Finite Group Databasebeta

← #30650296

Subgroups — #30650296

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523238C211yesmaximal, characteristic, complemented
3D4×Dic254C211yesmaximal, characteristic, complemented
4#22550271C211yesmaximal, characteristic, complemented
5#22550132C211yesmaximal, characteristic, complemented
6#22550103C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550086C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550032C211yesmaximal, characteristic, complemented
11#22550031C211yesmaximal, characteristic, complemented
12#22550014C211yesmaximal, characteristic, complemented
13#22549981C211yesmaximal, characteristic, complemented
14#22549948C211yesmaximal, characteristic, complemented
15#548×C127C211yesmaximal, characteristic
16C22×Dic254C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18#16861239—24no—
19C2×#14523238—24no—
20C2×#14523238C2211yescharacteristic
21C2×#14523238C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523236C2211yescharacteristic
29C2×#14523232C2211yescharacteristic
30C22×C127⋊C8C2211yescharacteristic
31#16861209C2211yescharacteristic
32#16861208—21no—
33#16861188C2211yescharacteristic
34#16861188C2211yescharacteristic
35#16861186—21no—
36#16861186—21no—
37C4×Dic254C2211yescharacteristic
38D4×Dic127—21no—
39#16861140C2211yescharacteristic
40C4⋊(C127⋊C8)C2211yescharacteristic
41#16861133C2211yescharacteristic
42#16861133C2211yescharacteristic
43#16861132C2211yescharacteristic
44#16861132C2211yescharacteristic
45#16861132C2211yescharacteristic
46#16861130C2211yescharacteristic
47C22⋊(C127⋊C8)C2211yescharacteristic
48C22⋊(C127⋊C8)C2211yescharacteristic
49#16861120C2211yescharacteristic
50C2×#41×C127C2211yescharacteristic
51C22×Q8×C127C2211yescharacteristic
52#124×C127—21no—
53#124×C127C2211yescharacteristic
54C22⋊Q8×C127—21no—
55C22⋊Q8×C127C2211yescharacteristic
56#121×C127—21no—
57#121×C127C2211yescharacteristic
58C4×Q8×C127C2211yescharacteristic
59C4×D4×C127—21no—
60C4×D4×C127C2211yescharacteristic
61C2×Dic254—22no—
62C2×Dic254—22no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254—41no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C22×Dic127—21no—
71C22×Dic127—21no—
72#14523238—22no—
73#14523238—22no—
74#14523238—22no—
75#14523238—22no—
76#14523238—41no—
77#14523238D414yescomplemented
78#14523236—22no—
79#14523236—22no—
80#14523236—41no—
81#14523232—22no—
82C2×C127⋊C8—22no—
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C2×C127⋊C8C2311yescharacteristic
87C4⋊Dic127—21no—
88C4⋊Dic127—21no—
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C4×Dic127—21no—
96#41×C127—22no—
97#41×C127—41no—
98C2×Q8×C127—21no—
99C2×Q8×C127—22no—
100C2×Q8×C127—41no—
101C2×Q8×C127C2311yescharacteristic
102C2×Q8×C127C2311yescharacteristic
103C2×D4×C127—21no—
104C2×D4×C127—21no—
105C2×D4×C127D411yescharacteristic
106C2×D4×C127C2311yescharacteristic
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127D411yescharacteristic
114C4⋊C4×C127C2311yescharacteristic
115C22⋊C4×C127—21no—
116C22⋊C4×C127—21no—
117C22⋊C4×C127—21no—
118C22⋊C4×C127—21no—
119C22⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123Dic254—21no—
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129Dic254—42no—
130Dic254C2×D412yes—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—42no—
136C2×Dic127—42no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C127⋊C8—21no—
140C127⋊C8C2×D412yes—
141C23×C127—21no—
142C23×C127C2×D411yescharacteristic
143C23×C127C2×D411yescharacteristic
144Q8×C127—21no—
145Q8×C127—21no—
146Q8×C127—21no—
147Q8×C127—42no—
148Q8×C127C2×D412yes—
149D4×C127—22no—
150D4×C127—21no—
151D4×C127—21no—
152D4×C127—42no—
153D4×C127—42no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—42no—
162C2×C4×C127—42no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2×D411yescharacteristic
167C2×C4×C127C2411yescharacteristic
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127#13712yescomplemented
177C22×C127C22×D411yescharacteristic
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—22no—
181C4×C127—21no—
182C4×C127—21no—
183C4×C127—41no—
184C4×C127—41no—
