Finite Group Databasebeta

← #30652346

Subgroups — #30652346

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#22550595C213yesmaximal, complemented
3#22550566C213yesmaximal, complemented
4#22550560C213yesmaximal, complemented
5Q8×C22⋊D127C211yesmaximal, characteristic, complemented
6#22550533C211yesmaximal, characteristic, complemented
7#22550526C213yesmaximal, complemented
8#22550428C213yesmaximal, complemented
9#22550407C213yesmaximal, complemented
10#22550405C213yesmaximal, complemented
11#41×Dic127C211yesmaximal, characteristic, complemented
12C4×#14523247C213yesmaximal, complemented
13C2×Q8×Dic127C211yesmaximal, characteristic, complemented
14C4×Q8×D127C211yesmaximal, characteristic, complemented
15#548×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2213yes—
17C2×Q8×D127C2211yescharacteristic
18#16861208C2213yes—
19#16861207C2213yes—
20#16861202C2216yes—
21#16861202C2216yes—
22#16861201C2213yes—
23#16861200C2213yes—
24#16861199C2216yes—
25#16861196C2213yes—
26#16861191C2216yes—
27#16861190C2216yes—
28#16861188C2213yes—
29#16861188C2213yes—
30#16861186C2216yes—
31#16861185C2213yes—
32#16861180C2213yes—
33Q8×Dic127C2211yescharacteristic
34Q8×Dic127C2211yescharacteristic
35Q8×Dic127C2213yes—
36Q8×Dic127C2211yescharacteristic
37Q8×Dic127C2213yes—
38C4×Dic254C2213yes—
39C4×Dic254C2213yes—
40#16861174C2213yes—
41#16861174C2213yes—
42D4×Dic127C2213yes—
43D4×Dic127C2213yes—
44D4×Dic127C2213yes—
45C4×C22⋊D127C2213yes—
46C4×C22⋊D127C2213yes—
47(C22⋊D127)⋊C4C2211yescharacteristic
48(C22⋊D127)⋊C4C2213yes—
49#16861168C2213yes—
50#16861167C2213yes—
51#16861165C2213yes—
52#16861165C2213yes—
53#16861165C2216yes—
54C2×C4⋊Dic127C2213yes—
55C4⋊C4×D127C2213yes—
56C2×C4×Dic127C2213yes—
57C2×C4×Dic127C2213yes—
58C42×D127C2213yes—
59C2×#41×C127C2211yescharacteristic
60C22×Q8×C127C2211yescharacteristic
61#124×C127C2213yes—
62C22⋊Q8×C127C2213yes—
63#121×C127C2213yes—
64C4×Q8×C127C2211yescharacteristic
65C4×D4×C127C2213yes—
66#14523247—212no—
67C2×Dic254C2313yes—
68C2×Dic254C2313yes—
69Q8×D127—24no—
70C2×C22⋊D127C2313yes—
71C2×C22⋊D127C2311yescharacteristic
72C22×Dic127C2313yes—
73C22×Dic127C2313yes—
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2313yes—
76C2×C4×D127C2313yes—
77C2×C4×D127C2313yes—
78C2×C4×D127C2311yescharacteristic
79C4⋊Dic127C2313yes—
80C4⋊Dic127C2316yes—
81C4⋊Dic127C2313yes—
82C4⋊Dic127C2313yes—
83C4⋊Dic127C2313yes—
84Dic127⋊C4C2316yes—
85Dic127⋊C4C2313yes—
86Dic127⋊C4C2316yes—
87Dic127⋊C4C2313yes—
88Dic127⋊C4C2316yes—
89Dic127⋊C4C2313yes—
90Dic127⋊C4C2311yescharacteristic
91C22⋊Dic127C2316yes—
92C22⋊Dic127C2313yes—
93C22⋊Dic127C2316yes—
94C22⋊Dic127C2313yes—
95(C2×D127)⋊C4C2313yes—
96(C2×D127)⋊C4C2311yescharacteristic
97C4×Dic127C2313yes—
98C4×Dic127C2313yes—
99C4×Dic127C2313yes—
100C4×Dic127C2313yes—
