Finite Group Databasebeta

← #30652806

Subgroups — #30652806

Order 16256.

1104 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#22550608C213yesmaximal, complemented
2Q8×Dic254C211yesmaximal, characteristic, complemented
3#22550584C213yesmaximal, complemented
4#22550545C211yesmaximal, characteristic, complemented
5#22550537C211yesmaximal, characteristic, complemented
6#22550530C213yesmaximal, complemented
7#22550526C213yesmaximal, complemented
8#22550499C213yesmaximal, complemented
9#22550495C213yesmaximal, complemented
10#22550421C213yesmaximal, complemented
11#22550406C213yesmaximal, complemented
12#22550359C211yesmaximal, characteristic, complemented
13#22550358C211yesmaximal, characteristic, complemented
14C2×Q8×Dic127C211yesmaximal, characteristic, complemented
15#548×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523247C2216yes—
18#16861209C2213yes—
19#16861208C2213yes—
20#16861208C2213yes—
21#16861205C2213yes—
22#16861204C2213yes—
23#16861202C2213yes—
24#16861199C2213yes—
25#16861198C2213yes—
26#16861195C2213yes—
27#16861195C2216yes—
28#16861193C2213yes—
29#16861193C2216yes—
30#16861191C2216yes—
31#16861190C2213yes—
32#16861189C2216yes—
33#16861186C2216yes—
34#16861186C2211yescharacteristic
35#16861185C2213yes—
36#16861183C2213yes—
37#16861183C2213yes—
38#16861181C2213yes—
39#16861180C2213yes—
40Q8×Dic127C2211yescharacteristic
41Q8×Dic127C2211yescharacteristic
42Q8×Dic127C2213yes—
43Q8×Dic127C2211yescharacteristic
44Q8×Dic127C2213yes—
45C4×Dic254C2213yes—
46C4×D508C2213yes—
47D4×Dic127C2213yes—
48D4×Dic127C2213yes—
49#16861171C2213yes—
50C4×C22⋊D127C2213yes—
51C4×C22⋊D127C2213yes—
52(C22⋊D127)⋊C4C2216yes—
53#16861168C2213yes—
54#16861166C2213yes—
55C2×C4⋊Dic127C2213yes—
56C2×C4⋊Dic127C2213yes—
57C4⋊C4×D127C2213yes—
58C2×C4×Dic127C2213yes—
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—412no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2316yes—
70C2×D508C2313yes—
71C2×C22⋊D127C2313yes—
72C2×C22⋊D127C2313yes—
73C2×C22⋊D127C2316yes—
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2313yes—
76C22×Dic127C2316yes—
77C2×C4×D127C2313yes—
78C2×C4×D127C2316yes—
79C4⋊Dic127C2313yes—
80C4⋊Dic127C2316yes—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2313yes—
83C4⋊Dic127C2313yes—
84C4⋊Dic127C2313yes—
85C4⋊Dic127C2313yes—
86Dic127⋊C4C2313yes—
87Dic127⋊C4C2313yes—
88Dic127⋊C4C2311yescharacteristic
89Dic127⋊C4C2311yescharacteristic
90Dic127⋊C4C2313yes—
91Dic127⋊C4C2313yes—
92C22⋊Dic127C2313yes—
93C22⋊Dic127C2311yescharacteristic
94C22⋊Dic127C2313yes—
95C22⋊Dic127C2313yes—
96(C2×D127)⋊C4C2316yes—
97(C2×D127)⋊C4C2313yes—
98(C2×D127)⋊C4C2313yes—
99(C2×D127)⋊C4C2313yes—
100(C2×D127)⋊C4C2313yes—
101C4×Dic127C2313yes—
102C4×Dic127C2313yes—
103C4×Dic127C2313yes—
104C4×Dic127C2313yes—
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×C127C2311yescharacteristic
118C22⋊C4×C127C2313yes—
119C42×C127C2313yes—
120C22×D127C2413yes—
121Dic254—22no—
122Dic254—46no—
123D508—26no—
124C22⋊D127—26no—
125C22⋊D127—26no—
126C22⋊D127—46no—
127C2×Dic127—26no—
128C2×Dic127—22no—
129C2×Dic127—46no—
130C2×Dic127C2411yescharacteristic
131C2×Dic127C2411yescharacteristic
132C2×Dic127C2413yes—
133C2×Dic127C2413yes—
134C2×Dic127C2413yes—
135C2×Dic127C2411yescharacteristic
136C2×Dic127C2411yescharacteristic
137C4×D127—26no—
138C4×D127—46no—
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×C127H(5,𝔽2)12yes—
158C22×C127C2511yescharacteristic
159C2×D127—23no—
160C2×D127—23no—
161C2×D127—23no—
162C4×C127—21no—
163C4×C127—23no—
164C4×C127—23no—
165C4×C127—21no—
166C4×C127C2×#4116yes—
167Dic127—21no—
168Dic127—23no—
169Dic127—23no—
170Dic127—23no—
171Dic127—21no—
172Dic127—21no—
173Dic1272−1+412yes—
174C2×C127—22no—
175C2×C127—43no—
176C2×C127C22×#4111yescharacteristic
177C2×C127C2×H(5,𝔽2)11yescharacteristic
178C2×C127C2×2−1+411yescharacteristic
179D127—43no—
180#3522—1271nomaximal, Sylow
181C127#352211yescharacteristic, complemented, Sylow
182#559—1273no—
