Finite Group Databasebeta

← #30647961

Subgroups — #30647961

Order 16256.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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×Q8×D127C211yesmaximal, characteristic, complemented
2C2×#16861238C212yesmaximal, complemented
3#22550386C211yesmaximal, characteristic
4#22550034C211yesmaximal, characteristic, complemented
5#22550033C212yesmaximal, complemented
6#22550032C211yesmaximal, characteristic, complemented
7#22550014C211yesmaximal, characteristic, complemented
8#22550013C212yesmaximal, complemented
9#22549883C212yesmaximal, complemented
10#22549838C211yesmaximal, characteristic, complemented
11#448×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2211yescharacteristic
13C22×Dic254—21no—
14C22×Dic254C2211yescharacteristic
15C2×Q8×D127—28no—
16C2×Q8×D127—21no—
17C2×Q8×D127—21no—
18C2×Q8×D127C2212yes—
19C2×Q8×D127C2212yes—
20C22×C4×D127—21no—
21C22×C4×D127C2211yescharacteristic
22#16861238—28no—
23C2×Dic508C2212yescomplemented
24C2×#14523238C2212yescomplemented
25C2×#14523236C2212yescomplemented
26C2×#14523231C2212yes—
27#16861191—21no—
28#16861189C2211yescharacteristic
29#16861188C2211yescharacteristic
30#16861180C2211yescharacteristic
31#16861177—21no—
32D4×Dic127—21no—
33(C22⋊D127)⋊C4—21no—
34#16861167C2211yescharacteristic
35#16861134C2212yescomplemented
36#16861133C2212yescomplemented
37#16861132C2212yescomplemented
38C22⋊(C127⋊C8)C2211yescharacteristic
39(C2×D127)⋊C8C2212yes—
40C22×Q8×C127C2211yescharacteristic
41C2×SD16×C127C2212yescomplemented
42#121×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44C22⋊C8×C127C2211yescharacteristic
45C23×D127D411yescharacteristic
46#14523247—24no—
47C2×Dic254—28no—
48C2×Dic254—24no—
49C2×Dic254—22no—
50C2×Dic254—22no—
51C2×Dic254—21no—
52C2×Dic254C2311yescharacteristic
53C2×Dic254C2312yes—
54Q8×D127—28no—
55Q8×D127—28no—
56Q8×D127—24no—
57Q8×D127—24no—
58Q8×D127—24no—
59Q8×D127D418yescomplemented
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C22×Dic127—21no—
63C22×Dic127—21no—
64C22×Dic127D411yescharacteristic
65C2×C4×D127—28no—
66C2×C4×D127—22no—
67C2×C4×D127—22no—
68C2×C4×D127—24no—
69C2×C4×D127D411yescharacteristic
70C2×C4×D127D411yescharacteristic
71C2×C4×D127C2311yescharacteristic
72C2×C4×D127C2311yescharacteristic
73Dic508—24no—
74#14523238—24no—
75#14523236—24no—
76#14523231—24no—
77C2×C127⋊C8C2312yes—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127—21no—
83(C2×D127)⋊C4—21no—
84(C2×D127)⋊C4—21no—
85C4×Dic127—21no—
86C2×Q8×C127—24no—
87C2×Q8×C127—21no—
88C2×Q8×C127C2312yes—
89C2×D4×C127—21no—
90C2×D4×C127C2311yescharacteristic
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93SD16×C127—24no—
94C2×C8×C127C2312yes—
95C4⋊C4×C127C2311yescharacteristic
96C22⋊C4×C127—21no—
97C22×D127—22no—
98C22×D127—22no—
99C22×D127—22no—
100C22×D127C2×D411yescharacteristic
101C22×D127C2×D411yescharacteristic
102Dic254—24no—
103Dic254—28no—
104Dic254—22no—
105Dic254—28no—
106Dic254—22no—
107Dic254C2×D414yes—
108C22⋊D127—42no—
109C22⋊D127—42no—
110C2×Dic127—24no—
111C2×Dic127—22no—
112C2×Dic127—22no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—42no—
116C2×Dic127C2×D411yescharacteristic
117C2×Dic127C2×D411yescharacteristic
118C4×D127—28no—
119C4×D127—28no—
120C4×D127—22no—
121C4×D127—22no—
122C4×D127—22no—
123C4×D127C2×D414yes—
124C127⋊C8—22no—
125C23×C127—21no—
126C23×C127C2×D411yescharacteristic
127Q8×C127—24no—
128Q8×C127—22no—
129Q8×C127C2×D414yes—
130D4×C127—22no—
131D4×C127—42no—
132C2×C4×C127—24no—
133C2×C4×C127—22no—
134C2×C4×C127—22no—
135C2×C4×C127—21no—
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2411yescharacteristic
138C8×C127—22no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127#13712yescomplemented
145C22×C127C22×D411yescharacteristic
146C2×D127—28no—
147C2×D127—21no—
148C2×D127—21no—
149C2×D127—21no—
150C2×D127—21no—
151C2×D127#12012yes—
152C2×D127#13712yescomplemented
153C4×C127—24no—
154C4×C127—21no—
155C4×C127—41no—
156C4×C127C22×D412yes—
157Dic127—24no—
158Dic127—21no—
159Dic127—41no—
160Dic127—41no—
161Dic127#12012yes—
162C2×C127—22no—
163C2×C127—81no—
164C2×C127C2×#12011yescharacteristic