101C4×Dic127C2313yes—
102C4×Dic127C2313yes—
103C4×Dic127C2313yes—
104C4×Dic127C2311yescharacteristic
105C4×Dic127C2313yes—
106#41×C127—24no—
107C2×Q8×C127—24no—
108C2×Q8×C127C2313yes—
109C2×Q8×C127C2311yescharacteristic
110C2×D4×C127C2313yes—
111C2×D4×C127C2313yes—
112C22×C4×C127C2313yes—
113C22×C4×C127C2313yes—
114C4⋊C4×C127C2313yes—
115C4⋊C4×C127C2313yes—
116C22⋊C4×C127C2316yes—
117C22⋊C4×C127C2313yes—
118C22⋊C4×C127C2311yescharacteristic
119C42×C127C2313yes—
120C22×D127C2411yescharacteristic
121Dic254—26no—
122Dic254—26no—
123C22⋊D127—26no—
124C22⋊D127#4114yescomplemented
125C2×Dic127—26no—
126C2×Dic127—26no—
127C2×Dic127—22no—
128C2×Dic127—26no—
129C2×Dic127C2411yescharacteristic
130C2×Dic127C2413yes—
131C2×Dic127C2413yes—
132C2×Dic127C2413yes—
133C2×Dic127C2413yes—
134C2×Dic127C2411yescharacteristic
135C2×Dic127C2411yescharacteristic
136C4×D127—26no—
137C4×D127—26no—
138C4×D127—22no—
139C23×C127C2413yes—
140C23×C127C2411yescharacteristic
141Q8×C127—26no—
142Q8×C127#4114yescomplemented
143D4×C127—26no—
144D4×C127—26no—
145C2×C4×C127—26no—
146C2×C4×C127—26no—
147C2×C4×C127C2411yescharacteristic
148C2×C4×C127C2413yes—
149C2×C4×C127C2413yes—
150C2×C4×C127C2411yescharacteristic
151C2×C4×C127C2413yes—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—23no—
155C22×C127—23no—
156C22×C127—23no—
157C22×C127C2×#4112yes—
158C22×C127C2511yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127C2×#4112yes—
162C4×C127—21no—
163C4×C127—23no—
164C4×C127—23no—
165C4×C127—21no—
166C4×C127C2×#4116yes—
167Dic127—23no—
168Dic127—23no—
169Dic127—23no—
170Dic127—21no—
171Dic127—23no—
172Dic127C2×#4112yes—
173Dic1272−1+412yes—
174C2×C127—22no—
175C2×C127—43no—
176C2×C127C22×#4111yescharacteristic
177C2×C127C22×#4111yescharacteristic
178C2×C127C2×2−1+411yescharacteristic
179D127—22no—
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#548D12711yescharacteristic, complemented
189#547—1273no—
190#533—1273no—
191#529—1273no—
192#529—1273no—
193C4×#41—1271no—
194C4×#41—1273no—
195C2×C4×Q8—1271no—
196C2×C4×Q8—1271no—
197C2×#41—1273no—
198C2×#41C2×D12711yescharacteristic
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#124C2×D12713yes—
210#123—1276no—
211#123—1276no—
212C22⋊Q8—1273no—
213C22⋊Q8—1273no—
214C22⋊Q8—1273no—
215C22⋊Q8—1276no—
216C22⋊Q8C2×D12713yes—
217#121—1273no—
218#121C2×D12713yes—
219C4×Q8—1271no—
220C4×Q8—1273no—
221C4×Q8—1271no—
222C4×Q8—1273no—
223C4×Q8—1271no—
224C4×Q8—1273no—
225C4×Q8—1273no—
226C4×Q8—1273no—
227C4×Q8—1273no—
228C4×Q8C2×D12711yescharacteristic
229C4×D4—1273no—
230C4×D4—1273no—
231C4×D4—1273no—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1271no—
235C4×D4—1273no—
236C4×D4C2×D12713yes—
237#117—1273no—
238#117—1273no—
239#117—1273no—
240#117—1273no—
241#117—1276no—
242C2×C4⋊C4—1273no—
243C2×C4⋊C4—1273no—
