Finite Group Databasebeta

← #30639736

Subgroups — #30639736

Order 16256.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14523237C211yesmaximal, characteristic, complemented
2C2×#16861189C211yesmaximal, characteristic, complemented
3C2×#16861184C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic, complemented
5C2×#16861129C211yesmaximal, characteristic, complemented
6C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×Dic254C2211yescharacteristic
9C22×D508—21no—
10C22×D508C2211yescharacteristic
11C22×C4×D127C2211yescharacteristic
12C2×#14523237—22no—
13C2×#14523237—22no—
14C2×#14523237—22no—
15C22×C127⋊C8C2211yescharacteristic
16#16861189—22no—
17#16861189—22no—
18#16861184—22no—
19#16861184—22no—
20C2×C4⋊Dic127—21no—
21C2×(C2×D127)⋊C4—21no—
22C2×(C2×D127)⋊C4—21no—
23#16861132—22no—
24#16861129—22no—
25(C2×D127)⋊C8—22no—
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C4⋊C4×C127C2211yescharacteristic
29C23×D127—21no—
30C23×D127D411yescharacteristic
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254D412yescomplemented
37C2×D508—24no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508—21no—
42C2×D508—21no—
43C2×D508—42no—
44C2×D508—42no—
45C2×D508D412yes—
46C22×Dic127—21no—
47C22×Dic127D411yescharacteristic
48C2×C4×D127—21no—
49C2×C4×D127—21no—
50C2×C4×D127—21no—
51C2×C4×D127—21no—
52C2×C4×D127—21no—
53C2×C4×D127—21no—
54#14523237—28no—
55C2×C127⋊C8—21no—
56C2×C127⋊C8—21no—
57C2×C127⋊C8—21no—
58C4⋊Dic127—24no—
59(C2×D127)⋊C4—24no—
60(C2×D127)⋊C4—24no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64C2×C8×C127—21no—
65C2×C8×C127—21no—
66C2×C8×C127D412yescomplemented
67C4⋊C4×C127—21no—
68C4⋊C4×C127—21no—
69C4⋊C4×C127D412yes—
70C4⋊C4×C127D412yescomplemented
71C22×D127—21no—
72C22×D127—21no—
73C22×D127—21no—
74C22×D127—21no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—22no—
78C22×D127—41no—
79C22×D127—41no—
80C22×D127—41no—
81C22×D127—41no—
82C22×D127—41no—
83C22×D127—41no—
84C22×D127SD1612yescomplemented
85Dic254—22no—
86Dic254—22no—
87Dic254—41no—
88Dic254—41no—
89D508—22no—
90D508—22no—
91D508—48no—
92D508—41no—
93D508—41no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—41no—
99C2×Dic127—41no—
100C4×D127—41no—
101C4×D127—41no—
102C4×D127—41no—
103C4×D127—41no—
104C127⋊C8—41no—
105C23×C127C2×D411yescharacteristic
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4111yescharacteristic
118C8×C127—22no—
119C22×C127#12011yescharacteristic
120C22×C127#12011yescharacteristic
121C22×C127#12111yescharacteristic
122C22×C127C2×SD1611yescharacteristic
123C22×C127#13511yescharacteristic
124C22×C127AGL(1,ℤ/8)11yescharacteristic
125C22×C127#13711yescharacteristic
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—41no—
130C2×D127—41no—
131C2×D127—41no—
132C2×D127—41no—
133C2×D127—42no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—81no—
137C2×D127—81no—
138C2×D127—81no—
139C2×D127—81no—
140C4×C127—42no—
141C4×C127—42no—
142C4×C127#12011yescharacteristic
143C4×C127#12111yescharacteristic
144C4×C127#12111yescharacteristic
145C4×C127#12711yescharacteristic
146Dic127—41no—
147Dic127—42no—
148C2×C127#39211yescharacteristic
149C2×C127#44811yescharacteristic
150C2×C127#44911yescharacteristic
151C2×C127#45011yescharacteristic
152C2×C127#45211yescharacteristic
153C2×C127#46511yescharacteristic
154C2×C127#46811yescharacteristic
155D127—42no—
156D127—82no—
157#2000—1271nomaximal, Sylow
158C127#200011yescharacteristic, complemented, Sylow
159C22×SD16—1271no—
160C2×C22⋊Q8—1271no—
161C2×#121—1271no—
162C2×#103—1271no—
163C2×#102—1271no—
164C2×C22⋊C8—1271no—
