—2544no—
210C4⋊C4—21no—
211C4⋊C4—1271no—
212C4⋊C4—2542no—
213C4⋊C4—2541no—
214C4⋊C4—2541no—
215C4⋊C4C22⋊D12711yescharacteristic
216C4⋊C4C22×D12711yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—2542no—
219C22⋊C4—2541no—
220C22⋊C4C22⋊D12712yes—
221C42—21no—
222C42C22×D12711yescharacteristic
223C23—22no—
224C23—2541no—
225C23C2×C22⋊D12712yes—
226Q8—21no—
227Q8—2542no—
228Q8—2542no—
229Q8D4×D12712yes—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—82no—
237D4—1272no—
238D4—5084no—
239D4—2541no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—5084no—
246C2×C4—5082no—
247C2×C4—2541no—
248C2×C4—2542no—
249C2×C4C2×C22⋊D12711yescharacteristic
250C2×C4C2×C22⋊D12712yes—
251C2×C4C2×C22⋊D12711yescharacteristic
252C2×C4C23×D12711yescharacteristic
253C8—1272no—
254C8—2542no—
255C8—2541no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—5081no—
263C22—5081no—
264C22—2542no—
265C22C22×C22⋊D12711yescharacteristic
266C4—21no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—5081no—
271C4—5082no—
272C4#1686122912yescomplemented
273C4C2×D4×D12711yescharacteristic
274C4#1686125411yescharacteristic
275C2—42no—
276C2—82no—
277C2—5082no—
278C2D4×C22⋊D12711yescharacteristic
279C2C2×#1686122911yescharacteristic
280C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#550×C127C2
socleC22×C127C22×D4
hypercenter#550D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550531C2semidirect
3. #22550248C2semidirect
4. #22550103C2semidirect
5. #22550086C2semidirect
6. #22550068C2semidirect
7. #22550038C2semidirect
8. #22550019C2semidirect
9. C4×#14523235C2semidirect
10. D4×C127⋊C8C2semidirect
11. #550×C127C2semidirect
52. #14523235D4semidirect
129. C4×C127#135semidirect
141. C127#3284semidirect
143. #550D127semidirect
272. C4#16861229semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225505312
#225502482
#225501032
#225500862
#225500682
#225500382
#225500192
C4×#145232352
D4×C127⋊C82
#550×C1272
#3284127