Finite Group Databasebeta

← #30652820

Subgroups — #30652820

Order 16256.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550604C213yesmaximal, complemented
2#22550602C211yesmaximal, characteristic, complemented
3#22550594C213yesmaximal, complemented
4#22550567C213yesmaximal, complemented
5#22550565C213yesmaximal, complemented
6Q8×D508C211yesmaximal, characteristic, complemented
7Q8×C22⋊D127C211yesmaximal, characteristic, complemented
8Q8×C22⋊D127C211yesmaximal, characteristic, complemented
9#22550488C213yesmaximal, complemented
10#22550484C213yesmaximal, complemented
11#22550430C213yesmaximal, complemented
12#22550404C213yesmaximal
13#22550352C211yesmaximal, characteristic, complemented
14C4×Q8×D127C211yesmaximal, characteristic, complemented
15#519×C127C211yesmaximal, characteristic, complemented
16C2×Q8×D127C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18#16861209C2213yes—
19#16861208C2213yes—
20#16861207C2213yes—
21#16861207C2213yes—
22#16861204C2213yes—
23#16861203C2216yes—
24#16861202C2216yes—
25#16861202C2213yes—
26#16861201C2216yes—
27#16861201C2213yes—
28#16861200C2213yes—
29#16861199C2216yes—
30#16861199C2213yes—
31#16861199C2216yes—
32#16861195C2213yes—
33#16861194C2213yes—
34#16861191C2213yes—
35#16861189C2213yes—
36#16861189C2213yes—
37#16861188C2213yes—
38#16861188C2213yes—
39#16861187C2213yes—
40#16861185C2213yes—
41#16861185C2213yes—
42#16861182C2211yescharacteristic
43Q8×Dic127C2211yescharacteristic
44Q8×Dic127C2211yescharacteristic
45C4×Dic254C2216yes—
46C4×Dic254C2213yes—
47C4×Dic254C2213yes—
48C4×Dic254C2213yes—
49#16861174C2213yes—
50#16861174C2213yes—
51C4×D508C2213yes—
52C4×C22⋊D127C2213yes—
53C4×C22⋊D127C2213yes—
54#16861167C2213yes—
55#16861166C2216yes—
56#16861166C2213yes—
57C4⋊C4×D127C2213yes—
58C42×D127C2213yes—
59C22×Q8×C127C2211yescharacteristic
60C4×Q8×C127C2211yescharacteristic
61C4×Q8×C127C2213yes—
62C4×Q8×C127C2211yescharacteristic
63C4×Q8×C127C2213yes—
64#117×C127C2213yes—
65#117×C127C2213yes—
66C2×Dic254C2313yes—
67C2×Dic254C2313yes—
68C2×Dic254C2313yes—
69Q8×D127—24no—
70Q8×D127—24no—
71C2×D508C2311yescharacteristic
72C2×C22⋊D127C2311yescharacteristic
73C2×C22⋊D127C2311yescharacteristic
74C2×C4×D127C2313yes—
75C2×C4×D127C2313yes—
76C2×C4×D127C2311yescharacteristic
77C2×C4×D127C2313yes—
78C4⋊Dic127C2313yes—
79C4⋊Dic127C2313yes—
80C4⋊Dic127C2316yes—
81C4⋊Dic127C2313yes—
82C4⋊Dic127C2313yes—
83Dic127⋊C4C2316yes—
84Dic127⋊C4C2316yes—
85Dic127⋊C4C2313yes—
86Dic127⋊C4C2313yes—
87Dic127⋊C4C2313yes—
88Dic127⋊C4C2313yes—
89Dic127⋊C4C2311yescharacteristic
90Dic127⋊C4C2316yes—
91Dic127⋊C4C2313yes—
92Dic127⋊C4C2316yes—
93C22⋊Dic127C2313yes—
94C22⋊Dic127C2313yes—
95(C2×D127)⋊C4C2313yes—
96(C2×D127)⋊C4C2313yes—
97(C2×D127)⋊C4C2313yes—
98(C2×D127)⋊C4C2313yes—
99(C2×D127)⋊C4C2313yes—
100(C2×D127)⋊C4C2311yescharacteristic
101C4×Dic127C2313yes—
102C4×Dic127C2313yes—
