Finite Group Databasebeta

← #30639137

Subgroups — #30639137

Order 16256.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861179C211yesmaximal, characteristic
2#22550349C211yesmaximal, characteristic, complemented
3C2×#16861123C211yesmaximal, characteristic, complemented
4#22549731C211yesmaximal, characteristic, complemented
5#22549731C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C22×D508—21no—
10C22×C22⋊D127C2211yescharacteristic
11#16861179—22no—
12#16861179C414yescomplemented
13C4×C22⋊D127—22no—
14(C22⋊D127)⋊C4—22no—
15#16861165—21no—
16C2×C22⋊Dic127C2211yescharacteristic
17C2×C22⋊Dic127C2211yescharacteristic
18C2×(C2×D127)⋊C4—21no—
19C2×(C2×D127)⋊C4—22no—
20C2×(C2×D127)⋊C4—22no—
21C22⋊C4×D127—21no—
22#16861123—22no—
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
26C23×D127—21no—
27C23×D127—21no—
28C23×D127—21no—
29C23×D127—21no—
30C23×D127—21no—
31C23×D127—21no—
32C23×D127—44no—
33C23×D127C2×C412yescomplemented
34C23×D127D411yescharacteristic
35C2×D508—24no—
36C2×D508—44no—
37C2×C22⋊D127—21no—
38C2×C22⋊D127—21no—
39C2×C22⋊D127—21no—
40C2×C22⋊D127—21no—
41C2×C22⋊D127C2×C412yes—
42C2×C22⋊D127D412yes—
43C22×Dic127—21no—
44C22×Dic127D411yescharacteristic
45C22×Dic127D412yescomplemented
46C2×C4×D127—22no—
47Dic127⋊C4—22no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127—21no—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—24no—
54(C2×D127)⋊C4—44no—
55(C2×D127)⋊C4—44no—
56C4×Dic127—22no—
57C24×C127C2311yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127D412yescomplemented
61C22⋊C4×C127—21no—
62C22⋊C4×C127—21no—
63C22⋊C4×C127—21no—
64C22⋊C4×C127—21no—
65C22⋊C4×C127—21no—
66C22⋊C4×C127C2×C412yescomplemented
67C22×D127—24no—
68C22×D127—24no—
69C22×D127—24no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—22no—
73C22×D127—21no—
74C22×D127—21no—
75C22×D127—44no—
76C22×D127—44no—
77C22×D127—44no—
78C22×D127—44no—
79C22×D127—42no—
80C22×D127—42no—
81C22×D127—42no—
82C22×D127—41no—
83C22×D127—41no—
84C22×D127—41no—
85C22×D127—41no—
86C22×D127—42no—
87C22×D127C22⋊C412yescomplemented
88D508—48no—
89C22⋊D127—22no—
90C22⋊D127—22no—
91C22⋊D127—22no—
92C22⋊D127—22no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127—42no—
98C2×Dic127—41no—
99C2×Dic127C22⋊C412yescomplemented
100C4×D127—82no—
101C23×C127—42no—
102C23×C127C22×C411yescharacteristic
103C23×C127C2×D411yescharacteristic
104C23×C127C2×D411yescharacteristic
105C23×C127C2×D411yescharacteristic
106C23×C127C2×D411yescharacteristic
107C23×C127#4111yescharacteristic
108C23×C127#4111yescharacteristic
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C22×C127—22no—
116C22×C127—22no—
117C22×C127—22no—
118C22×C127—41no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127C2×C22⋊C411yescharacteristic
123C22×C127C4×D411yescharacteristic
124C22×C127C4×D411yescharacteristic
125C22×C127#12011yescharacteristic
126C22×C127#12111yescharacteristic
127C22×C127#12111yescharacteristic
128C22×C127#12311yescharacteristic
129C2×D127—24no—
130C2×D127—24no—
131C2×D127—24no—
132C2×D127—22no—
133C2×D127—44no—
134C2×D127—42no—
135C2×D127—44no—
136C2×D127—42no—
137C2×D127—42no—
138C2×D127—44no—
139C2×D127—42no—
140C2×D127—42no—
141C2×D127—44no—
142C2×D127—42no—
143C2×D127—81no—
144C4×C127—42no—
145C4×C127—42no—
146C4×C127—81no—
147Dic127—22no—
148Dic127—42no—
149Dic127—81no—
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—21no—
153C2×C127—21no—
154C2×C127—21no—
155C2×C127—21no—
156C2×C127#38611yescharacteristic
157C2×C127B(4,𝔽2)12yes—
158D127—44no—
159D127—42no—
160#1889—1271nomaximal, Sylow
161C127#188911yescharacteristic, complemented, Sylow
