Finite Group Databasebeta

← #30650611

Subgroups — #30650611

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#22550535C211yesmaximal, characteristic
2#22550233C211yesmaximal, characteristic, complemented
3#22550217C211yesmaximal, characteristic, complemented
4#22550189C211yesmaximal, characteristic, complemented
5#22550180C211yesmaximal, characteristic, complemented
6#22550115C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550036C211yesmaximal, characteristic, complemented
10#22550031C211yesmaximal, characteristic, complemented
11C4×#14523238C211yesmaximal, characteristic, complemented
12D4×C127⋊C8C211yesmaximal, characteristic, complemented
13#22549870C211yesmaximal, characteristic, complemented
14C2×#16861132C211yesmaximal, characteristic, complemented
15#552×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×#14523238C2211yescharacteristic
19C2×#14523238C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861197C2211yescharacteristic
24#16861195—21no—
25#16861188C2211yescharacteristic
26#16861187—21no—
27#16861183C2211yescharacteristic
28#16861181—21no—
29C4×Dic254C2211yescharacteristic
30(C22⋊D127)⋊C4—21no—
31#16861142C2211yescharacteristic
32#16861142C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35#16861133C2211yescharacteristic
36#16861132C2211yescharacteristic
37#16861132C2211yescharacteristic
38#16861132C2211yescharacteristic
39#16861132C2211yescharacteristic
40#16861132C2211yescharacteristic
41#16861132C2211yescharacteristic
42#16861129C2211yescharacteristic
43#16861129C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47C4×C127⋊C8C2211yescharacteristic
48#126×C127—21no—
49#125×C127—21no—
50#125×C127C2211yescharacteristic
51#123×C127—21no—
52C22⋊Q8×C127C2211yescharacteristic
53C22⋊Q8×C127C2211yescharacteristic
54C4×Q8×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56#117×C127—21no—
57#117×C127C2211yescharacteristic
58C2×C4⋊C4×C127C2211yescharacteristic
59#14523248—22no—
60#14523248—41no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—41no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C22×Dic127—21no—
70C2×C4×D127—21no—
71#14523238—22no—
72#14523238—41no—
73#14523235—41no—
74C2×C127⋊C8—41no—
75C2×C127⋊C8—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83C22⋊Dic127—21no—
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86(C2×D127)⋊C4—21no—
87C4×Dic127—21no—
88C2×Q8×C127C2311yescharacteristic
89C2×D4×C127D411yescharacteristic
90C22×C4×C127—21no—
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127—21no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127D411yescharacteristic
100C4⋊C4×C127C2311yescharacteristic
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C22⋊C4×C127—21no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127D411yescharacteristic
107C22⋊C4×C127D411yescharacteristic
108C42×C127—21no—
109C42×C127—21no—
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254#4112yes—
117D508—21no—
118D508—21no—
119C22⋊D127—42no—
120C22⋊D127—42no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—42no—
125C4×D127—42no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127C2×D411yescharacteristic
131C23×C127C2×D411yescharacteristic
132Q8×C127—21no—
133Q8×C127—21no—
134D4×C127—22no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—42no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
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×C127—41no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127C2×#4111yescharacteristic
168C4×C127H(5,𝔽2)11yescharacteristic
169Dic127—22no—
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#552D12711yescharacteristic, complemented
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×D4—1271no—
193#416—1271no—
194C2×#103—1271no—
195C2×#41—1271no—
196C22×Q8—1271no—
197C2×Q16—1271no—
198C2×Q16—1271no—
199C2×SD16—1271no—
200C22×C8—1271no—
201C22×C8—1271no—
202#126—21no—
203#125—21no—
204#125C2×D12711yescharacteristic
205#124—1271no—
206#124—2541no—
207#123—21no—
208C22⋊Q8—1271no—
209C22⋊Q8—2541no—
210C22⋊Q8C2×D12711yescharacteristic
211C22⋊Q8C2×D12711yescharacteristic
212#121—1271no—
213#121—2541no—
214C4×Q8—1271no—
215C4×Q8C2×D12711yescharacteristic
216C4×D4—2541no—
217C4×D4C2×D12711yescharacteristic
218#117—21no—
219#117C2×D12711yescharacteristic
220C2×C4⋊C4C2×D12711yescharacteristic
221#107—1271no—
222#107—1271no—
223#106—1271no—
224C4⋊C8—1271no—
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⋊C8—1271no—
236C22⋊C8—1271no—
237C4×C8—1271no—
238#41—5081no—
239#41—2542no—
240C2×Q8—1271no—
241C2×Q8—1271no—
242C2×Q8—5081no—
243C2×Q8—2542no—
244C2×Q8—2541no—
245C2×Q8C22×D12711yescharacteristic
246C2×D4—1271no—
247C2×D4—2541no—
248C2×D4—2541no—
249C2×D4C22⋊D12711yescharacteristic
250C22×C4—21no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4C22×D12711yescharacteristic
254C22×C4C22×D12711yescharacteristic
255Q16—5081no—
256Q16—2542no—
257SD16—5081no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8—1271no—
261C2×C8—1271no—
262C2×C8—5081no—
263C2×C8—5081no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—1271no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22⋊D12711yescharacteristic
274C4⋊C4C22×D12711yescharacteristic
275C4⋊C4C22×D12711yescharacteristic
276C4⋊C4C22×D12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C22⋊C4C22⋊D12711yescharacteristic
285C22⋊C4C22⋊D12711yescharacteristic
286C42—21no—
287C42—21no—
288C42—2541no—
289C42C22×D12711yescharacteristic
290C23—2541no—
291C23C2×C22⋊D12711yescharacteristic
292C23C2×C22⋊D12711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1272no—
296Q8—5082no—
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—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—5082no—
315C2×C4—5082no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4C2×C22⋊D12711yescharacteristic
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4C2×C22⋊D12711yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C8—2541no—
325C8—2541no—
326C8—2541no—
327C8—2541no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—5081no—
334C22—5081no—
335C22—5081no—
336C22#1686122912yes—
337C22C22×C22⋊D12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5081no—
347C4—5081no—
348C4—2542no—
349C4#41×D12711yescharacteristic
350C4#1686125511yescharacteristic
351C2—22no—
352C2—41no—
353C2—10161no—
354C2#2255051411yescharacteristic
355C2C2×#1686122911yescharacteristic
356C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#552×C127C2
socleC22×C127C22×D4
prime radical#22550535C2
hypercenter#552D127
hypocenterC127#3300
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #22550233C2semidirect
3. #22550217C2semidirect
4. #22550189C2semidirect
5. #22550180C2semidirect
6. #22550115C2semidirect
7. #22550107C2semidirect
8. #22550098C2semidirect
9. #22550036C2semidirect
10. #22550031C2semidirect
11. C4×#14523238C2semidirect
12. D4×C127⋊C8C2semidirect
13. #22549870C2semidirect
14. C2×#16861132C2semidirect
15. #552×C127C2semidirect
179. C127#3300semidirect
180. #552D127semidirect

Maximal subgroups

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

subgroupindex
#225505352
#225502332
#225502172
#225501892
#225501802
#225501152
#225501072
#225500982
#225500362
#225500312
C4×#145232382
D4×C127⋊C82
#225498702
C2×#168611322
#552×C1272
#3300127