165#340D12711yescharacteristic, complemented
166C22×Q8—1271no—
167C22×D4—1271no—
168C22×D4—2541no—
169C23×C4—1271no—
170C2×SD16—2542no—
171C2×SD16—2542no—
172C2×SD16—2542no—
173C22×C8—1271no—
174C22×C8C2×D12711yescharacteristic
175C22⋊Q8—2542no—
176C22⋊Q8—2542no—
177#121—2542no—
178#121—2542no—
179C2×C4⋊C4—2541no—
180C2×C4⋊C4C2×D12711yescharacteristic
181C2×C4⋊C4C2×D12711yescharacteristic
182C2×C22⋊C4—2541no—
183C2×C22⋊C4—2541no—
184#103—2542no—
185#102—2542no—
186C22⋊C8—2542no—
187C24—1271no—
188C24—2541no—
189C2×Q8—1272no—
190C2×Q8—2541no—
191C2×Q8—2541no—
192C2×Q8—2541no—
193C2×Q8—2541no—
194C2×Q8—2541no—
195C2×D4—1272no—
196C2×D4—5082no—
197C2×D4—5082no—
198C2×D4—2544no—
199C2×D4—2541no—
200C2×D4—2541no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4—2541no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4C22×D12711yescharacteristic
215SD16—2548no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—2541no—
219C2×C8—2541no—
220C2×C8—2541no—
221C2×C8D50812yescomplemented
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—2544no—
225C4⋊C4D50812yes—
226C4⋊C4D50812yescomplemented
227C22⋊C4—2544no—
228C22⋊C4—2544no—
229C23—1272no—
230C23—5081no—
231C23—5081no—
232C23—5081no—
233C23—5081no—
234C23—5081no—
235C23—5081no—
236C23—2542no—
237C23—2541no—
238C23—2541no—
239C23—2541no—
240C23—2541no—
241C23—2541no—
242C23—2541no—
243C23D4×D12711yescharacteristic
244Q8—5081no—
245Q8—5081no—
246Q8—2542no—
247Q8—2542no—
248D4—5088no—
249D4—5081no—
250D4—5081no—
251D4—2542no—
252D4—2542no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—5081no—
260C2×C4—5081no—
261C2×C4—5081no—
262C2×C4—5081no—
263C2×C4—5081no—
264C2×C4—5081no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4—2542no—
268C2×C4—2541no—
269C2×C4D4×D12711yescharacteristic
270C2×C4D4×D12711yescharacteristic
271C2×C4C2×D50811yescharacteristic
272C2×C4C2×D50811yescharacteristic
273C2×C4C2×D50811yescharacteristic
274C2×C4#1452324911yescharacteristic
275C8—22no—
276C8—5081no—
277C22—10161no—
278C22—10161no—
279C22—10161no—
280C22—10161no—
281C22—5082no—
282C22—5082no—
283C22—5082no—
284C22—5081no—
285C22—5081no—
286C22—5081no—
287C22—5081no—
288C22—2542no—
289C22—2542no—
290C22—2542no—
291C22#1686117911yescharacteristic
292C22#1686117911yescharacteristic
293C22#1686118411yescharacteristic
294C22SD16×D12711yescharacteristic
295C22#1686123011yescharacteristic
296C22#1686123311yescharacteristic
297C22#1686124111yescharacteristic
298C4—42no—
299C4—42no—
300C4—5081no—
301C4—5082no—
302C4#1686117911yescharacteristic
303C4#1686118411yescharacteristic
304C4#1686118411yescharacteristic
305C4#1686120611yescharacteristic
306C2—10162no—
307C2—5082no—
308C2#2254978611yescharacteristic
309C2#2255001611yescharacteristic
310C2#2255002211yescharacteristic
311C2#2255002811yescharacteristic
312C2#2255004111yescharacteristic
313C2#2255010611yescharacteristic
314C2#2255012311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861184C2
hypercenter#340D127
hypocenterC127#2000
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C22×#14523237C2semidirect
2. C2×#16861189C2semidirect
4. C2×#16861132C2semidirect
5. C2×#16861129C2semidirect
6. C2×(C2×D127)⋊C8C2semidirect
7. #340×C127C2semidirect
36. C2×Dic254D4semidirect
66. C2×C8×C127D4semidirect
70. C4⋊C4×C127D4semidirect
84. C22×D127SD16semidirect
158. C127#2000semidirect
165. #340D127semidirect
221. C2×C8D508semidirect
226. C4⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
C22×#145232372
C2×#168611892
C2×#168611842
C2×#168611322
C2×#168611292
C2×(C2×D127)⋊C82
#340×C1272
#2000127