244C2×C42—1273no—
245C2×C42—1273no—
246C2×C42—1273no—
247#41—24no—
248#41—25412no—
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×D4C22×D12713yes—
258C2×D4C22×D12713yes—
259C22×C4—1271no—
260C22×C4—1273no—
261C22×C4—1273no—
262C22×C4—1273no—
263C22×C4—1271no—
264C22×C4—1273no—
265C22×C4—1273no—
266C22×C4C22×D12713yes—
267C22×C4C22×D12713yes—
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—1276no—
278C4⋊C4—1273no—
279C4⋊C4—1273no—
280C4⋊C4C22×D12713yes—
281C4⋊C4C22×D12713yes—
282C22⋊C4—1273no—
283C22⋊C4—1276no—
284C22⋊C4—1273no—
285C22⋊C4—1276no—
286C22⋊C4—1271no—
287C22⋊C4—1273no—
288C22⋊C4C22×D12711yescharacteristic
289C22⋊C4C22×D12713yes—
290C22⋊C4C22×D12716yes—
291C42—1271no—
292C42—1273no—
293C42—1273no—
294C42—1273no—
295C42—1273no—
296C42—1273no—
297C42—1273no—
298C42—1273no—
299C42—1273no—
300C42C22×D12713yes—
301C23—1271no—
302C23C23×D12713yes—
303C23C23×D12711yescharacteristic
304Q8—26no—
305Q8—2546no—
306Q8—2546no—
307Q8#1452324714yescomplemented
308D4—26no—
309D4—26no—
310D4—1274no—
311D4—2546no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—1271no—
315C2×C4—1271no—
316C2×C4—1271no—
317C2×C4—1273no—
318C2×C4—1273no—
319C2×C4—1273no—
320C2×C4—1273no—
321C2×C4—2542no—
322C2×C4—2546no—
323C2×C4—2546no—
324C2×C4—2542no—
325C2×C4—2546no—
326C2×C4—2546no—
327C2×C4—2546no—
328C2×C4C23×D12711yescharacteristic
329C2×C4C23×D12711yescharacteristic
330C2×C4C23×D12713yes—
331C2×C4C23×D12713yes—
332C2×C4C23×D12713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1272no—
339C22—2541no—
340C22—2541no—
341C22#41×D12712yes—
342C22C24×D12711yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—1272no—
348C4—1272no—
349C4—2543no—
350C4—2543no—
351C4—2543no—
352C4—2541no—
353C4—2543no—
354C4C2×#1452324716yes—
355C2—22no—
356C2—43no—
357C2—2542no—
358C2C2×#41×D12711yescharacteristic
359C2C22×#1452324711yescharacteristic
360C22−1+4×D12711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#548×C127C2
hypercenter#548D127
hypocenterC127#3482
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22550602C2semidirect
2. #22550595C2semidirect
3. #22550566C2semidirect
4. #22550560C2semidirect
5. Q8×C22⋊D127C2semidirect
6. #22550533C2semidirect
7. #22550526C2semidirect
8. #22550428C2semidirect
9. #22550407C2semidirect
10. #22550405C2semidirect
11. #41×Dic127C2semidirect
12. C4×#14523247C2semidirect
13. C2×Q8×Dic127C2semidirect
14. C4×Q8×D127C2semidirect
15. #548×C127C2semidirect
124. C22⋊D127#41semidirect
142. Q8×C127#41semidirect
181. C127#3482semidirect
188. #548D127semidirect
307. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225506022
#225505952
#225505662
#225505602
Q8×C22⋊D1272
#225505332
#225505262
#225504282
#225504072
#225504052
#41×Dic1272
C4×#145232472
C2×Q8×Dic1272
C4×Q8×D1272
#548×C1272
#3482127