162C2×#120—1271no—
163#518—1271no—
164C2×#99—1271no—
165#379—1271no—
166#379—1271no—
167#342—1271no—
168#342D12711yescharacteristic, complemented
169C25—1271no—
170C22×D4—1271no—
171C22×D4—2541no—
172#120—1274no—
173#120—2542no—
174C4×D4—2542no—
175C4×D4—2542no—
176#117—2541no—
177C2×C22⋊C4—1271no—
178C2×C22⋊C4—1271no—
179C2×C22⋊C4—2541no—
180C2×C22⋊C4—2542no—
181C2×C22⋊C4—2542no—
182C2×C22⋊C4—2541no—
183C2×C22⋊C4C2×D12711yescharacteristic
184C2×C22⋊C4C2×D12711yescharacteristic
185C2×C22⋊C4C2×D12711yescharacteristic
186#99—2542no—
187C24—1271no—
188C24—1272no—
189C24—5084no—
190C24—2541no—
191C24—2541no—
192C24—2541no—
193C24—2541no—
194C24—2541no—
195C24—2541no—
196C24C22×D12711yescharacteristic
197C2×D4—1272no—
198C2×D4—1272no—
199C2×D4—5084no—
200C2×D4—2544no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C2×D4—2541no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—1271no—
208C22×C4—1272no—
209C22×C4—2542no—
210C22×C4—2541no—
211C22×C4D50812yescomplemented
212C4⋊C4—2542no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—5084no—
219C22⋊C4—5084no—
220C22⋊C4—2544no—
221C22⋊C4—2542no—
222C22⋊C4—2541no—
223C22⋊C4—2541no—
224C22⋊C4—2541no—
225C22⋊C4—2541no—
226C22⋊C4C4×D12712yescomplemented
227C42—2542no—
228C23—42no—
229C23—1272no—
230C23—5082no—
231C23—5082no—
232C23—5082no—
233C23—5082no—
234C23—5084no—
235C23—5084no—
236C23—5084no—
237C23—5084no—
238C23—5081no—
239C23—5081no—
240C23—5081no—
241C23—5081no—
242C23—2544no—
243C23—2544no—
244C23—2544no—
245C23—2541no—
246C23—2542no—
247C23—2542no—
248C23—2542no—
249C23—2541no—
250C23C2×C4×D12711yescharacteristic
251C23D4×D12711yescharacteristic
252C23D4×D12711yescharacteristic
253C23D4×D12711yescharacteristic
254C23C2×D50811yescharacteristic
255C23#1452324811yescharacteristic
256C23#1452324911yescharacteristic
257D4—5088no—
258D4—2542no—
259D4—2542no—
260D4—2542no—
261D4—2542no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—1272no—
269C2×C4—10162no—
270C2×C4—5082no—
271C2×C4—5081no—
272C2×C4—5082no—
273C2×C4—2542no—
274C2×C4—2541no—
275C2×C4—2541no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—10161no—
284C22—5082no—
285C22—5082no—
286C22—5084no—
287C22—5084no—
288C22—5084no—
289C22—5082no—
290C22—5082no—
291C22—5082no—
292C22—5084no—
293C22—5082no—
294C22—2544no—
295C22—2544no—
296C22—2544no—
297C22—2542no—
298C22C22⋊C4×D12711yescharacteristic
299C22#1686117111yescharacteristic
300C22C4×D50811yescharacteristic
301C22#1686117911yescharacteristic
302C22#1686118211yescharacteristic
303C22#1686118411yescharacteristic
304C22#1686119411yescharacteristic
305C4—42no—
306C4—42no—
307C4—81no—
308C4—10161no—
309C4—5082no—
310C4—2542no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—5082no—
318C2—5084no—
319C2#2254975411yescharacteristic
320C2#2255006412yes—
subgroupG/H
centerC22#16861179
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
prime radicalC2×#16861179C2
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1889
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #22550349C2semidirect
3. C2×#16861123C2semidirect
4. #22549731C2semidirect
5. #22549731C2semidirect
6. #22549642C2semidirect
7. #342×C127C2semidirect
12. #16861179C4semidirect
33. C23×D127C2×C4semidirect
45. C22×Dic127D4semidirect
60. C22×C4×C127D4semidirect
66. C22⋊C4×C127C2×C4semidirect
87. C22×D127C22⋊C4semidirect
99. C2×Dic127C22⋊C4semidirect
161. C127#1889semidirect
168. #342D127semidirect
211. C22×C4D508semidirect
226. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611792
#225503492
C2×#168611232
#225497312
#225496422
#342×C1272
#1889127