Finite Group Databasebeta

← #30650456

Subgroups — #30650456

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#22550547C211yesmaximal, characteristic, complemented
3#22550250C211yesmaximal, characteristic, complemented
4#22550107C212yesmaximal, complemented
5#22550083C211yesmaximal, characteristic, complemented
6#22550071C211yesmaximal, characteristic, complemented
7#22550036C212yesmaximal, complemented
8#22550021C212yesmaximal, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10D4×C127⋊C8C211yesmaximal, characteristic, complemented
11#547×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yes—
13#16861229—28no—
14C2×#14523238C2212yes—
15C2×#14523236C2211yescharacteristic
16C2×#14523235C2212yes—
17C2×#14523235C2212yes—
18C2×#14523233C2212yes—
19C22×C127⋊C8C2212yes—
20#16861206C2211yescharacteristic
21#16861205—21no—
22#16861183C2212yes—
23#16861182—22no—
24C4×Dic254C2211yescharacteristic
25#16861171—21no—
26#16861140C2211yescharacteristic
27C4⋊(C127⋊C8)C2211yescharacteristic
28#16861133C2212yes—
29#16861132C2211yescharacteristic
30#16861130C2211yescharacteristic
31#16861129C2212yes—
32C22⋊(C127⋊C8)C2212yes—
33C4×C127⋊C8C2211yescharacteristic
34C2×#41×C127C2212yes—
35C4⋊Q8×C127C2211yescharacteristic
36#123×C127—22no—
37C22⋊Q8×C127C2212yes—
38#121×C127—21no—
39C4×D4×C127C2211yescharacteristic
40C4×D4×C127C2211yescharacteristic
41C2×C4⋊C4×C127—21no—
42#14523248—24no—
43#14523248—42no—
44C2×Dic254C2311yescharacteristic
45C2×D508—21no—
46C2×D508—21no—
47C2×D508C2312yes—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C4×D127—22no—
51#14523238—24no—
52#14523236D414yescomplemented
53#14523235—24no—
54#14523235—42no—
55#14523233—24no—
56C2×C127⋊C8—24no—
57C2×C127⋊C8C2312yes—
58C2×C127⋊C8C2311yescharacteristic
59C2×C127⋊C8C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—21no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—21no—
64#41×C127—24no—
65#41×C127—42no—
66C2×Q8×C127C2312yes—
67C2×D4×C127—22no—
68C2×D4×C127D411yescharacteristic
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—22no—
72C22×C4×C127C2312yes—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—22no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127D411yescharacteristic
77C4⋊C4×C127C2311yescharacteristic
78C22⋊C4×C127—22no—
79C22⋊C4×C127—21no—
80C22⋊C4×C127D412yes—
81C42×C127C2311yescharacteristic
82C22×D127—22no—
83Dic254—21no—
84Dic254C2×D412yes—
85D508—22no—
86D508—22no—
87D508—42no—
88D508—81no—
89C22⋊D127—44no—
90C22⋊D127—82no—
91C2×Dic127—21no—
92C2×Dic127—21no—
93C4×D127—44no—
94C127⋊C8—22no—
95C127⋊C8—21no—
96C127⋊C8C2×D412yes—
97C23×C127—21no—
98C23×C127C2×D412yes—
99Q8×C127—22no—
100Q8×C127—22no—
101D4×C127—22no—
102D4×C127—21no—
103D4×C127—44no—
104D4×C127C2×D412yes—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—44no—
110C2×C4×C127—42no—
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—22no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127C22×D411yescharacteristic
122C2×D127—42no—
123C2×D127—42no—
124C2×D127—42no—
125C4×C127—22no—
126C4×C127—21no—
127C4×C127—42no—
128C4×C127—41no—
129C4×C127#13512yescomplemented
130C4×C127C22×D411yescharacteristic
131C4×C127H(5,𝔽2)11yescharacteristic
132Dic127—22no—
133Dic127—41no—
134C2×C127—42no—
135C2×C127—42no—
136C2×C127D4211yescharacteristic
137C2×C127C2×#13511yescharacteristic
138C2×C127#57711yescharacteristic
139D127—82no—
140#3284—1271nomaximal, Sylow
141C127#328411yescharacteristic, complemented, Sylow
142C2×#135—1272no—
143#550—1271no—
144#547D12711yescharacteristic, complemented
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⋊Q8C2×D12711yescharacteristic
163#127—1271no—
164#127—2541no—
165#123—22no—
166C22⋊Q8C2×D12712yes—
167#121—21no—
168#121—1272no—
169#121—2542no—
170C4×Q8—1271no—
171C4×D4—2541no—
172C4×D4C2×D12711yescharacteristic
173C4×D4C2×D12711yescharacteristic
174C2×C4⋊C4—21no—
175#106—1271no—
176C4⋊C8—1271no—
177#103—1272no—
178#103—1271no—
179#102—1271no—
180#102—1272no—
181C22⋊C8—1272no—
182C4×C8—1271no—
183#41—24no—
184#41—42no—
185#41—5082no—
186#41—2544no—
187C2×Q8—1271no—
188C2×Q8C22×D12712yes—
189C2×D4—22no—
190C2×D4—1272no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4—2541no—
194C2×D4—2541no—
195C2×D4C22⋊D12711yescharacteristic
196C2×D4C22×D12711yescharacteristic
197C22×C4—21no—
198C22×C4—22no—
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—22no—
212C4⋊C4—21no—
213C4⋊C4—1271no—
214C4⋊C4—2541no—
215C4⋊C4C22⋊D12711yescharacteristic
216C4⋊C4C22×D12711yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—2542no—
220C22⋊C4C22⋊D12712yes—
221C42—2541no—
222C42C22×D12711yescharacteristic
223C23—21no—
224C23—2542no—
225C23C2×C22⋊D12712yes—
226Q8—22no—
227Q8—22no—
228Q8—1272no—
229Q8—2541no—
230D4—22no—
231D4—21no—
232D4—44no—
233D4—10161no—
234D4—10162no—
235D4—5084no—
236D4—5082no—
237D4—2542no—
238D4—2542no—
239D4D4×D12712yes—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—5084no—
247C2×C4—2541no—
248C2×C4—2541no—
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—22no—
260C22—41no—
261C22—41no—
262C22—5082no—
263C22—5082no—
264C22—5082no—
265C22C22×C22⋊D12711yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—5081no—
271C4—2542no—
272C4#1686122912yescomplemented
273C4C2×D4×D12711yescharacteristic
274C4#1686125411yescharacteristic
275C2—42no—
276C2—42no—
277C2—10162no—
278C2D4×C22⋊D12711yescharacteristic
279C2C2×#1686122911yescharacteristic
280C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. #22550547C2semidirect
3. #22550250C2semidirect
4. #22550107C2semidirect
5. #22550083C2semidirect
6. #22550071C2semidirect
7. #22550036C2semidirect
8. #22550021C2semidirect
9. C4×#14523236C2semidirect
10. D4×C127⋊C8C2semidirect
11. #547×C127C2semidirect
52. #14523236D4semidirect
129. C4×C127#135semidirect
141. C127#3284semidirect
144. #547D127semidirect
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
#225505472
#225502502
#225501072
#225500832
#225500712
#225500362
#225500212
C4×#145232362
D4×C127⋊C82
#547×C1272
#3284127