165C2×C127C2×#13711yescharacteristic
166C2×C127C2×#13711yescharacteristic
167D127—22no—
168D127—22no—
169#3004—1271nomaximal, Sylow
170C127#300411yescharacteristic, complemented, Sylow
171C23×Q8—1271no—
172C2×#137—1272no—
173#525—1271no—
174#451—1271no—
175#451—1272no—
176#451—1271no—
177#448—1272no—
178#448—1271no—
179#448D12711yescharacteristic, complemented
180#419—1272no—
181#407—1271no—
182C2×#41—1271no—
183C22×Q8—1272no—
184C22×Q8—1272no—
185C22×Q8—1271no—
186C22×Q8—2541no—
187C22×Q8—2541no—
188C22×Q8—2541no—
189C22×Q8—2548no—
190C22×Q8C2×D12711yescharacteristic
191C23×C4—1271no—
192C23×C4—2541no—
193#137—2548no—
194C2×Q16—1272no—
195C2×Q16—1272no—
196C2×SD16—1272no—
197C2×SD16C2×D12712yescomplemented
198C2×M16—1272no—
199#123—2541no—
200C22⋊Q8—1271no—
201C22⋊Q8—1271no—
202#121—1271no—
203#121C2×D12711yescharacteristic
204#120—2541no—
205C4×D4—2541no—
206C4×D4—2541no—
207#117—1271no—
208#103—1272no—
209#103—1272no—
210#103—1272no—
211#103C2×D12712yescomplemented
212C22⋊C8—1271no—
213C22⋊C8—1272no—
214C22⋊C8C2×D12711yescharacteristic
215C24—1271no—
216#41—2544no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—1278no—
220C2×Q8—1272no—
221C2×Q8—1271no—
222C2×Q8—2542no—
223C2×Q8—2542no—
224C2×Q8—2544no—
225C2×Q8—2548no—
226C2×Q8—2544no—
227C2×Q8—2544no—
228C2×Q8—2548no—
229C2×Q8—2548no—
230C2×Q8—2544no—
231C2×Q8—2541no—
232C2×Q8C22×D12712yes—
233C2×D4—21no—
234C2×D4—2541no—
235C2×D4—2541no—
236C2×D4C22×D12711yescharacteristic
237C22×C4—21no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—2544no—
244C22×C4—2542no—
245C22×C4—2542no—
246C22×C4—2548no—
247C22×C4—2541no—
248C22×C4—2541no—
249C22×C4C22×D12711yescharacteristic
250Q16—2544no—
251Q16—2544no—
252SD16—24no—
253SD16—2544no—
254M16—2544no—
255C2×C8—1272no—
256C2×C8C22×D12712yes—
257C4⋊C4—1271no—
258C4⋊C4—2541no—
259C4⋊C4—2541no—
260C4⋊C4C22×D12711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2541no—
263C22⋊C4—2541no—
264C22⋊C4—2541no—
265C22⋊C4—2541no—
266C42—2541no—
267C23—21no—
268C23—1271no—
269C23—1271no—
270C23—2542no—
271C23—2542no—
272C23—2542no—
273C23D4×D12711yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8—1274no—
277Q8—2548no—
278Q8—2548no—
279Q8—2544no—
280Q8—2542no—
281Q8—2542no—
282Q8D4×D12714yes—
283D4—22no—
284D4—42no—
285D4—5082no—
286D4—5082no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—1271no—
292C2×C4—1274no—
293C2×C4—1271no—
294C2×C4—5082no—
295C2×C4—2542no—
296C2×C4—2542no—
297C2×C4—2548no—
298C2×C4—2548no—
299C2×C4—2542no—
300C2×C4—2541no—
301C2×C4—2541no—
302C2×C4—2544no—
303C2×C4—2542no—
304C2×C4—2542no—
305C2×C4D4×D12711yescharacteristic
306C2×C4C23×D12711yescharacteristic
307C8—22no—
308C8—2542no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—1272no—
315C22—1272no—
316C22—2548no—
317C22—2541no—
318C22—2541no—
319C22—2541no—
320C22—2541no—
321C22#1686123812yescomplemented
322C22C2×D4×D12711yescharacteristic
323C4—24no—
324C4—21no—
325C4—41no—
326C4—1272no—
327C4—5081no—
328C4—5081no—
329C4—2541no—
330C4—2544no—
331C4C2×D4×D12712yes—
332C2—22no—
333C2—81no—
334C2—2542no—
335C2—2542no—
336C2#120×D12711yescharacteristic
337C2#137×D12711yescharacteristic
338C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#22550386C2
hypercenter#448D127
hypocenterC127#3004
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. C2×#16861238C2semidirect
4. #22550034C2semidirect
5. #22550033C2semidirect
6. #22550032C2semidirect
7. #22550014C2semidirect
8. #22550013C2semidirect
9. #22549883C2semidirect
10. #22549838C2semidirect
11. #448×C127C2semidirect
23. C2×Dic508C22semidirect
24. C2×#14523238C22semidirect
25. C2×#14523236C22semidirect
35. #16861134C22semidirect
36. #16861133C22semidirect
37. #16861132C22semidirect
41. C2×SD16×C127C22semidirect
43. #103×C127C22semidirect
59. Q8×D127D4semidirect
144. C22×C127#137semidirect
152. C2×D127#137semidirect
170. C127#3004semidirect
179. #448D127semidirect
197. C2×SD16C2×D127semidirect
211. #103C2×D127semidirect
321. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C2×#168612382
#225503862
#225500342
#225500332
#225500322
#225500142
#225500132
#225498832
#225498382
#448×C1272
#3004127