Finite Group Databasebeta

← #30652347

Subgroups — #30652347

Order 16256.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550602C211yesmaximal, characteristic, complemented
2#22550598C213yesmaximal, complemented
3#22550567C213yesmaximal, complemented
4#22550561C213yesmaximal, complemented
5Q8×C22⋊D127C211yesmaximal, characteristic, complemented
6#22550538C211yesmaximal, characteristic, complemented
7#22550537C211yesmaximal, characteristic, complemented
8#22550527C213yesmaximal, complemented
9#22550430C213yesmaximal, complemented
10#22550413C213yesmaximal, complemented
11#22550410C213yesmaximal, complemented
12C4×#14523249C211yesmaximal, characteristic, complemented
13C4×#14523248C213yesmaximal, complemented
14C4×Q8×D127C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2213yes—
18C2×Q8×D127C2211yescharacteristic
19#16861208C2213yes—
20#16861207C2213yes—
21#16861204C2216yes—
22#16861203C2216yes—
23#16861201C2213yes—
24#16861200C2213yes—
25#16861199C2216yes—
26#16861198C2213yes—
27#16861197C2213yes—
28#16861194C2216yes—
29#16861192C2216yes—
30#16861189C2213yes—
31#16861188C2213yes—
32#16861187C2216yes—
33#16861185C2213yes—
34#16861185C2213yes—
35#16861184C2213yes—
36#16861183C2213yes—
37Q8×Dic127C2211yescharacteristic
38Q8×Dic127C2211yescharacteristic
39C4×Dic254C2213yes—
40C4×Dic254C2213yes—
41C4×Dic254C2213yes—
42#16861174C2213yes—
43C4×D508C2213yes—
44C4×D508C2213yes—
45C4×D508C2213yes—
46#16861171C2213yes—
47C4×C22⋊D127C2213yes—
48C4×C22⋊D127C2213yes—
49C4×C22⋊D127C2211yescharacteristic
50C4×C22⋊D127C2213yes—
51#16861168C2213yes—
52#16861167C2213yes—
53#16861166C2213yes—
54#16861166C2213yes—
55#16861166C2216yes—
56C4⋊C4×D127C2213yes—
57C42×D127C2213yes—
58C42×D127C2213yes—
59C22×Q8×C127C2211yescharacteristic
60C4×Q8×C127C2211yescharacteristic
61C4×Q8×C127C2213yes—
62C4×Q8×C127C2211yescharacteristic
63C4×Q8×C127C2213yes—
64C2×C4⋊C4×C127C2213yes—
65C2×C42×C127C2213yes—
66#14523249—24no—
67#14523248—212no—
68C2×Dic254C2313yes—
69C2×Dic254C2313yes—
70Q8×D127—24no—
71C2×D508C2313yes—
72C2×D508C2313yes—
73C2×C22⋊D127C2313yes—
74C2×C22⋊D127C2311yescharacteristic
75C2×C4×D127C2313yes—
76C2×C4×D127C2313yes—
77C2×C4×D127C2313yes—
78C2×C4×D127C2313yes—
79C2×C4×D127C2311yescharacteristic
80C2×C4×D127C2313yes—
81C4⋊Dic127C2313yes—
82C4⋊Dic127C2313yes—
83C4⋊Dic127C2313yes—
84C4⋊Dic127C2313yes—
85Dic127⋊C4C2313yes—
86Dic127⋊C4C2316yes—
87Dic127⋊C4C2313yes—
88Dic127⋊C4C2316yes—
89Dic127⋊C4C2316yes—
90Dic127⋊C4C2313yes—
91Dic127⋊C4C2311yescharacteristic
92C22⋊Dic127C2313yes—
93C22⋊Dic127C2311yescharacteristic
94(C2×D127)⋊C4C2316yes—
95(C2×D127)⋊C4C2313yes—
96(C2×D127)⋊C4C2316yes—
97(C2×D127)⋊C4C2316yes—
98(C2×D127)⋊C4C2313yes—
99(C2×D127)⋊C4C2313yes—
100(C2×D127)⋊C4C2311yescharacteristic
101C4×Dic127C2313yes—
102C4×Dic127C2313yes—
103C4×Dic127C2313yes—
104C4×Dic127C2313yes—
105C4×Dic127C2311yescharacteristic
106C4×Dic127C2313yes—
107C2×Q8×C127—24no—
108C2×Q8×C127C2313yes—
109C2×Q8×C127C2311yescharacteristic
