Finite Group Databasebeta

← #30639796

Subgroups — #30639796

Order 16256.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550674C211yesmaximal, characteristic, complemented
2#22550388C211yesmaximal, characteristic, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22549892C211yesmaximal, characteristic, complemented
5#22549839C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×C4×D127C2211yescharacteristic
10#16861240—21no—
11#16861239—21no—
12#16861235—21no—
13#16861234—21no—
14#16861229—22no—
15C2×#14523232C2211yescharacteristic
16#16861193—21no—
17#16861190—21no—
18#16861189—21no—
19#16861185—21no—
20#16861184—21no—
21#16861180—21no—
22#16861179—21no—
23#16861177—21no—
24C4×D508—22no—
25C4×C22⋊D127—22no—
26#16861168C2211yescharacteristic
27#16861135—22no—
28(C2×D127)⋊C8—22no—
29C2×#41×C127C2211yescharacteristic
30C2×M16×C127C2211yescharacteristic
31#117×C127C2211yescharacteristic
32#104×C127—22no—
33C23×D127D411yescharacteristic
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523248—21no—
38C2×Dic254D411yescharacteristic, complemented
39C2×D508—22no—
40C2×D508D411yescharacteristic, complemented
41C2×C22⋊D127—22no—
42C2×C22⋊D127—21no—
43C22×Dic127D411yescharacteristic
44C2×C4×D127—21no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48C2×C4×D127—21no—
49C2×C4×D127—21no—
50C2×C4×D127—21no—
51#14523238—22no—
52#14523236—22no—
53#14523235—22no—
54#14523233—22no—
55#14523232—21no—
56#14523232—21no—
57C2×C127⋊C8D412yescomplemented
58C4⋊Dic127—22no—
59C4⋊Dic127—21no—
60Dic127⋊C4—22no—
61C22⋊Dic127—22no—
62C22⋊Dic127—21no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4—22no—
66C4×Dic127D412yescomplemented
67#41×C127—21no—
68#41×C127—21no—
69#41×C127—21no—
70#41×C127—21no—
71C2×Q8×C127D411yescharacteristic, complemented
72C2×D4×C127—21no—
73C2×D4×C127D411yescharacteristic, complemented
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76M16×C127—21no—
77M16×C127—21no—
78C2×C8×C127—21no—
79C4⋊C4×C127—21no—
80C22⋊C4×C127—21no—
81C42×C127D412yescomplemented
82C22×D127—21no—
83C22×D127—21no—
84C22×D127—21no—
85C22×D127—21no—
86C22×D127—42no—
87Dic254—21no—
88Dic254—21no—
89D508—21no—
90D508—21no—
91D508—44no—
92C22⋊D127—44no—
93C22⋊D127—41no—
94C22⋊D127—41no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C4×D127—41no—
101C4×D127—41no—
102C4×D127—41no—
103C4×D127—41no—
104C4×D127—41no—
105C4×D127—41no—
106C127⋊C8—22no—
107C23×C127—21no—
108C23×C127C2×D411yescharacteristic
109Q8×C127—21no—
110Q8×C127—21no—
111D4×C127—21no—
112D4×C127—21no—
113D4×C127—41no—
114D4×C127—41no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D411yescharacteristic
124C2×C4×C127#4111yescharacteristic
125C8×C127—41no—
126C22×C127—41no—
127C22×C127—42no—
128C22×C127—41no—
129C22×C127#12011yescharacteristic
130C22×C127#12011yescharacteristic
131C22×C127#12111yescharacteristic
132C2×D127—22no—
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—41no—
136C2×D127—41no—
137C2×D127—41no—
138C2×D127—41no—
139C2×D127—42no—
140C2×D127—41no—
141C4×C127—42no—
142C4×C127—41no—
143C4×C127#12011yescharacteristic
144C4×C127#12111yescharacteristic
145C4×C127#12111yescharacteristic
146C4×C127#12711yescharacteristic
147Dic127—42no—
148Dic127—41no—
149Dic127—41no—
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—21no—
153C2×C127—81no—
154C2×C127#39211yescharacteristic
155D127—42no—
156D127—81no—
157#2010—1271nomaximal, Sylow
158C127#201011yescharacteristic, complemented, Sylow
159#575—1271no—
160#525—1271no—
161#525—1271no—
