Finite Group Databasebeta

← #30650706

Subgroups — #30650706

Order 16256.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550572C211yesmaximal, characteristic, complemented
2#22550244C211yesmaximal, characteristic, complemented
3#22550214C211yesmaximal, characteristic, complemented
4#22550204C211yesmaximal, characteristic, complemented
5#22550176C211yesmaximal, characteristic, complemented
6#22550124C211yesmaximal, characteristic, complemented
7#22550107C211yesmaximal, characteristic, complemented
8#22550078C211yesmaximal, characteristic, complemented
9#22550021C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22549986C211yesmaximal, characteristic, complemented
12#22549948C211yesmaximal, characteristic, complemented
13#22549877C211yesmaximal, characteristic, complemented
14#22549872C211yesmaximal, characteristic, complemented
15#547×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20C22×C127⋊C8C2211yescharacteristic
21#16861204—21no—
22#16861201C2211yescharacteristic
23#16861191—21no—
24#16861183C2211yescharacteristic
25#16861183C2211yescharacteristic
26#16861181—21no—
27C4×D508C2211yescharacteristic
28(C22⋊D127)⋊C4—21no—
29#16861168C2211yescharacteristic
30C2×C4⋊Dic127C2211yescharacteristic
31#16861142C2211yescharacteristic
32#16861140C2211yescharacteristic
33#16861140C2211yescharacteristic
34C4⋊(C127⋊C8)C2211yescharacteristic
35#16861133C2211yescharacteristic
36#16861133C2211yescharacteristic
37#16861133C2211yescharacteristic
38#16861133C2211yescharacteristic
39#16861133C2211yescharacteristic
40#16861133C2211yescharacteristic
41#16861130C2211yescharacteristic
42#16861130C2211yescharacteristic
43#16861130C2211yescharacteristic
44#16861129C2211yescharacteristic
45C22⋊(C127⋊C8)C2211yescharacteristic
46C22⋊(C127⋊C8)C2211yescharacteristic
47#16861120C2211yescharacteristic
48C2×#41×C127C2211yescharacteristic
49C2×#41×C127C2211yescharacteristic
50C4⋊Q8×C127C2211yescharacteristic
51#123×C127—21no—
52#123×C127—21no—
53C22⋊Q8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127—21no—
56C4×D4×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58C2×C4⋊C4×C127—21no—
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C22×Dic127—21no—
63C2×C4×D127—21no—
64#14523235—41no—
65#14523235—41no—
66#14523233—22no—
67#14523232—41no—
68C2×C127⋊C8—41no—
69C2×C127⋊C8C2311yescharacteristic
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C2×C127⋊C8C2311yescharacteristic
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79C22⋊Dic127—21no—
80(C2×D127)⋊C4—21no—
81(C2×D127)⋊C4—21no—
82(C2×D127)⋊C4—21no—
83C4×Dic127—21no—
84#41×C127—22no—
85#41×C127—22no—
86#41×C127—41no—
87#41×C127—41no—
88C2×Q8×C127C2311yescharacteristic
89C2×Q8×C127C2311yescharacteristic
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127D411yescharacteristic
93C2×D4×C127C2311yescharacteristic
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127D411yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C22⋊C4×C127—21no—
108C22⋊C4×C127D411yescharacteristic
109C22⋊C4×C127D411yescharacteristic
110C42×C127C2311yescharacteristic
111C22×D127—21no—
112D508—21no—
113D508—21no—
114C22⋊D127—42no—
115C22⋊D127—81no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C4×D127—81no—
121C127⋊C8—21no—
122C127⋊C8—21no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127C2×D411yescharacteristic
127C23×C127C2×D411yescharacteristic
128Q8×C127—21no—
129Q8×C127—21no—
130Q8×C127—21no—
131Q8×C127—21no—
132D4×C127—21no—
133D4×C127—41no—
134D4×C127—42no—
135D4×C127—42no—
136D4×C127#4112yes—
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—21no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2×D411yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—22no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127C22×D411yescharacteristic
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C2×#4111yescharacteristic
173C4×C127H(5,𝔽2)11yescharacteristic
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127—42no—
