Finite Group Databasebeta

← #30650588

Subgroups — #30650588

Order 16256.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D127C211yesmaximal, characteristic, complemented
2#22550526C211yesmaximal, characteristic
3#22550225C211yesmaximal, characteristic, complemented
4#22550207C211yesmaximal, characteristic, complemented
5#22550205C211yesmaximal, characteristic, complemented
6#22550159C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550013C211yesmaximal, characteristic, complemented
11SD16×Dic127C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16861133C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#448×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×Q8×D127C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C22×C127⋊C8C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861207—21no—
23#16861207C2211yescharacteristic
24#16861191—21no—
25#16861190—21no—
26#16861188—21no—
27#16861188C2211yescharacteristic
28#16861186—21no—
29#16861185—21no—
30#16861185C2211yescharacteristic
31#16861180C2211yescharacteristic
32Q8×Dic127C2211yescharacteristic
33D4×Dic127C2211yescharacteristic
34C4×C22⋊D127—21no—
35C4×C22⋊D127C2211yescharacteristic
36C2×C4⋊Dic127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861141C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861130C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C22×Q8×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—41no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127D411yescharacteristic
67C22×Dic127—21no—
68C22×Dic127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127C2311yescharacteristic
72#14523236—41no—
73C2×C127⋊C8—41no—
74C2×C127⋊C8C2311yescharacteristic
75C2×C127⋊C8C2311yescharacteristic
76C8×D127—41no—
77C4⋊Dic127—21no—
78C4⋊Dic127—21no—
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127D411yescharacteristic
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127—21no—
98C2×Q8×C127—22no—
99C2×Q8×C127—41no—
100C2×Q8×C127C2311yescharacteristic
101C2×Q8×C127C2311yescharacteristic
102C2×D4×C127—21no—
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106SD16×C127—22no—
107SD16×C127—41no—
108C2×C8×C127C2311yescharacteristic
109C2×C8×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127C2×D411yescharacteristic
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254—42no—
117C22⋊D127—42no—
118C22⋊D127SD1614yescomplemented
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127—42no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—21no—
129C4×D127—21no—
130C4×D127—42no—
131C4×D127—42no—
132C127⋊C8—21no—
133C127⋊C8—21no—
134C23×C127—21no—
135C23×C127C2×D411yescharacteristic
136Q8×C127—21no—
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—42no—
140Q8×C127#4112yescomplemented
141D4×C127—21no—
142D4×C127—21no—
143D4×C127—42no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—42no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C8×C127—21no—
153C8×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127C2×SD1612yes—
160C22×C127C22×D411yescharacteristic
161C2×D127—21no—
162C2×D127—21no—
163C2×D127C2×SD1612yes—
164C4×C127—21no—
165C4×C127—22no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127C2×#4111yescharacteristic
169C4×C127H(5,𝔽2)11yescharacteristic
170Dic127—21no—
171Dic127—21no—
172Dic127—41no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176Dic127C2×SD1612yes—
177C2×C127—22no—
178C2×C127—81no—
179C2×C127#54611yescharacteristic
180C2×C127C22×SD1611yescharacteristic
181C2×C127#57811yescharacteristic
182D127—22no—
183#3298—1271nomaximal, Sylow
184C127#329811yescharacteristic, complemented, Sylow
185D4×Q8—1271no—
186#547—1271no—
187#487—1271no—
188#483—1271no—
189#483—1271no—
190#475—1271no—
191#465—1271no—
192#465—1271no—
193#461—1271no—
194#448—1271no—
195#448D12711yescharacteristic, complemented
196C4×SD16—1271no—
197C8×D4—1271no—
198C2×#103—1271no—
199C2×#103—1271no—
200C2×#41—2541no—
201C22×Q8—1271no—
202C22×Q8C2×D12711yescharacteristic
203C2×SD16—1271no—
204C2×SD16C2×D12711yescharacteristic
205C2×SD16C2×D12711yescharacteristic
206C22×C8—1271no—
207C22×C8—1271no—
208C4⋊Q8—1271no—
209C4⋊Q8—1271no—
210C4⋊Q8—2541no—
211#123—2541no—
212#123—2541no—
213C22⋊Q8—1271no—
214C22⋊Q8—1271no—
215C22⋊Q8—2541no—
216C22⋊Q8—2541no—
217C22⋊Q8—2541no—
218#121—1271no—
219#121C2×D12711yescharacteristic
220C4×Q8—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223C4×D4—2541no—
224C2×C4⋊C4—1271no—
225C2×C4⋊C4—1271no—
226#106—1271no—
227#106—1271no—
228#106—1271no—
229C4⋊C8—1271no—
230#103—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#103C2×D12711yescharacteristic
238#103C2×D12711yescharacteristic
239#102—1271no—
240C22⋊C8—1271no—
241C22⋊C8C2×D12711yescharacteristic
242C4×C8—1271no—
243#41—5084no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—1271no—
248C2×Q8—5081no—
249C2×Q8—2541no—
250C2×Q8—2541no—
251C2×Q8—2542no—
252C2×Q8C22×D12711yescharacteristic
253C2×Q8C22×D12711yescharacteristic
254C2×D4—21no—
255C2×D4—1271no—
256C2×D4—2541no—
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264C22×C4C22×D12711yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—5081no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—5081no—
271C2×C8—5081no—
272C2×C8C22×D12711yescharacteristic
273C2×C8C22×D12711yescharacteristic
274C4⋊C4—1271no—
275C4⋊C4—1271no—
276C4⋊C4—1271no—
277C4⋊C4—1271no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4—2541no—
283C4⋊C4—2541no—
284C4⋊C4—2541no—
285C4⋊C4—2541no—
286C4⋊C4C22×D12711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—1271no—
289C22⋊C4—1271no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—1271no—
295C42—2541no—
296C23—21no—
297C23—1271no—
298C23D4×D12711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—5082no—
304Q8—5082no—
305Q8—2541no—
306Q8—2541no—
307Q8#1452324712yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—1274no—
312D4—5082no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—1271no—
322C2×C4—5082no—
323C2×C4—5082no—
324C2×C4—5082no—
325C2×C4—5082no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4D4×D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2541no—
337C8—2541no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1272no—
344C22—2541no—
345C22—2541no—
346C22SD16×D12712yes—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—1272no—
353C4—5081no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4C2×#1452324711yescharacteristic
360C4#1686125411yescharacteristic
361C2—22no—
362C2—81no—
363C2—2542no—
364C2#2255051311yescharacteristic
365C2C2×SD16×D12711yescharacteristic
366C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#22550526C2
hypercenter#448D127
hypocenterC127#3298
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. Q8×C22⋊D127C2semidirect
3. #22550225C2semidirect
4. #22550207C2semidirect
5. #22550205C2semidirect
6. #22550159C2semidirect
7. #22550103C2semidirect
8. #22550102C2semidirect
9. #22550079C2semidirect
10. #22550013C2semidirect
11. SD16×Dic127C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. C2×#16861133C2semidirect
14. #103×D127C2semidirect
15. #448×C127C2semidirect
118. C22⋊D127SD16semidirect
140. Q8×C127#41semidirect
184. C127#3298semidirect
195. #448D127semidirect
307. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1272
#225505262
#225502252
#225502072
#225502052
#225501592
#225501032
#225501022
#225500792
#225500132
SD16×Dic1272
(C22⋊D127)⋊C82
C2×#168611332
#103×D1272
#448×C1272
#3298127