185C4×C127C22×D411yescharacteristic
186C4×C127H(5,𝔽2)11yescharacteristic
187Dic127—22no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—22no—
192C2×C127—41no—
193C2×C127—81no—
194C2×C127D4211yescharacteristic
195C2×C127C2×#13711yescharacteristic
196C2×C127#57811yescharacteristic
197#3270—1271nomaximal, Sylow
198C127#327011yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C22×Q16—1271no—
201D4×Q8—1271no—
202#548D12711yescharacteristic
203#497—1271no—
204#470—1271no—
205#465—1271no—
206#464—1271no—
207#462—1271no—
208#452—1271no—
209#451—1271no—
210#451—1271no—
211#448—1271no—
212#441—1271no—
213#435—1271no—
214C2×#41C2×D12711yescharacteristic
215C22×Q8—1271no—
216C22×Q8—1271no—
217C22×Q8C2×D12711yescharacteristic
218#137—2544no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16—1271no—
225C2×Q16—2544no—
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16—1271no—
229C2×M16—1271no—
230C22×C8—1271no—
231C4⋊Q8—1271no—
232C4⋊Q8—2541no—
233#124—21no—
234#124C2×D12711yescharacteristic
235C22⋊Q8—21no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—2541no—
239C22⋊Q8—2541no—
240C22⋊Q8C2×D12711yescharacteristic
241#121—21no—
242#121C2×D12711yescharacteristic
243C4×Q8—1271no—
244C4×Q8C2×D12711yescharacteristic
245C4×D4—21no—
246C4×D4—2541no—
247C4×D4C2×D12711yescharacteristic
248#106—1271no—
249C4⋊C8—1271no—
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#102—1271no—
256C22⋊C8—1271no—
257C22⋊C8—1271no—
258#97—1271no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—1271no—
265C2×Q8—1271no—
266C2×Q8—1271no—
267C2×Q8—5081no—
268C2×Q8—5081no—
269C2×Q8—2542no—
270C2×Q8—2542no—
271C2×Q8—2541no—
272C2×Q8—2541no—
273C2×Q8C22×D12711yescharacteristic
274C2×Q8C22×D12711yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊D12711yescharacteristic
278C2×D4C22×D12711yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4C22×D12711yescharacteristic
284C22×C4C22×D12711yescharacteristic
285Q16—1274no—
286Q16—5081no—
287Q16—2542no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—5081no—
292SD16—2542no—
293SD16—2542no—
294M16—2542no—
295C2×C8—1271no—
296C2×C8—1271no—
297C2×C8—1271no—
298C2×C8—1271no—
299C2×C8—2542no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1271no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4C22⋊D12711yescharacteristic
309C4⋊C4C22×D12711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4C22⋊D12711yescharacteristic
317C22⋊C4C22⋊D12711yescharacteristic
318C42—21no—
319C42—2541no—
320C42C22×D12711yescharacteristic
321C23—21no—
322C23C2×C22⋊D12711yescharacteristic
323C23C2×C22⋊D12711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—42no—
328Q8—1272no—
329Q8—5082no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336Q8D4×D12712yes—
337D4—22no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4C2×C22⋊D12711yescharacteristic
358C2×C4C2×C22⋊D12711yescharacteristic
359C2×C4C2×C22⋊D12711yescharacteristic
360C2×C4C2×C22⋊D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—1272no—
363C8—2541no—
364C8—2541no—
365C8—2541no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22#1686123912yescomplemented
375C22C22×C22⋊D12711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—22no—
390C2—41no—
391C2—81no—
392C2D4×C22⋊D12711yescharacteristic
393C2C2×#1686123911yescharacteristic
394C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#548×C127C2
socleC22×C127C22×D4
hypercenter#548D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C22×#14523238C2semidirect
3. D4×Dic254C2semidirect
4. #22550271C2semidirect
5. #22550132C2semidirect
6. #22550103C2semidirect
7. #22550095C2semidirect
8. #22550086C2semidirect
9. #22550038C2semidirect
10. #22550032C2semidirect
11. #22550031C2semidirect
12. #22550014C2semidirect
13. #22549981C2semidirect
14. #22549948C2semidirect
77. #14523238D4semidirect
176. C22×C127#137semidirect
198. C127#3270semidirect
374. C22#16861239semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232382
D4×Dic2542
#225502712
#225501322
#225501032
#225500952
#225500862
#225500382
#225500322
#225500312
#225500142
#225499812
#225499482
#548×C1272
#3270127