103C4×Dic127C2316yes—
104C4×Dic127C2313yes—
105C4×Dic127C2313yes—
106C2×Q8×C127—24no—
107C2×Q8×C127C2313yes—
108C2×Q8×C127C2311yescharacteristic
109C22×C4×C127C2313yes—
110C4⋊C4×C127C2313yes—
111C4⋊C4×C127C2313yes—
112C4⋊C4×C127C2313yes—
113C4⋊C4×C127C2313yes—
114C22⋊C4×C127C2313yes—
115C22⋊C4×C127C2311yescharacteristic
116C42×C127C2316yes—
117C42×C127C2313yes—
118C42×C127C2313yes—
119C22×D127C2411yescharacteristic
120C22×D127C2411yescharacteristic
121Dic254—26no—
122Dic254—26no—
123Dic254—26no—
124D508—22no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C2×Dic127C2413yes—
128C2×Dic127C2411yescharacteristic
129C2×Dic127C2413yes—
130C2×Dic127C2413yes—
131C2×Dic127C2413yes—
132C2×Dic127C2411yescharacteristic
133C4×D127—26no—
134C4×D127—26no—
135C4×D127—26no—
136C4×D127—22no—
137C23×C127C2411yescharacteristic
138Q8×C127—26no—
139Q8×C127#4114yes—
140C2×C4×C127—26no—
141C2×C4×C127C2413yes—
142C2×C4×C127C2411yescharacteristic
143C2×C4×C127C2413yes—
144C2×C4×C127C2411yescharacteristic
145C2×C4×C127C2413yes—
146C2×C4×C127C2413yes—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127C2511yescharacteristic
151C2×D127—21no—
152C2×D127—21no—
153C2×D127—21no—
154C2×D127—21no—
155C2×D127—21no—
156C2×D1272−1+412yes—
157C4×C127—23no—
158C4×C127—23no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—23no—
162C4×C127C2×#4116yes—
163Dic127—23no—
164Dic127—23no—
165Dic127—23no—
166Dic127—21no—
167Dic127—23no—
168Dic1272−1+412yes—
169C2×C127—41no—
170C2×C127C22×#4111yescharacteristic
171C2×C127C2×2−1+411yescharacteristic
172C2×C127C2×2−1+411yescharacteristic
173D127—22no—
174D127—41no—
175#3523—1271nomaximal, Sylow
176C127#352311yescharacteristic, complemented, Sylow
177#557—1271no—
178#557—1273no—
179#556—1273no—
180#552—1273no—
181#552—1273no—
182D4×Q8—1271no—
183D4×Q8—1271no—
184D4×Q8—1271no—
185#541—1273no—
186#541—1273no—
187#533—1273no—
188#529—1273no—
189#519—1271no—
190#519D12711yescharacteristic, complemented
191C2×C4×Q8—1271no—
192C22×Q8—1271no—
193C22×Q8—1271no—
194C22×Q8C2×D12711yescharacteristic
195C4⋊Q8—1273no—
196C4⋊Q8—1273no—
197C4⋊Q8—1273no—
198C4⋊Q8—1273no—
199#126—1273no—
200#126—1273no—
201#126—1276no—
202#126—1276no—
203#125—1273no—
204#125—1276no—
205#125—1276no—
206#125—1273no—
207#125—1276no—
208#125—1273no—
209#124—1273no—
210#123—1273no—
211#123—1273no—
212C22⋊Q8—1273no—
213C22⋊Q8—1273no—
214C22⋊Q8—1273no—
215C22⋊Q8—1273no—
216C22⋊Q8—1273no—
217C22⋊Q8—1273no—
218C22⋊Q8—1273no—
219#121—1271no—
220C4×Q8—1271no—
221C4×Q8—1271no—
222C4×Q8—1273no—
223C4×Q8—1273no—
224C4×Q8—1273no—
225C4×Q8—1273no—
226C4×Q8—1273no—
227C4×Q8—1276no—
228C4×Q8C2×D12711yescharacteristic
229C4×Q8C2×D12713yes—
230C4×Q8C2×D12711yescharacteristic
231C4×Q8C2×D12713yes—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1273no—
235#117—1273no—
236#117—1273no—
237#117—1276no—
238#117C2×D12713yes—
239#117C2×D12713yes—