162#421—1271no—
163#421D12711yescharacteristic, complemented
164#407—1271no—
165H(3,ℤ/4)—1271no—
166C2×#41—1271no—
167C2×#41C2×D12711yescharacteristic
168C23×C4—1271no—
169#137—2541no—
170#137—2541no—
171AGL(1,ℤ/8)—2541no—
172AGL(1,ℤ/8)—2541no—
173#135—2542no—
174C2×M16—1271no—
175C2×M16C2×D12711yescharacteristic
176#123—2541no—
177#123—2541no—
178C22⋊Q8—2541no—
179C22⋊Q8—2541no—
180#121—2541no—
181#121—2541no—
182#120—2541no—
183#120—2541no—
184C4×D4—2542no—
185C4×D4—2542no—
186#117—1271no—
187#117C2×D12711yescharacteristic
188#104—22no—
189#104—2542no—
190C22⋊C8—2542no—
191C24—1271no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—2541no—
197#41—2541no—
198#41—2541no—
199#41—2541no—
200C2×Q8—1271no—
201C2×Q8D50811yescharacteristic, complemented
202C2×D4—21no—
203C2×D4—1271no—
204C2×D4—2542no—
205C2×D4—2542no—
206C2×D4—2541no—
207C2×D4D50811yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—1271no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4—2541no—
215C22×C4—2541no—
216C22×C4—2541no—
217C22×C4C22×D12711yescharacteristic
218Q16—2542no—
219SD16—2542no—
220SD16—2542no—
221D8—2542no—
222M16—21no—
223M16—21no—
224M16—2541no—
225M16—2541no—
226C2×C8—21no—
227C2×C8—1272no—
228C4⋊C4—21no—
229C4⋊C4—2542no—
230C4⋊C4—2542no—
231C4⋊C4—2541no—
232C22⋊C4—21no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C22⋊C4—2542no—
237C22⋊C4—2541no—
238C42—1272no—
239C42C22⋊D12712yescomplemented
240C23—21no—
241C23—5082no—
242C23—2541no—
243C23—2541no—
244C23—2541no—
245C23—2541no—
246C23D4×D12711yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—2541no—
250Q8—2541no—
251D4—21no—
252D4—21no—
253D4—41no—
254D4—41no—
255D4—5084no—
256D4—5084no—
257D4—5081no—
258D4—5081no—
259D4—2541no—
260D4—2541no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—5081no—
266C2×C4—5081no—
267C2×C4—5081no—
268C2×C4—5081no—
269C2×C4—5081no—
270C2×C4—5081no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4—2542no—
274C2×C4—2541no—
275C2×C4—2541no—
276C2×C4D4×D12711yescharacteristic
277C2×C4D4×D12711yescharacteristic
278C2×C4D4×D12711yescharacteristic
279C2×C4C2×C22⋊D12711yescharacteristic
280C2×C4C2×D50811yescharacteristic
281C2×C4#1452324811yescharacteristic
282C8—41no—
283C8—2542no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—5082no—
288C22—5081no—
289C22—5081no—
290C22—5081no—
291C22—5081no—
292C22—5081no—
293C22—2542no—
294C22—2542no—
295C22—2542no—
296C22#1686117711yescharacteristic
297C22#1686117911yescharacteristic
298C22#1686118211yescharacteristic
299C4—42no—
300C4—41no—
301C4—5082no—
302C4—5081no—
303C4—5081no—
304C4#1686117911yescharacteristic
305C4#1686118211yescharacteristic
306C4#1686118311yescharacteristic
307C4#1686120511yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2—10161no—
313C2—5082no—
314C2#2254978511yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#392
hypercenter#421D127
hypocenterC127#2010
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550388C2semidirect
3. #22550383C2semidirect
4. #22549892C2semidirect
5. #22549839C2semidirect
6. #22549632C2semidirect
7. #421×C127C2semidirect
38. C2×Dic254D4semidirect
40. C2×D508D4semidirect
57. C2×C127⋊C8D4semidirect
66. C4×Dic127D4semidirect
71. C2×Q8×C127D4semidirect
73. C2×D4×C127D4semidirect
81. C42×C127D4semidirect
158. C127#2010semidirect
163. #421D127semidirect
201. C2×Q8D508semidirect
207. C2×D4D508semidirect
239. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225503882
#225503832
#225498922
#225498392
#225496322
#421×C1272
#2010127