Finite Group Databasebeta

← #30648694

Subgroups — #30648694

Order 16256.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550449C211yesmaximal, characteristic
2#22550356C211yesmaximal, characteristic
3#22550219C211yesmaximal, characteristic, complemented
4#22550217C211yesmaximal, characteristic, complemented
5#22550216C211yesmaximal, characteristic, complemented
6#22550212C211yesmaximal, characteristic, complemented
7#22550210C211yesmaximal, characteristic, complemented
8#22550208C211yesmaximal, characteristic, complemented
9#22550204C211yesmaximal, characteristic, complemented
10#22549915C211yesmaximal, characteristic, complemented
11#22549913C211yesmaximal, characteristic, complemented
12#22549883C211yesmaximal, characteristic, complemented
13#103×D127C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic
15#485×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861208—21no—
21#16861195—21no—
22#16861194—21no—
23#16861191—21no—
24#16861189C2211yescharacteristic
25#16861188C2211yescharacteristic
26#16861185C2211yescharacteristic
27C4×Dic254—21no—
28C4×D508—21no—
29(C22⋊D127)⋊C4—21no—
30(C22⋊D127)⋊C4—21no—
31#16861167C2211yescharacteristic
32#16861167C2211yescharacteristic
33#16861167C2211yescharacteristic
34C2×C4⋊Dic127C2211yescharacteristic
35C4⋊C4×D127C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861143C2211yescharacteristic
39#16861142C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861140C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861132C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C22⋊Q8×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#106×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61C2×Dic254—21no—
62C2×Dic254D411yescharacteristic
63C2×Dic254C2311yescharacteristic
64Q8×D127—22no—
65Q8×D127—22no—
66C2×D508D411yescharacteristic
67C2×C22⋊D127D411yescharacteristic
68C2×C22⋊D127D411yescharacteristic
69C22×Dic127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74C2×C4×D127C2311yescharacteristic
75#14523231—22no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—22no—
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C2×Q8×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127—21no—
108C22×D127C2×D411yescharacteristic
109C22×D127C2×D411yescharacteristic
110Dic254—22no—
111Dic254—21no—
112Dic254—21no—
113Dic254—42no—
114D508—22no—
115C22⋊D127—22no—
116C22⋊D127—22no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—81no—
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C4×D127—21no—
125C4×D127—21no—
126C4×D127—42no—
127C4×D127—81no—
128C4×D127—81no—
129C4×D127#4112yes—
130C4×D127#4112yes—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2411yescharacteristic
144C8×C127—21no—
145C8×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127C22×D411yescharacteristic
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—21no—
155C2×D127#12312yes—
156C4×C127—21no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127C2×#4111yescharacteristic
162C4×C127C2×#4111yescharacteristic
163Dic127—21no—
164Dic127—41no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168Dic127#12312yes—
169C2×C127—41no—
170C2×C127C2×#12311yescharacteristic
171C2×C127#57711yescharacteristic
172C2×C127#57811yescharacteristic
173D127—22no—
174D127—41no—
175#3097—1271nomaximal, Sylow
176C127#309711yescharacteristic, complemented, Sylow
177#536—1271no—
178#520—1271no—
179#485—1271no—
180#485—1271no—
181#485—1271no—
182#485D12711yescharacteristic, complemented
183#484—1271no—
184#484—1271no—
185#483—1271no—
186#483—1271no—
187#428—1271no—
188#427—1271no—
189#419—1271no—
190C2×#103—1271no—
191#408—1271no—
192C2×#41—1271no—
193C22×Q8—1271no—
194C2×M16—1271no—
195C22×C8—1271no—
196C4⋊Q8—2541no—
197#124—2541no—
198#123—2541no—
199#123—2541no—
200C22⋊Q8—1271no—
201C22⋊Q8—1271no—
202C22⋊Q8—1271no—
203C22⋊Q8C2×D12711yescharacteristic
204C4×Q8—2541no—
205C4×D4—2541no—
206C4×D4—2541no—
207C4×D4—2541no—
208#117—1271no—
209#117—1271no—
210#117—1271no—
211#117C2×D12711yescharacteristic
212C2×C4⋊C4—1271no—
213C2×C4⋊C4—1271no—
214C2×C4⋊C4—1271no—
215#107—1271no—
216#107—1271no—
217#107—1271no—
218#107C2×D12711yescharacteristic
219#106—1271no—
220#106—1271no—
221#106—1271no—
222#106C2×D12711yescharacteristic
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103C2×D12711yescharacteristic
230#103C2×D12711yescharacteristic
231C22⋊C8—1271no—
232C22⋊C8—1271no—
233C22⋊C8—1271no—
234C22⋊C8C2×D12711yescharacteristic
235#41—2542no—
236#41—2542no—
237C2×Q8—1271no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×Q8—2542no—
241C2×Q8—2542no—
242C2×Q8C22×D12711yescharacteristic
243C2×D4—1271no—
244C2×D4—1271no—
245C2×D4—1271no—
246C22×C4—1271no—
247C22×C4—1271no—
248C22×C4—2541no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4C22×D12711yescharacteristic
253M16—2542no—
254C2×C8—1271no—
255C2×C8—1271no—
256C2×C8—2542no—
257C2×C8C22×D12711yescharacteristic
258C2×C8C22×D12711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—1271no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4C22×D12711yescharacteristic
270C4⋊C4C22×D12711yescharacteristic
271C4⋊C4C22×D12711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C42—21no—
281C42—2541no—
282C42—2541no—
283C42—2541no—
284C23—1271no—
285C23—1271no—
286C23D4×D12711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—5082no—
290Q8—2542no—
291Q8—2541no—
292Q8—2541no—
293D4—2542no—
294D4—2542no—
295D4—2542no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1271no—
303C2×C4—1272no—
304C2×C4—1272no—
305C2×C4—1271no—
306C2×C4—10161no—
307C2×C4—10161no—
308C2×C4—10161no—
309C2×C4—5082no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4D4×D12711yescharacteristic
317C2×C4C23×D12711yescharacteristic
318C8—21no—
319C8—21no—
320C8—2541no—
321C8—2541no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—1272no—
326C22—2541no—
327C22—2541no—
328C22—2541no—
329C22—2541no—
330C22—2541no—
331C22C2×D4×D12711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—1272no—
338C4—5081no—
339C4—5081no—
340C4—5081no—
341C4—5081no—
342C4—2541no—
343C4#41×D12711yescharacteristic
344C4#41×D12711yescharacteristic
345C2—41no—
346C2—5081no—
347C2—2542no—
348C2#123×D12711yescharacteristic
349C2#2255068511yescharacteristic
350C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#485×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#485D127
hypocenterC127#3097
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #22550219C2semidirect
4. #22550217C2semidirect
5. #22550216C2semidirect
6. #22550212C2semidirect
7. #22550210C2semidirect
8. #22550208C2semidirect
9. #22550204C2semidirect
10. #22549915C2semidirect
11. #22549913C2semidirect
12. #22549883C2semidirect
13. #103×D127C2semidirect
15. #485×C127C2semidirect
176. C127#3097semidirect
182. #485D127semidirect

Maximal subgroups

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

subgroupindex
#225504492
#225503562
#225502192
#225502172
#225502162
#225502122
#225502102
#225502082
#225502042
#225499152
#225499132
#225498832
#103×D1272
#225498412
#485×C1272
#3097127