Finite Group Databasebeta

← #30652320

Subgroups — #30652320

Order 16256.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550608C213yesmaximal, complemented
2#22550602C211yesmaximal, characteristic, complemented
3#22550597C213yesmaximal, complemented
4#22550578C213yesmaximal, complemented
5#22550572C213yesmaximal, complemented
6#22550545C211yesmaximal, characteristic, complemented
7#22550537C211yesmaximal, characteristic, complemented
8#22550421C213yesmaximal, complemented
9#22550412C213yesmaximal, complemented
10#22550405C213yesmaximal, complemented
11#41×Dic127C211yesmaximal, characteristic, complemented
12C4×#14523249C211yesmaximal, characteristic, complemented
13C4×#14523247C213yesmaximal, complemented
14C2×Q8×Dic127C211yesmaximal, characteristic, complemented
15#548×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2213yes—
18#16861208C2213yes—
19#16861205C2213yes—
20#16861204C2216yes—
21#16861202C2216yes—
22#16861201C2213yes—
23#16861200C2213yes—
24#16861198C2213yes—
25#16861195C2216yes—
26#16861193C2216yes—
27#16861191C2216yes—
28#16861185C2213yes—
29#16861183C2213yes—
30#16861183C2213yes—
31#16861181C2216yes—
32#16861180C2213yes—
33Q8×Dic127C2211yescharacteristic
34Q8×Dic127C2211yescharacteristic
35Q8×Dic127C2213yes—
36Q8×Dic127C2211yescharacteristic
37Q8×Dic127C2213yes—
38C4×Dic254C2213yes—
39#16861174C2213yes—
40C4×D508C2213yes—
41D4×Dic127C2213yes—
42D4×Dic127C2213yes—
43D4×Dic127C2213yes—
44#16861171C2213yes—
45C4×C22⋊D127C2213yes—
46C4×C22⋊D127C2213yes—
47(C22⋊D127)⋊C4C2213yes—
48(C22⋊D127)⋊C4C2213yes—
49(C22⋊D127)⋊C4C2216yes—
50#16861168C2213yes—
51#16861167C2213yes—
52#16861167C2213yes—
53#16861165C2211yescharacteristic
54#16861165C2213yes—
55C2×C4⋊Dic127C2213yes—
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#14523249—24no—
67#14523247—212no—
68C2×Dic254C2313yes—
69C2×D508C2313yes—
70C2×C22⋊D127C2313yes—
71C2×C22⋊D127C2313yes—
72C2×C22⋊D127C2316yes—
73C22×Dic127C2313yes—
74C22×Dic127C2313yes—
75C22×Dic127C2313yes—
76C22×Dic127C2311yescharacteristic
77C2×C4×D127C2313yes—
78C2×C4×D127C2313yes—
79C2×C4×D127C2313yes—
80C4⋊Dic127C2313yes—
81C4⋊Dic127C2316yes—
82C4⋊Dic127C2313yes—
83C4⋊Dic127C2313yes—
84C4⋊Dic127C2313yes—
85Dic127⋊C4C2313yes—
86Dic127⋊C4C2313yes—
87Dic127⋊C4C2316yes—
88Dic127⋊C4C2311yescharacteristic
89Dic127⋊C4C2311yescharacteristic
90C22⋊Dic127C2316yes—
91C22⋊Dic127C2313yes—
92C22⋊Dic127C2311yescharacteristic
93(C2×D127)⋊C4C2316yes—
94(C2×D127)⋊C4C2313yes—
95(C2×D127)⋊C4C2316yes—
96(C2×D127)⋊C4C2313yes—
97C4×Dic127C2313yes—
98C4×Dic127C2313yes—
99C4×Dic127C2313yes—
100C4×Dic127C2313yes—
101C4×Dic127C2313yes—
102C4×Dic127C2311yescharacteristic
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×C127C2313yes—
117C22⋊C4×C127C2316yes—
118C22⋊C4×C127C2311yescharacteristic
