Finite Group Databasebeta

← #30648841

Subgroups — #30648841

Order 16256.

392 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#22550410C211yesmaximal, characteristic, complemented
2C4×#14523248C211yesmaximal, characteristic, complemented
3#22550218C211yesmaximal, characteristic, complemented
4#22550201C211yesmaximal, characteristic, complemented
5#22550184C211yesmaximal, characteristic, complemented
6#22550177C211yesmaximal, characteristic, complemented
7#22550099C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550040C211yesmaximal, characteristic, complemented
10#22550023C211yesmaximal, characteristic
11#22549989C211yesmaximal, characteristic, complemented
12#22549984C211yesmaximal, characteristic, complemented
13#22549978C211yesmaximal, characteristic, complemented
14#22549978C211yesmaximal, characteristic, complemented
15#432×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×Dic508C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×D1016C2211yescharacteristic
21#16861203—21no—
22#16861201C2211yescharacteristic
23#16861197C2211yescharacteristic
24#16861192—21no—
25#16861188C2211yescharacteristic
26#16861183C2211yescharacteristic
27C4×Dic254C2211yescharacteristic
28C4×Dic254C2211yescharacteristic
29C4×Dic254C2211yescharacteristic
30C4×D508C2211yescharacteristic
31C4×D508C2211yescharacteristic
32C4×D508C2211yescharacteristic
33C4×C22⋊D127—21no—
34C4×C22⋊D127—21no—
35C4×C22⋊D127—21no—
36#16861168C2211yescharacteristic
37#16861166—21no—
38#16861166—21no—
39C42×D127—21no—
40#16861143C2211yescharacteristic
41#16861143C2211yescharacteristic
42#16861141C2211yescharacteristic
43#16861141C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic
52C2×C42×C127C2211yescharacteristic
53C4⋊C8×C127C2211yescharacteristic
54C4⋊C8×C127C2211yescharacteristic
55C22⋊C8×C127C2211yescharacteristic
56C22⋊C8×C127C2211yescharacteristic
57#97×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—22no—
61C2×Dic254C2311yescharacteristic
62C2×Dic254C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71Dic508—22no—
72#14523237—22no—
73#14523237—22no—
74D1016—22no—
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93C4×Dic127—21no—
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C22×C4×C127D411yescharacteristic
98C22×C4×C127D411yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C42×C127D411yescharacteristic
105C42×C127D411yescharacteristic
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110Dic254—21no—
111Dic254—21no—
112Dic254—21no—
113Dic254#4112yes—
114D508—21no—
115D508—21no—
116D508—21no—
117D508#4112yes—
118C22⋊D127—24no—
119C22⋊D127—42no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C4×D127—24no—
127C4×D127—42no—
128C4×D127—42no—
129C4×D127—42no—
130C23×C127C2×D411yescharacteristic
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127C2×D411yescharacteristic
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2411yescharacteristic
141C8×C127—21no—
142C8×C127—21no—
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127C22×D411yescharacteristic
149C2×D127—22no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—21no—
160C4×C127C2×#4111yescharacteristic
161C4×C127C2×#4111yescharacteristic
162Dic127—22no—
163Dic127—41no—
164Dic127—41no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—41no—
169C2×C127#52511yescharacteristic
170C2×C127#57511yescharacteristic
171C2×C127#57511yescharacteristic
172D127—42no—
173D127—81no—
174#3111—1271nomaximal, Sylow
175C127#311111yescharacteristic, complemented, Sylow
176#529—1271no—
177C4×#41—1271no—
178#485—1271no—
179#482—1271no—