240C2×C4⋊C4—1273no—
241C2×C42—1273no—
242C2×Q8—24no—
243C2×Q8—1273no—
244C2×Q8—1273no—
245C2×Q8—1273no—
246C2×Q8—2544no—
247C2×Q8—2544no—
248C2×Q8C22×D12713yes—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—1271no—
251C2×D4—1271no—
252C2×D4—1271no—
253C22×C4—1271no—
254C22×C4—1273no—
255C22×C4—1273no—
256C22×C4—1273no—
257C22×C4C22×D12713yes—
258C4⋊C4—1276no—
259C4⋊C4—1273no—
260C4⋊C4—1273no—
261C4⋊C4—1276no—
262C4⋊C4—1276no—
263C4⋊C4—1273no—
264C4⋊C4—1273no—
265C4⋊C4—1271no—
266C4⋊C4—1276no—
267C4⋊C4—1273no—
268C4⋊C4—1273no—
269C4⋊C4—1273no—
270C4⋊C4—1273no—
271C4⋊C4—1276no—
272C4⋊C4—1273no—
273C4⋊C4C22×D12713yes—
274C4⋊C4C22×D12713yes—
275C4⋊C4C22×D12713yes—
276C4⋊C4C22×D12713yes—
277C22⋊C4—1273no—
278C22⋊C4—1273no—
279C22⋊C4—1271no—
280C22⋊C4—1273no—
281C22⋊C4—1273no—
282C22⋊C4—1273no—
283C22⋊C4—1273no—
284C22⋊C4—1273no—
285C22⋊C4C22×D12711yescharacteristic
286C22⋊C4C22×D12713yes—
287C42—1273no—
288C42—1273no—
289C42—1273no—
290C42—1276no—
291C42—1273no—
292C42C22×D12713yes—
293C42C22×D12716yes—
294C42C22×D12713yes—
295C23—1271no—
296C23—1271no—
297C23C23×D12711yescharacteristic
298Q8—26no—
299Q8—2546no—
300Q8—2546no—
301Q8—2546no—
302Q8#1452324814yes—
303D4—2542no—
304D4—2542no—
305D4—2542no—
306C2×C4—26no—
307C2×C4—1271no—
308C2×C4—1273no—
309C2×C4—1271no—
310C2×C4—1273no—
311C2×C4—1273no—
312C2×C4—1273no—
313C2×C4—2542no—
314C2×C4—2546no—
315C2×C4—2546no—
316C2×C4—2546no—
317C2×C4C23×D12713yes—
318C2×C4C23×D12711yescharacteristic
319C2×C4C23×D12713yes—
320C2×C4C23×D12711yescharacteristic
321C2×C4C23×D12713yes—
322C2×C4C23×D12713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—1272no—
327C22—2541no—
328C22—2541no—
329C22—2541no—
330C22—2541no—
331C22—2541no—
332C22C24×D12711yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—1272no—
339C4—2543no—
340C4—2543no—
341C4—2543no—
342C4—2541no—
343C4—2543no—
344C4C2×#1452324816yes—
345C2—41no—
346C2—5081no—
347C2—2542no—
348C2C22×#1452324811yescharacteristic
349C22−1+4×D12711yescharacteristic
350C22−1+4×D12711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#519×C127C2
prime radical#16861182C22
hypercenter#519D127
hypocenterC127#3523
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #22550604C2semidirect
2. #22550602C2semidirect
3. #22550594C2semidirect
4. #22550567C2semidirect
5. #22550565C2semidirect
6. Q8×D508C2semidirect
7. Q8×C22⋊D127C2semidirect
8. Q8×C22⋊D127C2semidirect
9. #22550488C2semidirect
10. #22550484C2semidirect
11. #22550430C2semidirect
13. #22550352C2semidirect
14. C4×Q8×D127C2semidirect
15. #519×C127C2semidirect
176. C127#3523semidirect
190. #519D127semidirect

Maximal subgroups

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

subgroupindex
#225506042
#225506022
#225505942
#225505672
#225505652
Q8×D5082
Q8×C22⋊D1272
#225504882
#225504842
#225504302
#225504042
#225503522
C4×Q8×D1272
#519×C1272
#3523127