183Q82—1271no—
184#554—1273no—
185#550—1271no—
186#548—1271no—
187#548D12711yescharacteristic, complemented
188#547—1273no—
189#547—1273no—
190#542—1273no—
191#542—1273no—
192#532—1273no—
193#529—1273no—
194#520—1271no—
195#520—1271no—
196C2×C4×Q8—1271no—
197C2×#41—1271no—
198C2×#41—1276no—
199C2×#41C2×D12711yescharacteristic
200C22×Q8C2×D12711yescharacteristic
201C4⋊Q8—1273no—
202C4⋊Q8—1273no—
203C4⋊Q8—1273no—
204#127—1273no—
205#126—1273no—
206#126—1273no—
207#125—1273no—
208#124—1273no—
209#124—1273no—
210#124—1276no—
211#124C2×D12713yes—
212#123—1273no—
213#123—1276no—
214#123—1273no—
215#123—1276no—
216C22⋊Q8—1273no—
217C22⋊Q8—1276no—
218C22⋊Q8—1271no—
219C22⋊Q8—1276no—
220C22⋊Q8C2×D12713yes—
221#121—1273no—
222#121—1273no—
223#121—1273no—
224#121—1273no—
225#121C2×D12713yes—
226C4×Q8—1271no—
227C4×Q8—1273no—
228C4×Q8—1271no—
229C4×Q8—1273no—
230C4×Q8—1271no—
231C4×Q8—1273no—
232C4×Q8C2×D12711yescharacteristic
233C4×D4—1273no—
234C4×D4—1273no—
235C4×D4—1273no—
236C4×D4—1273no—
237C4×D4—1273no—
238C4×D4—1276no—
239C4×D4—1273no—
240C4×D4C2×D12713yes—
241#117—1273no—
242#117—1273no—
243C2×C4⋊C4—1273no—
244C2×C4⋊C4—1273no—
245C2×C4⋊C4—1273no—
246C2×C42—1273no—
247#41—24no—
248#41—50812no—
249#41—2544no—
250C2×Q8—24no—
251C2×Q8—1276no—
252C2×Q8—1271no—
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—1276no—
262C22×C4—1271no—
263C22×C4—1273no—
264C22×C4—1276no—
265C22×C4—1273no—
266C22×C4C22×D12713yes—
267C22×C4C22×D12713yes—
268C4⋊C4—1273no—
269C4⋊C4—1271no—
270C4⋊C4—1271no—
271C4⋊C4—1273no—
272C4⋊C4—1273no—
273C4⋊C4—1273no—
274C4⋊C4—1273no—
275C4⋊C4—1273no—
276C4⋊C4—1273no—
277C4⋊C4—1273no—
278C4⋊C4—1276no—
279C4⋊C4—1271no—
280C4⋊C4—1273no—
281C4⋊C4C22×D12713yes—
282C4⋊C4C22×D12713yes—
283C22⋊C4—1273no—
284C22⋊C4—1273no—
285C22⋊C4—1271no—
286C22⋊C4—1273no—
287C22⋊C4—1273no—
288C22⋊C4—1273no—
289C22⋊C4—1273no—
290C22⋊C4—1276no—
291C22⋊C4—1273no—
292C22⋊C4C22×D12716yes—
293C22⋊C4C22×D12711yescharacteristic
294C22⋊C4C22×D12713yes—
295C42—1273no—
296C42—1273no—
297C42—1273no—
298C42—1273no—
299C42—1273no—
300C42C22×D12713yes—
301C23—1273no—
302C23C23×D12713yes—
303C23C23×D12711yescharacteristic
304Q8—26no—
305Q8—5086no—
306Q8—2542no—
307Q8#1452324714yescomplemented
308D4—26no—
309D4—26no—
310D4—5086no—
311D4—2546no—
312D4—2546no—
313D4—2546no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—1271no—
317C2×C4—1271no—
318C2×C4—1271no—
319C2×C4—1273no—
320C2×C4—1273no—
321C2×C4—1273no—
322C2×C4—1271no—
323C2×C4—5086no—
324C2×C4—5086no—
325C2×C4—2542no—
326C2×C4—2546no—
327C2×C4—2546no—
328C2×C4C23×D12713yes—
329C2×C4C23×D12711yescharacteristic
330C2×C4C23×D12713yes—
331C2×C4C23×D12711yescharacteristic
332C2×C4C23×D12713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2543no—
339C22—2543no—
340C22—2543no—
341C22#1686125512yes—
342C22C24×D12711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—1272no—
348C4—2541no—
349C4—2543no—
350C4—2543no—
351C4—2543no—
352C4—2541no—
353C4—2541no—
354C4C2×#1452324716yes—
355C2—22no—
356C2—43no—
357C2—5083no—
358C2C22×#1452324711yescharacteristic
359C2C2×#1686125511yescharacteristic
360C22−1+4×D12711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
Fitting#548×C127C2
hypercenter#548D127
hypocenterC127#3522
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22550608C2semidirect
2. Q8×Dic254C2semidirect
3. #22550584C2semidirect
4. #22550545C2semidirect
5. #22550537C2semidirect
6. #22550530C2semidirect
7. #22550526C2semidirect
8. #22550499C2semidirect
9. #22550495C2semidirect
10. #22550421C2semidirect
11. #22550406C2semidirect
12. #22550359C2semidirect
13. #22550358C2semidirect
14. C2×Q8×Dic127C2semidirect
15. #548×C127C2semidirect
142. Q8×C127#41semidirect
181. C127#3522semidirect
187. #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
Q8×Dic2542
#225505842
#225505452
#225505372
#225505302
#225505262
#225504992
#225504952
#225504212
#225504062
#225503592
#225503582
C2×Q8×Dic1272
#548×C1272
#3522127