Finite Group Databasebeta

← #30650591

Subgroups — #30650591

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#22550527C211yesmaximal, characteristic
3#22550225C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550205C211yesmaximal, characteristic, complemented
6#22550159C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550080C211yesmaximal, characteristic, complemented
10#22550016C211yesmaximal, characteristic, complemented
11#22550015C211yesmaximal, characteristic, complemented
12#22549961C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15C2×#103×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—21no—
17C2×Q8×D127C2211yescharacteristic
18C2×#14523237C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861207—21no—
24#16861207C2211yescharacteristic
25#16861194—21no—
26#16861192—21no—
27#16861188—21no—
28#16861188C2211yescharacteristic
29#16861187—21no—
30#16861185—21no—
31#16861185C2211yescharacteristic
32#16861184C2211yescharacteristic
33#16861183C2211yescharacteristic
34Q8×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic
36C4×C22⋊D127—21no—
37C4×C22⋊D127C2211yescharacteristic
38C4⋊C4×D127C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41#16861140C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861132C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861131C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C22×Q8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#103×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#14523248—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63Q8×D127—22no—
64Q8×D127—41no—
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127D411yescharacteristic
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73#14523237—41no—
74#14523235—22no—
75#14523235—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—41no—
79C4⋊Dic127—21no—
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⋊Dic127D411yescharacteristic
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C2×Q8×C127—21no—
99C2×Q8×C127—22no—
100C2×Q8×C127—41no—
101C2×Q8×C127C2311yescharacteristic
102C2×Q8×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106C2×C8×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C4⋊C4×C127C2311yescharacteristic
112C22×D127—21no—
113C22×D127C2×D411yescharacteristic
114Dic254—21no—
115Dic254—21no—
116Dic254—42no—
117Dic254—42no—
118D508—21no—
119D508—21no—
120D508—42no—
121C22⋊D127—42no—
122C22⋊D127SD1614yescomplemented
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—42no—
132C4×D127—42no—
133C4×D127—42no—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127C2×D411yescharacteristic
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—21no—
140Q8×C127—42no—
141Q8×C127#4112yescomplemented
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C8×C127—21no—
153C8×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127C2×SD1612yes—
157C22×C127C22×D411yescharacteristic
158C2×D127—21no—
159C2×D127—21no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127C2×SD1612yes—
164C4×C127—21no—
165C4×C127—22no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127H(5,𝔽2)11yescharacteristic
171Dic127—21no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127—41no—
176Dic127C2×SD1612yes—
177C2×C127—22no—
178C2×C127#54611yescharacteristic
179C2×C127C22×SD1611yescharacteristic
180C2×C127#57811yescharacteristic
181D127—22no—
182D127—81no—
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#448—1271no—
196C4×SD16—1271no—
197C8×D4—1271no—
198C2×#103—1271no—
199C2×#103D12711yescharacteristic, complemented
200C2×#41—2541no—
201C22×Q8—1271no—
202C22×Q8C2×D12711yescharacteristic
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×SD16—1271no—
206C22×C8—1271no—
207C22×C8C2×D12711yescharacteristic
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#121—1271no—
220C4×Q8—1271no—
221C4×D4—1271no—
222C4×D4—1271no—
223C4×D4—2541no—
224C2×C4⋊C4—1271no—
225C2×C4⋊C4C2×D12711yescharacteristic
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#103C2×D12711yescharacteristic
236#103C2×D12711yescharacteristic
237#103C2×D12711yescharacteristic
238#103C2×D12711yescharacteristic
239#102—1271no—
240C22⋊C8—1271no—
241C22⋊C8—1271no—
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—1271no—
255C2×D4—1271no—
256C2×D4—2541no—
257C2×D4—2541no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264C22×C4C22×D12711yescharacteristic
265SD16—5081no—
266SD16—5081no—
267SD16—2542no—
268C2×C8—41no—
269C2×C8—1271no—
270C2×C8—1271no—
271C2×C8—5081no—
272C2×C8C22×D12711yescharacteristic
273C2×C8C22×D12711yescharacteristic
274C4⋊C4—21no—
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⋊C4C22×D12711yescharacteristic
286C4⋊C4C22×D12711yescharacteristic
287C22⋊C4—1271no—
288C22⋊C4—1271no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—1271no—
295C42—2541no—
296C23—1271no—
297C23—2541no—
298C23D4×D12711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—5082no—
304Q8—5082no—
305Q8—2541no—
306Q8—2541no—
307Q8#1452324812yescomplemented
308D4—1274no—
309D4—5082no—
310D4—5082no—
311D4—2541no—
312D4—2541no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—5082no—
325C2×C4—5082no—
326C2×C4—5082no—
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—1272no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—2541no—
345C22—2541no—
346C22SD16×D12712yes—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1272no—
354C4—5081no—
355C4—5081no—
356C4—5081no—
357C4—2541no—
358C4—2541no—
359C4C2×#1452324811yescharacteristic
360C4#1686125411yescharacteristic
361C2—22no—
362C2—10161no—
363C2—2542no—
364C2#2255051611yescharacteristic
365C2C2×SD16×D12711yescharacteristic
366C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
prime radical#22550527C2
hypercenterC2×#103D127
hypocenterC127#3298
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D127C2semidirect
3. #22550225C2semidirect
4. #22550206C2semidirect
5. #22550205C2semidirect
6. #22550159C2semidirect
7. #22550108C2semidirect
8. #22550105C2semidirect
9. #22550080C2semidirect
10. #22550016C2semidirect
11. #22550015C2semidirect
12. #22549961C2semidirect
13. C8×C22⋊D127C2semidirect
14. #103×D127C2semidirect
15. C2×#103×C127C2semidirect
122. C22⋊D127SD16semidirect
141. Q8×C127#41semidirect
184. C127#3298semidirect
199. C2×#103D127semidirect
307. Q8#14523248semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1272
#225505272
#225502252
#225502062
#225502052
#225501592
#225501082
#225501052
#225500802
#225500162
#225500152
#225499612
C8×C22⋊D1272
#103×D1272
C2×#103×C1272
#3298127