180#479—1271no—
181#478—1271no—
182#464—1271no—
183#463—1271no—
184#452—1271no—
185#449—1271no—
186#442—1271no—
187#441—1271no—
188#440—1271no—
189#440—1271no—
190#432D12711yescharacteristic, complemented
191C2×#41—1271no—
192C2×Q16—1271no—
193C2×SD16—1271no—
194C2×SD16—1271no—
195C2×D8—1271no—
196#126—2541no—
197#125—1271no—
198#124—1271no—
199#123—2541no—
200C22⋊Q8—1271no—
201#121—1271no—
202C4×Q8—1271no—
203C4×Q8—1271no—
204C4×Q8—1271no—
205C4×D4—1271no—
206C4×D4—1271no—
207C4×D4—1271no—
208C4×D4—2541no—
209C4×D4—2541no—
210C4×D4—2541no—
211#117—1271no—
212#117—2541no—
213#117—2541no—
214C2×C42—2541no—
215C2×C42C2×D12711yescharacteristic
216#107—1271no—
217#107—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8C2×D12711yescharacteristic
221C4⋊C8C2×D12711yescharacteristic
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⋊C8C2×D12711yescharacteristic
231C22⋊C8C2×D12711yescharacteristic
232#97C2×D12711yescharacteristic
233#97C2×D12711yescharacteristic
234#41—2542no—
235#41—2542no—
236C2×Q8—1271no—
237C2×Q8—1271no—
238C2×D4—1271no—
239C2×D4—1271no—
240C2×D4—2541no—
241C2×D4—2541no—
242C22×C4—2541no—
243C22×C4—2541no—
244C22×C4—2541no—
245C22×C4—2541no—
246C22×C4D50811yescharacteristic
247C22×C4D50811yescharacteristic
248C22×C4C22×D12711yescharacteristic
249Q16—2542no—
250SD16—2542no—
251SD16—2542no—
252D8—2542no—
253C2×C8C22×D12711yescharacteristic
254C2×C8C22×D12711yescharacteristic
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C22⋊C4—2541no—
268C22⋊C4—2541no—
269C22⋊C4—2541no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C42—2541no—
276C42—2541no—
277C42—2541no—
278C42—2541no—
279C42D50811yescharacteristic
280C42D50811yescharacteristic
281C42C22×D12711yescharacteristic
282C42C22×D12711yescharacteristic
283C23—2541no—
284C23—2541no—
285C23C2×D50811yescharacteristic
286Q8—1272no—
287Q8—2541no—
288Q8—2541no—
289Q8—2541no—
290D4—1272no—
291D4—5082no—
292D4—2544no—
293D4—2541no—
294D4—2541no—
295D4—2541no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—5082no—
301C2×C4—5082no—
302C2×C4—5082no—
303C2×C4—2544no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4C2×D50811yescharacteristic
311C2×C4C2×D50811yescharacteristic
312C2×C4C2×D50811yescharacteristic
313C2×C4C2×D50811yescharacteristic
314C2×C4C2×D50811yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—5081no—
324C22—5081no—
325C22—5081no—
326C22—5081no—
327C22—5081no—
328C22—2542no—
329C22C22×D50811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—5081no—
336C4—5081no—
337C4—5081no—
338C4—5081no—
339C4—5081no—
340C4—2542no—
341C4#41×D12711yescharacteristic
342C4#41×D12711yescharacteristic
343C2—41no—
344C2—10161no—
345C2—5082no—
346C2#2255038811yescharacteristic
347C2#2255067611yescharacteristic
348C2#2255067611yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#432×C127C2
socleC22×C127C22×D4
prime radical#22550023C2
hypercenter#432D127
hypocenterC127#3111
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #22550410C2semidirect
2. C4×#14523248C2semidirect
3. #22550218C2semidirect
4. #22550201C2semidirect
5. #22550184C2semidirect
6. #22550177C2semidirect
7. #22550099C2semidirect
8. #22550092C2semidirect
9. #22550040C2semidirect
11. #22549989C2semidirect
12. #22549984C2semidirect
13. #22549978C2semidirect
14. #22549978C2semidirect
15. #432×C127C2semidirect
175. C127#3111semidirect
190. #432D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
C4×#145232482
#225502182
#225502012
#225501842
#225501772
#225500992
#225500922
#225500402
#225500232
#225499892
#225499842
#225499782
#432×C1272
#3111127