Finite Group Databasebeta

← #30648986

Subgroups — #30648986

Order 16256.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550356C211yesmaximal, characteristic, complemented
2#22550209C211yesmaximal, characteristic, complemented
3#22550187C211yesmaximal, characteristic, complemented
4#22550168C211yesmaximal, characteristic, complemented
5#22550153C211yesmaximal, characteristic, complemented
6#22550095C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550036C211yesmaximal, characteristic, complemented
9#22550019C211yesmaximal, characteristic
10#22549976C211yesmaximal, characteristic, complemented
11#22549974C211yesmaximal, characteristic, complemented
12C4×#14523238C211yesmaximal, characteristic, complemented
13C4×#14523233C211yesmaximal, characteristic, complemented
14#22549937C211yesmaximal, characteristic, complemented
15#542×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523238C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21C4×Dic254C2211yescharacteristic
22C4×Dic254C2211yescharacteristic
23C4×D508C2211yescharacteristic
24C4×D508C2211yescharacteristic
25(C22⋊D127)⋊C4—21no—
26(C22⋊D127)⋊C4—21no—
27#16861167—21no—
28C2×C4⋊Dic127C2211yescharacteristic
29C4⋊C4×D127—21no—
30#16861142C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861142C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35C4⋊(C127⋊C8)C2211yescharacteristic
36#16861133C2211yescharacteristic
37#16861133C2211yescharacteristic
38#16861132C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861130C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44C22⋊(C127⋊C8)C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46#16861120C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48C4⋊Q8×C127C2211yescharacteristic
49#126×C127—21no—
50#124×C127—21no—
51#124×C127C2211yescharacteristic
52#123×C127—21no—
53C22⋊Q8×C127—21no—
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56C4×Q8×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58#117×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61C2×Dic254C2311yescharacteristic
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C2×C4×D127—21no—
66C2×C4×D127—21no—
67C2×C4×D127—21no—
68#14523238—22no—
69#14523236—22no—
70#14523235—22no—
71#14523233—22no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83C4×Dic127—21no—
84C4×Dic127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127C2311yescharacteristic
87C2×D4×C127—21no—
88C2×D4×C127C2311yescharacteristic
89C22×C4×C127—21no—
90C22×C4×C127C2311yescharacteristic
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127D411yescharacteristic
95C4⋊C4×C127D411yescharacteristic
96C4⋊C4×C127C2311yescharacteristic
97C4⋊C4×C127C2311yescharacteristic
98C22⋊C4×C127—21no—
99C22⋊C4×C127—21no—
100C22⋊C4×C127—21no—
101C22⋊C4×C127—21no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C42×C127—21no—
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109Dic254—21no—
110Dic254#4112yes—
111D508—21no—
112D508#4112yes—
113C22⋊D127—24no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—81no—
118C4×D127—24no—
119C4×D127—81no—
120C4×D127—81no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127C2×D411yescharacteristic
127Q8×C127—21no—
128Q8×C127—21no—
129Q8×C127—81no—
130D4×C127—21no—
131D4×C127—21no—
132D4×C127—81no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—81no—
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—22no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—21no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127C2×#4111yescharacteristic
164C4×C127C2×#4111yescharacteristic
165Dic127—22no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—41no—
169C2×C127—81no—
170C2×C127#52511yescharacteristic
171C2×C127#57611yescharacteristic
172C2×C127#57811yescharacteristic
173D127—42no—
174#3124—1271nomaximal, Sylow
175C127#312411yescharacteristic, complemented, Sylow
176#542D12711yescharacteristic, complemented
177#520—1271no—
178#484—1271no—
179#480—1271no—
180#477—1271no—
181#474—1271no—
182#464—1271no—
183#463—1271no—
184#452—1271no—
185#449—1271no—
186#440—1271no—
187#440—1271no—
188C4×Q16—1271no—
189C4×D8—1271no—
190#433—1271no—
191C2×#41—1271no—
192C2×Q16—1271no—
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×D8—1271no—
196C4⋊Q8C2×D12711yescharacteristic
197#126—21no—
198#124—21no—
199#124C2×D12711yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C2×D12711yescharacteristic
203#121C2×D12711yescharacteristic
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8C2×D12711yescharacteristic
207C4×D4—1271no—
208C4×D4—1271no—
209C4×D4—2541no—
210C4×D4—2541no—
211C4×D4C2×D12711yescharacteristic
212#117—2541no—
213#117C2×D12711yescharacteristic
214C2×C4⋊C4—1271no—
215C2×C4⋊C4—2541no—
216#107—1271no—
217#107—1271no—
218#107—1271no—
219#107—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230C22⋊C8—1271no—
231C22⋊C8—1271no—
232#97—1271no—
233C4×C8—1271no—
234#41—2542no—
235#41—2542no—
236C2×Q8—21no—
237C2×Q8—1271no—
238C2×Q8C22×D12711yescharacteristic
239C2×D4—21no—
240C2×D4—1271no—
241C2×D4—2541no—
242C2×D4C22×D12711yescharacteristic
243C22×C4—21no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4C22×D12711yescharacteristic
249Q16—2542no—
250SD16—2542no—
251SD16—2542no—
252D8—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8—1271no—
256C2×C8—1271no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4C22⋊D12711yescharacteristic
266C4⋊C4C22⋊D12711yescharacteristic
267C4⋊C4C22×D12711yescharacteristic
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4C22⋊D12711yescharacteristic
277C22⋊C4C22⋊D12711yescharacteristic
278C42—21no—
279C42—2541no—
280C42—2541no—
281C42C22×D12711yescharacteristic
282C42C22×D12711yescharacteristic
283C23—21no—
284C23—2541no—
285C23C2×C22⋊D12711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—81no—
289Q8—1272no—
290Q8—2541no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—1272no—
295D4—2544no—
296D4—2541no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—10161no—
304C2×C4—10161no—
305C2×C4—10161no—
306C2×C4—2544no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4C2×C22⋊D12711yescharacteristic
311C2×C4C2×C22⋊D12711yescharacteristic
312C2×C4C2×C22⋊D12711yescharacteristic
313C2×C4C2×C22⋊D12711yescharacteristic
314C2×C4C2×C22⋊D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—2541no—
317C8—2541no—
318C8—2541no—
319C8—2541no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5081no—
327C22—5081no—
328C22—2542no—
329C22C22×C22⋊D12711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—5081no—
339C4—5081no—
340C4—2542no—
341C4#41×D12711yescharacteristic
342C4#41×D12711yescharacteristic
343C2—41no—
344C2—81no—
345C2—5082no—
346C2#2255038311yescharacteristic
347C2#2255067911yescharacteristic
348C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#542×C127C2
socleC22×C127C22×D4
prime radical#22550019C2
hypercenter#542D127
hypocenterC127#3124
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #22550356C2semidirect
2. #22550209C2semidirect
3. #22550187C2semidirect
4. #22550168C2semidirect
5. #22550153C2semidirect
6. #22550095C2semidirect
7. #22550088C2semidirect
8. #22550036C2semidirect
10. #22549976C2semidirect
11. #22549974C2semidirect
12. C4×#14523238C2semidirect
13. C4×#14523233C2semidirect
14. #22549937C2semidirect
15. #542×C127C2semidirect
175. C127#3124semidirect
176. #542D127semidirect

Maximal subgroups

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

subgroupindex
#225503562
#225502092
#225501872
#225501682
#225501532
#225500952
#225500882
#225500362
#225500192
#225499762
#225499742
C4×#145232382
C4×#145232332
#225499372
#542×C1272
#3124127