119C42×C127C2313yes—
120C22×D127C2413yes—
121Dic254—26no—
122D508—26no—
123C22⋊D127—26no—
124C22⋊D127—26no—
125C22⋊D127—46no—
126C2×Dic127—26no—
127C2×Dic127—43no—
128C2×Dic127—43no—
129C2×Dic127#4114yescomplemented
130C2×Dic127C2411yescharacteristic
131C2×Dic127C2413yes—
132C2×Dic127C2411yescharacteristic
133C2×Dic127C2413yes—
134C2×Dic127C2413yes—
135C2×Dic127C2411yescharacteristic
136C2×Dic127C2411yescharacteristic
137C4×D127—26no—
138C4×D127—26no—
139C4×D127—43no—
140C23×C127C2413yes—
141C23×C127C2411yescharacteristic
142Q8×C127—26no—
143Q8×C127#4114yescomplemented
144D4×C127—26no—
145D4×C127—26no—
146C2×C4×C127—26no—
147C2×C4×C127—26no—
148C2×C4×C127C2411yescharacteristic
149C2×C4×C127C2413yes—
150C2×C4×C127C2413yes—
151C2×C4×C127C2411yescharacteristic
152C2×C4×C127C2413yes—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—23no—
156C22×C127—23no—
157C22×C127—23no—
158C22×C127C2×#4112yes—
159C22×C127C2511yescharacteristic
160C2×D127—23no—
161C2×D127—23no—
162C2×D127—23no—
163C4×C127—21no—
164C4×C127—23no—
165C4×C127—23no—
166C4×C127—21no—
167C4×C127C2×#4116yes—
168Dic127—21no—
169Dic127—23no—
170Dic127—23no—
171Dic127—21no—
172Dic127—23no—
173Dic127C2×#4112yes—
174Dic127C2×#4112yes—
175C2×C127—22no—
176C2×C127—43no—
177C2×C127C22×#4111yescharacteristic
178C2×C127C22×#4111yescharacteristic
179C2×C127#58511yescharacteristic
180D127—43no—
181#3480—1271nomaximal, Sylow
182C127#348011yescharacteristic, complemented, Sylow
183#559—1273no—
184#557—1271no—
185#556—1273no—
186#553—1273no—
187#553—1273no—
188#550—1271no—
189#548—1271no—
190#548D12711yescharacteristic, complemented
191#532—1273no—
192#529—1273no—
193#529—1273no—
194C4×#41—1271no—
195C4×#41—1271no—
196C4×#41—1273no—
197C2×C4×Q8—1271no—
198C2×#41—1271no—
199C2×#41—1273no—
200C2×#41C2×D12711yescharacteristic
201C22×Q8C2×D12711yescharacteristic
202C4⋊Q8—1273no—
203#127—1273no—
204#126—1276no—
205#126—1276no—
206#125—1273no—
207#125—1273no—
208#124—1273no—
209#124—1276no—
210#124C2×D12713yes—
211#123—1276no—
212#123—1276no—
213C22⋊Q8—1273no—
214C22⋊Q8C2×D12713yes—
215#121—1273no—
216#121—1276no—
217#121—1273no—
218#121—1273no—
219#121C2×D12713yes—
220C4×Q8—1271no—
221C4×Q8—1273no—
222C4×Q8—1271no—
223C4×Q8—1273no—
224C4×Q8—1271no—
225C4×Q8—1273no—
226C4×Q8—1273no—
227C4×Q8C2×D12711yescharacteristic
228C4×D4—1273no—
229C4×D4—1273no—
230C4×D4—1273no—
231C4×D4—1273no—
232C4×D4—1273no—
233C4×D4—1273no—
234C4×D4—1273no—
235C4×D4—1273no—
236C4×D4—1276no—
237C4×D4—1273no—
238C4×D4C2×D12713yes—
239#117—1273no—
240#117—1273no—
241#117—1273no—
242#117—1271no—
243#117—1273no—
244C2×C4⋊C4—1273no—
245C2×C42—1273no—
246C2×C42—1273no—
247C2×C42—1273no—
248#41—24no—
249#41—25412no—
250#41—2544no—
251C2×Q8—24no—
252C2×Q8—1273no—
253C2×Q8C22×D12713yes—
254C2×Q8C22×D12711yescharacteristic
