Finite Group Databasebeta

← #30650619

Subgroups — #30650619

Order 16256.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550566C211yesmaximal, characteristic, complemented
2#22550535C211yesmaximal, characteristic
3#22550231C211yesmaximal, characteristic, complemented
4#22550218C211yesmaximal, characteristic, complemented
5#22550190C211yesmaximal, characteristic, complemented
6#22550184C211yesmaximal, characteristic, complemented
7#22550117C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550099C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12C4×Dic508C211yesmaximal, characteristic, complemented
13#22549873C211yesmaximal, characteristic, complemented
14C2×#16861134C211yesmaximal, characteristic, complemented
15C8×D4×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×Dic508C2211yescharacteristic
19C2×Dic508C2211yescharacteristic
20C2×#14523237C2211yescharacteristic
21#16861202—21no—
22#16861201C2211yescharacteristic
23#16861199—21no—
24#16861197C2211yescharacteristic
25#16861195—21no—
26#16861190—21no—
27#16861188C2211yescharacteristic
28#16861188C2211yescharacteristic
29#16861188C2211yescharacteristic
30#16861187—21no—
31#16861183C2211yescharacteristic
32#16861181—21no—
33C4×Dic254C2211yescharacteristic
34C4×Dic254C2211yescharacteristic
35(C22⋊D127)⋊C4—21no—
36#16861168C2211yescharacteristic
37#16861165—21no—
38C2×C4⋊Dic127C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861143C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic
52C22×C8×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C4×D4×C127C2211yescharacteristic
55C4⋊C8×C127C2211yescharacteristic
56C22⋊C8×C127C2211yescharacteristic
57C22⋊C8×C127C2211yescharacteristic
58C4×C8×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—41no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—41no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73Dic508—22no—
74Dic508—41no—
75#14523237—41no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95C4×Dic127—21no—
96C4×Dic127—21no—
97C4×Dic127—21no—
98C2×D4×C127D411yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C2×C8×C127—41no—
102C2×C8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127D411yescharacteristic
108C22⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116Dic254—21no—
117Dic254—42no—
118Dic254#4112yes—
119D508—21no—
120D508—21no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—42no—
131C2×Dic127—42no—
132C4×D127—42no—
133C23×C127C2×D411yescharacteristic
134C23×C127C2×D411yescharacteristic
135D4×C127—22no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2411yescharacteristic
145C8×C127—21no—
146C8×C127—21no—
147C8×C127—21no—
148C8×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127#13512yes—
155C22×C127C22×D411yescharacteristic
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127C2×#4111yescharacteristic
164C4×C127H(5,𝔽2)11yescharacteristic
165Dic127—22no—
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172C2×C127—22no—
173C2×C127—41no—
174C2×C127#54611yescharacteristic
175C2×C127C2×#13511yescharacteristic
176C2×C127#57811yescharacteristic
177D127—81no—
178#3300—1271nomaximal, Sylow
179C127#330011yescharacteristic, complemented, Sylow
180#552—1271no—
181#548—1271no—
182#488—1271no—
183#485—1271no—
184#480—1271no—
185#479—1271no—
186#467—1271no—
187#465—1271no—
188#464—1271no—
189#452—1271no—
190#451—1271no—
191C4×Q16—1271no—
192C8×D4D12711yescharacteristic, complemented
193#416—1271no—
194C2×#103—1271no—
195C2×#41—1271no—
196C22×Q8—1271no—
197C2×Q16—1271no—
198C2×Q16—1271no—
199C2×SD16—1271no—
200C22×C8C2×D12711yescharacteristic
201C22×C8C2×D12711yescharacteristic
202#126—2541no—
203#125—1271no—
204#125—2541no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208C22⋊Q8—1271no—
209C22⋊Q8—1271no—
210C22⋊Q8—1271no—
211C22⋊Q8—2541no—
212#121—1271no—
213#121—2541no—
214C4×Q8—1271no—
215C4×Q8—1271no—
216C4×D4—2541no—
217C4×D4C2×D12711yescharacteristic
218#117—1271no—
219#117—2541no—
220C2×C4⋊C4—1271no—
221#107—1271no—
222#107—1271no—
223#106—1271no—
224C4⋊C8C2×D12711yescharacteristic
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103—1271no—
231#103—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235C22⋊C8C2×D12711yescharacteristic
236C22⋊C8C2×D12711yescharacteristic
237C4×C8C2×D12711yescharacteristic
238#41—5081no—
239#41—2542no—
240C2×Q8—1271no—
241C2×Q8—1271no—
242C2×Q8—1271no—
243C2×Q8—5081no—
244C2×Q8—2542no—
245C2×Q8—2541no—
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4D50811yescharacteristic
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4C22×D12711yescharacteristic
254C22×C4C22×D12711yescharacteristic
255Q16—5081no—
256Q16—2542no—
257SD16—5081no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C2×C8C22×D12711yescharacteristic
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4D50811yescharacteristic
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4D50811yescharacteristic
285C22⋊C4D50811yescharacteristic
286C42—2541no—
287C42—2541no—
288C42—2541no—
289C42C22×D12711yescharacteristic
290C23—2541no—
291C23C2×D50811yescharacteristic
292C23C2×D50811yescharacteristic
293Q8—1272no—
294Q8—5082no—
295Q8—2541no—
296Q8—2541no—
297Q8—2541no—
298Q8—2541no—
299Q8—2541no—
300D4—22no—
301D4—5082no—
302D4—5082no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—5082no—
310C2×C4—5082no—
311C2×C4—5082no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4C2×D50811yescharacteristic
320C2×C4C2×D50811yescharacteristic
321C2×C4C2×D50811yescharacteristic
322C2×C4C2×D50811yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—5081no—
334C22—5081no—
335C22—5081no—
336C22#1686123112yes—
337C22C22×D50811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—5081no—
343C4—5081no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—2542no—
349C4#41×D12711yescharacteristic
350C4#1686125411yescharacteristic
351C2—22no—
352C2—41no—
353C2—10161no—
354C2#2255052111yescharacteristic
355C2C2×#1686123111yescharacteristic
356C2#2255069311yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC8×D4×C127C2
socleC22×C127C22×D4
prime radical#22550535C2
hypercenterC8×D4D127
hypocenterC127#3300
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22550566C2semidirect
3. #22550231C2semidirect
4. #22550218C2semidirect
5. #22550190C2semidirect
6. #22550184C2semidirect
7. #22550117C2semidirect
8. #22550105C2semidirect
9. #22550099C2semidirect
10. #22550040C2semidirect
11. #22550034C2semidirect
12. C4×Dic508C2semidirect
13. #22549873C2semidirect
14. C2×#16861134C2semidirect
15. C8×D4×C127C2semidirect
179. C127#3300semidirect
192. C8×D4D127semidirect

Maximal subgroups

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

subgroupindex
#225505662
#225505352
#225502312
#225502182
#225501902
#225501842
#225501172
#225501052
#225500992
#225500402
#225500342
C4×Dic5082
#225498732
C2×#168611342
C8×D4×C1272
#3300127