Finite Group Databasebeta

← #30640712

Subgroups — #30640712

Order 16256.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550508C211yesmaximal, characteristic, complemented
2#22550063C211yesmaximal, characteristic, complemented
3#22550044C211yesmaximal, characteristic, complemented
4#22549668C211yesmaximal, characteristic, complemented
5#22549663C211yesmaximal, characteristic, complemented
6#22549662C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127—21no—
9C2×D4×D127—21no—
10#16861234—21no—
11#16861205C2211yescharacteristic
12#16861182—21no—
13#16861181—21no—
14#16861179—21no—
15#16861177—21no—
16#16861177C2211yescharacteristic
17(C22⋊D127)⋊C4—21no—
18#16861136—21no—
19#16861126C2211yescharacteristic, complemented
20#16861124—21no—
21#16861124C2211yescharacteristic, complemented
22H(5,𝔽2)×C127C2211yescharacteristic, complemented
23#120×C127—21no—
24#120×C127C2211yescharacteristic
25#99×C127—21no—
26#99×C127C2211yescharacteristic, complemented
27C23×D127—21no—
28C23×D127D411yescharacteristic, complemented
29C2×D508—21no—
30C2×D508D411yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—41no—
37C2×C22⋊D127—44no—
38D4×D127—44no—
39C22×Dic127—21no—
40C2×C4×D127—21no—
41#14523235—41no—
42#14523233—41no—
43#14523232—21no—
44Dic127⋊C4—21no—
45C22⋊Dic127—21no—
46C22⋊Dic127D411yescharacteristic, complemented
47(C2×D127)⋊C4—21no—
48(C2×D127)⋊C4—21no—
49(C2×D127)⋊C4—41no—
50C4×Dic127D411yescharacteristic, complemented
51C24×C127D411yescharacteristic, complemented
52#41×C127—21no—
53#41×C127—41no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127C2311yescharacteristic
60C22⋊C4×C127—21no—
61C22⋊C4×C127—21no—
62C22⋊C4×C127—41no—
63C22⋊C4×C127D411yescharacteristic, complemented
64C22×D127—21no—
65C22×D127—21no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127—41no—
69C22×D127—42no—
70C22×D127—81no—
71D508—41no—
72D508—42no—
73C22⋊D127—24no—
74C22⋊D127—42no—
75C22⋊D127—42no—
76C22⋊D127—42no—
77C22⋊D127—42no—
78C22⋊D127—82no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—42no—
82C2×Dic127—41no—
83C4×D127—42no—
84C127⋊C8—41no—
85C23×C127—21no—
86C23×C127—21no—
87C23×C127—21no—
88C23×C127—41no—
89C23×C127—42no—
90C23×C127C2×D411yescharacteristic
91C23×C127C2×D411yescharacteristic
92Q8×C127—21no—
93D4×C127—21no—
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—41no—
97D4×C127—42no—
98D4×C127—41no—
99D4×C127—82no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127C2×D411yescharacteristic
106C22×C127—22no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—42no—
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—81no—
116C22×C127—81no—
117C22×C127—81no—
118C22×C127#12011yescharacteristic
119C2×D127—22no—
120C2×D127—41no—
121C2×D127—41no—
122C2×D127—42no—
123C2×D127—41no—
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—81no—
127C2×D127—84no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—41no—
131C4×C127—81no—
132Dic127—22no—
133Dic127—41no—
134Dic127—81no—
135C2×C127—21no—
136C2×C127—41no—
137C2×C127—41no—
138C2×C127—42no—
139C2×C127—81no—
140C2×C127B(4,𝔽2)11yescharacteristic
141D127—42no—
142D127—81no—
143#2196—1271nomaximal, Sylow
144C127#219611yescharacteristic, complemented, Sylow
145D42—1271no—
146B(4,𝔽2)—1271no—
147B(4,𝔽2)D12711yescharacteristic, complemented
148#453—1271no—
149#353—1271no—
150AΣL(1,𝔽16)—1271no—
151AΣL(1,𝔽16)—1271no—
152H(5,𝔽2)C2×D12711yescharacteristic, complemented
153C22×D4—2541no—
154C22×D4—2541no—
155AGL(1,ℤ/8)—2541no—
156#127—1271no—
157#121—2541no—
158#121—2541no—
159#120—21no—
160#120—1271no—
161#120—2541no—
162#120—2541no—
163#120C2×D12711yescharacteristic
164C4×D4—2541no—
165#104—2541no—
166#100—1271no—
167#99—21no—
168#99—1271no—
169#99—2541no—
170#99C2×D12711yescharacteristic, complemented
171C24—1271no—
172C24—2541no—
173C24C22⋊D12711yescharacteristic, complemented
174#41—21no—
175#41—41no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—1271no—
182C2×D4—5084no—
183C2×D4—5081no—
184C2×D4—5084no—
185C2×D4—2541no—
186C2×D4—2541no—
187C2×D4—2541no—
188C2×D4—2541no—
189C2×D4—2541no—
190C2×D4—2541no—
191C2×D4C22×D12711yescharacteristic
192C22×C4—2541no—
193C22×C4—2541no—
194SD16—5081no—
195D8—5081no—
196M16—2541no—
197C4⋊C4—2541no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—41no—
201C22⋊C4—1271no—
202C22⋊C4—5081no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C22⋊C4C22⋊D12711yescharacteristic, complemented
207C42—1271no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—42no—
213C23—10161no—
214C23—5082no—
215C23—5081no—
216C23—5082no—
217C23—5082no—
218C23—2541no—
219C23—2541no—
220C23D4×D12711yescharacteristic
221C23C2×C22⋊D12711yescharacteristic
222Q8—21no—
223D4—21no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—42no—
228D4—41no—
229D4—82no—
230D4—10162no—
231D4—5081no—
232D4—5082no—
233D4—5082no—
234D4—5082no—
235D4—5082no—
236D4—5082no—
237D4—2544no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—5082no—
244C2×C4—5081no—
245C2×C4—5082no—
246C2×C4—2541no—
247C2×C4—2541no—
248C2×C4D4×D12711yescharacteristic
249C8—5081no—
250C22—22no—
251C22—21no—
252C22—21no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—10164no—
263C22—10161no—
264C22—5081no—
265C22—5081no—
266C22—5081no—
267C22—5082no—
268C22—5081no—
269C22—5081no—
270C22—2542no—
271C22#1686117711yescharacteristic
272C4—21no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—10161no—
277C4—5081no—
278C4—2542no—
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—81no—
284C2—10161no—
285C2—5082no—
286C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingB(4,𝔽2)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterB(4,𝔽2)D127
hypocenterC127#2196
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550508C2semidirect
2. #22550063C2semidirect
3. #22550044C2semidirect
4. #22549668C2semidirect
5. #22549663C2semidirect
6. #22549662C2semidirect
7. B(4,𝔽2)×C127C2semidirect
19. #16861126C22semidirect
21. #16861124C22semidirect
22. H(5,𝔽2)×C127C22semidirect
26. #99×C127C22semidirect
28. C23×D127D4semidirect
30. C2×D508D4semidirect
46. C22⋊Dic127D4semidirect
50. C4×Dic127D4semidirect
51. C24×C127D4semidirect
63. C22⋊C4×C127D4semidirect
144. C127#2196semidirect
147. B(4,𝔽2)D127semidirect
152. H(5,𝔽2)C2×D127semidirect
170. #99C2×D127semidirect
173. C24C22⋊D127semidirect
206. C22⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225505082
#225500632
#225500442
#225496682
#225496632
#225496622
B(4,𝔽2)×C1272
#2196127