180C2×C127#54611yescharacteristic
181C2×C127#57511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—81no—
184#3307—1271nomaximal, Sylow
185C127#330711yescharacteristic, complemented, Sylow
186#553—1271no—
187#547D12711yescharacteristic, complemented
188#491—1271no—
189#485—1271no—
190#483—1271no—
191#478—1271no—
192#468—1271no—
193#465—1271no—
194#461—1271no—
195#449—1271no—
196#449—1271no—
197#442—1271no—
198#435—1271no—
199#417—1271no—
200#416—1271no—
201C2×#41C2×D12711yescharacteristic
202C2×#41C2×D12711yescharacteristic
203C2×SD16—1271no—
204C2×SD16—1271no—
205C2×D8—1271no—
206C2×M16—1271no—
207C22×C8—1271no—
208C4⋊Q8C2×D12711yescharacteristic
209#126—2541no—
210#125—1271no—
211#123—21no—
212#123—21no—
213#123—2541no—
214C22⋊Q8C2×D12711yescharacteristic
215C22⋊Q8C2×D12711yescharacteristic
216#121—21no—
217#121—1271no—
218#121—1271no—
219#121—2541no—
220C4×D4—1271no—
221C4×D4—2541no—
222C4×D4C2×D12711yescharacteristic
223C4×D4C2×D12711yescharacteristic
224#117—1271no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1271no—
227#107—1271no—
228#106—1271no—
229#106—1271no—
230C4⋊C8—1271no—
231#103—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#103—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241C22⋊C8—1271no—
242C22⋊C8—1271no—
243#97—1271no—
244#41—22no—
245#41—22no—
246#41—41no—
247#41—41no—
248C2×Q8C22×D12711yescharacteristic
249C2×Q8C22×D12711yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—1271no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4C22⋊D12711yescharacteristic
256C2×D4C22×D12711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264SD16—5081no—
265SD16—5081no—
266D8—2542no—
267M16—5081no—
268C2×C8—1271no—
269C2×C8—1271no—
270C2×C8—1271no—
271C2×C8—1271no—
272C2×C8—5081no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—1271no—
278C4⋊C4—1271no—
279C4⋊C4—1271no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4—2541no—
283C4⋊C4C22⋊D12711yescharacteristic
284C4⋊C4C22×D12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4C22⋊D12711yescharacteristic
293C22⋊C4C22⋊D12711yescharacteristic
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—2541no—
298C23C2×C22⋊D12711yescharacteristic
299C23C2×C22⋊D12711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—41no—
306D4—42no—
307D4—42no—
308D4—10161no—
309D4—5082no—
310D4—2541no—
311D4—2541no—
312D4#1452324712yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—10161no—
324C2×C4—10161no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4C2×C22⋊D12711yescharacteristic
329C2×C4C2×C22⋊D12711yescharacteristic
330C2×C4C2×C22⋊D12711yescharacteristic
331C2×C4C2×C22⋊D12711yescharacteristic
332C2×C4C23×D12711yescharacteristic
333C8—2541no—
334C8—2541no—
335C8—2541no—
336C8—2541no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
349C22C22×C22⋊D12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5081no—
358C4—5081no—
359C4—5081no—
360C4C2×#1452324711yescharacteristic
361C4#1686125411yescharacteristic
362C2—41no—
363C2—41no—
364C2—42no—
365C2—10161no—
366C2#2255052011yescharacteristic
367C2#2255067411yescharacteristic
368C2#2255068411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#547×C127C2
socleC22×C127C22×D4
hypercenter#547D127
hypocenterC127#3307
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #22550572C2semidirect
2. #22550244C2semidirect
3. #22550214C2semidirect
4. #22550204C2semidirect
5. #22550176C2semidirect
6. #22550124C2semidirect
7. #22550107C2semidirect
8. #22550078C2semidirect
9. #22550021C2semidirect
10. #22550021C2semidirect
11. #22549986C2semidirect
12. #22549948C2semidirect
13. #22549877C2semidirect
14. #22549872C2semidirect
15. #547×C127C2semidirect
185. C127#3307semidirect
187. #547D127semidirect

Maximal subgroups

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

subgroupindex
#225505722
#225502442
#225502142
#225502042
#225501762
#225501242
#225501072
#225500782
#225500212
#225499862
#225499482
#225498772
#225498722
#547×C1272
#3307127