110C22×C4×C127C2313yes—
111C22×C4×C127C2313yes—
112C22×C4×C127C2311yescharacteristic
113C4⋊C4×C127C2313yes—
114C4⋊C4×C127C2316yes—
115C4⋊C4×C127C2313yes—
116C42×C127C2313yes—
117C42×C127C2313yes—
118C42×C127C2313yes—
119C42×C127C2313yes—
120C22×D127C2413yes—
121C22×D127C2411yescharacteristic
122Dic254—26no—
123Dic254—26no—
124D508—26no—
125D508—26no—
126C22⋊D127—26no—
127C22⋊D127#4114yescomplemented
128C2×Dic127C2411yescharacteristic
129C2×Dic127C2411yescharacteristic
130C2×Dic127C2413yes—
131C2×Dic127C2413yes—
132C2×Dic127C2413yes—
133C2×Dic127C2411yescharacteristic
134C4×D127—26no—
135C4×D127—26no—
136C4×D127—26no—
137C4×D127—22no—
138C4×D127—26no—
139C4×D127—26no—
140C23×C127C2411yescharacteristic
141Q8×C127—26no—
142Q8×C127#4114yescomplemented
143C2×C4×C127—26no—
144C2×C4×C127—26no—
145C2×C4×C127—22no—
146C2×C4×C127C2411yescharacteristic
147C2×C4×C127C2413yes—
148C2×C4×C127C2413yes—
149C2×C4×C127C2413yes—
150C2×C4×C127C2413yes—
151C2×C4×C127C2411yescharacteristic
152C22×C127—21no—
153C22×C127—21no—
154C22×C127C2×#4112yes—
155C22×C127C2511yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—23no—
159C2×D127—23no—
160C2×D127—23no—
161C2×D127C2×#4112yes—
162C4×C127—23no—
163C4×C127—23no—
164C4×C127—23no—
165C4×C127—21no—
166C4×C127C2×#4116yes—
167C4×C1272−1+412yes—
168Dic127—21no—
169Dic127—23no—
170Dic127—23no—
171Dic127—21no—
172Dic127—23no—
173Dic127C2×#4112yes—
174C2×C127—22no—
175C2×C127C22×#4111yescharacteristic
176C2×C127C22×#4111yescharacteristic
177C2×C127C2×2−1+411yescharacteristic
178D127—22no—
179D127—43no—
180#3482—1271nomaximal, Sylow
181C127#348211yescharacteristic, complemented, Sylow
182#557—1271no—
183#556—1273no—
184#552—1273no—
185#552—1273no—
186D4×Q8—1271no—
187#548—1271no—
188#548—1271no—
189#547—1273no—
190#533—1273no—
191#529—1273no—
192#529—1273no—
193C4×#41—1271no—
194C4×#41—1273no—
195C2×C4×Q8—1271no—
196C2×C4×Q8D12711yescharacteristic, complemented
197C2×#41—1271no—
198C2×#41—1273no—
199C22×Q8—1271no—
200C22×Q8C2×D12711yescharacteristic
201C4⋊Q8—1273no—
202C4⋊Q8—1273no—
203#126—1276no—
204#126—1276no—
205#125—1276no—
206#125—1273no—
207#125—1273no—
208#124—1273no—
209#124—1273no—
210#123—1276no—
211#123—1276no—
212C22⋊Q8—1273no—
213C22⋊Q8—1273no—
214C22⋊Q8—1273no—
215C22⋊Q8—1276no—
216C22⋊Q8—1273no—
217#121—1273no—
218#121—1273no—
219C4×Q8—1271no—
220C4×Q8—1271no—
221C4×Q8—1273no—
222C4×Q8—1273no—
223C4×Q8—1273no—
224C4×Q8—1273no—
225C4×Q8C2×D12711yescharacteristic
226C4×Q8C2×D12713yes—
227C4×Q8C2×D12711yescharacteristic
228C4×Q8C2×D12713yes—
229C4×D4—1271no—
230C4×D4—1273no—
231C4×D4—1273no—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1273no—
235C4×D4—1273no—
236C4×D4—1273no—
237#117—1273no—
238#117—1273no—
239#117—1273no—
240#117—1273no—
241#117—1276no—
242C2×C4⋊C4—1273no—
243C2×C4⋊C4C2×D12713yes—
244C2×C42—1273no—
245C2×C42—1273no—
246C2×C42C2×D12713yes—
247#41—25412no—
248#41—2544no—
249C2×Q8—24no—
250C2×Q8—1273no—
251C2×Q8—1273no—
252C2×Q8—2544no—