255C2×D4—1276no—
256C2×D4—1273no—
257C2×D4—1273no—
258C2×D4—1273no—
259C2×D4C22×D12713yes—
260C2×D4C22×D12713yes—
261C22×C4—1271no—
262C22×C4—1273no—
263C22×C4—1273no—
264C22×C4—1273no—
265C22×C4—1273no—
266C22×C4—1273no—
267C22×C4—1273no—
268C22×C4C22×D12713yes—
269C22×C4C22×D12713yes—
270C4⋊C4—1271no—
271C4⋊C4—1271no—
272C4⋊C4—1273no—
273C4⋊C4—1273no—
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—1271no—
283C22⋊C4—1273no—
284C22⋊C4—1276no—
285C22⋊C4—1273no—
286C22⋊C4—1276no—
287C22⋊C4—1273no—
288C22⋊C4—1276no—
289C22⋊C4C22×D12711yescharacteristic
290C22⋊C4C22×D12713yes—
291C22⋊C4C22×D12716yes—
292C42—1271no—
293C42—1271no—
294C42—1273no—
295C42—1273no—
296C42—1273no—
297C42—1273no—
298C42—1273no—
299C42—1273no—
300C42—1273no—
301C42C22×D12713yes—
302C23—1273no—
303C23C23×D12713yes—
304C23C23×D12711yescharacteristic
305Q8—26no—
306Q8—2546no—
307Q8#1452324714yescomplemented
308D4—26no—
309D4—26no—
310D4—5086no—
311D4—2546no—
312D4—2546no—
313D4—2546no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—1274no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—1271no—
320C2×C4—1271no—
321C2×C4—1273no—
322C2×C4—1273no—
323C2×C4—1273no—
324C2×C4—5083no—
325C2×C4—5083no—
326C2×C4—5083no—
327C2×C4—2546no—
328C2×C4—2546no—
329C2×C4—2546no—
330C2×C4C23×D12711yescharacteristic
331C2×C4C23×D12713yes—
332C2×C4C23×D12711yescharacteristic
333C2×C4C23×D12713yes—
334C2×C4C23×D12713yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—2543no—
341C22—2543no—
342C22—2543no—
343C22#41×D12712yes—
344C22C24×D12711yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—1272no—
350C4—1272no—
351C4—2543no—
352C4—2541no—
353C4—2543no—
354C4—2541no—
355C4—2543no—
356C4C2×#1452324716yes—
357C2—22no—
358C2—43no—
359C2—5083no—
360C2C2×#41×D12711yescharacteristic
361C2C22×#1452324711yescharacteristic
362C2#2255071311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#548×C127C2
hypercenter#548D127
hypocenterC127#3480
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22550608C2semidirect
2. #22550602C2semidirect
3. #22550597C2semidirect
4. #22550578C2semidirect
5. #22550572C2semidirect
6. #22550545C2semidirect
7. #22550537C2semidirect
8. #22550421C2semidirect
9. #22550412C2semidirect
10. #22550405C2semidirect
11. #41×Dic127C2semidirect
12. C4×#14523249C2semidirect
13. C4×#14523247C2semidirect
14. C2×Q8×Dic127C2semidirect
15. #548×C127C2semidirect
129. C2×Dic127#41semidirect
143. Q8×C127#41semidirect
182. C127#3480semidirect
190. #548D127semidirect
307. Q8#14523247semidirect

Maximal subgroups

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

subgroupindex
#225506082
#225506022
#225505972
#225505782
#225505722
#225505452
#225505372
#225504212
#225504122
#225504052
#41×Dic1272
C4×#145232492
C4×#145232472
C2×Q8×Dic1272
#548×C1272
#3480127