253C2×Q8C22×D12713yes—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—1271no—
256C2×D4—1273no—
257C2×D4—1273no—
258C2×D4—1273no—
259C22×C4—1273no—
260C22×C4—1273no—
261C22×C4—1273no—
262C22×C4—1273no—
263C22×C4—1271no—
264C22×C4—1273no—
265C22×C4C22×D12713yes—
266C22×C4C22×D12713yes—
267C22×C4C22×D12711yescharacteristic
268C4⋊C4—1276no—
269C4⋊C4—1271no—
270C4⋊C4—1273no—
271C4⋊C4—1273no—
272C4⋊C4—1273no—
273C4⋊C4—1276no—
274C4⋊C4—1276no—
275C4⋊C4—1273no—
276C4⋊C4—1273no—
277C4⋊C4—1273no—
278C4⋊C4—1273no—
279C4⋊C4C22×D12713yes—
280C4⋊C4C22×D12713yes—
281C4⋊C4C22×D12716yes—
282C22⋊C4—1271no—
283C22⋊C4—1273no—
284C22⋊C4—1271no—
285C22⋊C4—1273no—
286C22⋊C4—1276no—
287C22⋊C4—1273no—
288C22⋊C4—1273no—
289C22⋊C4—1276no—
290C22⋊C4—1276no—
291C42—1273no—
292C42—1273no—
293C42—1273no—
294C42—1273no—
295C42—1271no—
296C42—1273no—
297C42C22×D12713yes—
298C42C22×D12713yes—
299C42C22×D12713yes—
300C42C22×D12713yes—
301C23—1271no—
302C23—1273no—
303C23C23×D12711yescharacteristic
304Q8—26no—
305Q8—2546no—
306Q8—2546no—
307Q8#1452324814yescomplemented
308D4—1274no—
309D4—2546no—
310D4—2546no—
311D4—2546no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—1273no—
319C2×C4—1273no—
320C2×C4—1273no—
321C2×C4—2546no—
322C2×C4—2546no—
323C2×C4—2546no—
324C2×C4—2546no—
325C2×C4—2546no—
326C2×C4—2542no—
327C2×C4C23×D12713yes—
328C2×C4C23×D12711yescharacteristic
329C2×C4C23×D12713yes—
330C2×C4C23×D12713yes—
331C2×C4C23×D12713yes—
332C2×C4C23×D12711yescharacteristic
333C22—21no—
334C22—21no—
335C22—1272no—
336C22—2543no—
337C22—2543no—
338C22—2543no—
339C22—2541no—
340C22—2541no—
341C22#41×D12712yes—
342C22C24×D12711yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—1272no—
348C4—2543no—
349C4—2541no—
350C4—2543no—
351C4—2541no—
352C4—2543no—
353C4C2×#1452324816yes—
354C4#1686125712yes—
355C2—22no—
356C2—5083no—
357C2—2542no—
358C2C2×#41×D12711yescharacteristic
359C2C22×#1452324811yescharacteristic
360C2C2×#1686125711yescharacteristic
subgroupG/H
centerC2×C4C23×D127
commutator, socle, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×C4×Q8×C127C2
hypercenterC2×C4×Q8D127
hypocenterC127#3482
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22550602C2semidirect
2. #22550598C2semidirect
3. #22550567C2semidirect
4. #22550561C2semidirect
5. Q8×C22⋊D127C2semidirect
6. #22550538C2semidirect
7. #22550537C2semidirect
8. #22550527C2semidirect
9. #22550430C2semidirect
10. #22550413C2semidirect
11. #22550410C2semidirect
12. C4×#14523249C2semidirect
13. C4×#14523248C2semidirect
14. C4×Q8×D127C2semidirect
15. C2×C4×Q8×C127C2semidirect
127. C22⋊D127#41semidirect
142. Q8×C127#41semidirect
181. C127#3482semidirect
196. C2×C4×Q8D127semidirect
307. Q8#14523248semidirect

Maximal subgroups

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

subgroupindex
#225506022
#225505982
#225505672
#225505612
Q8×C22⋊D1272
#225505382
#225505372
#225505272
#225504302
#225504132
#225504102
C4×#145232492
C4×#145232482
C4×Q8×D1272
C2×C4×Q8×C1272
#3482127