Finite Group Databasebeta

← #30650299

Subgroups — #30650299

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×#16861238C211yesmaximal, characteristic, complemented
2C2×Q16×D127C211yesmaximal, characteristic, complemented
3#22550541C211yesmaximal, characteristic, complemented
4#22550534C211yesmaximal, characteristic
5#22550131C211yesmaximal, characteristic, complemented
6#22550102C211yesmaximal, characteristic, complemented
7#22550095C211yesmaximal, characteristic, complemented
8#22550083C211yesmaximal, characteristic, complemented
9#22550035C211yesmaximal, characteristic, complemented
10#22550030C211yesmaximal, characteristic, complemented
11#22550030C211yesmaximal, characteristic, complemented
12#22550013C211yesmaximal, characteristic, complemented
13#22549984C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#497×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861238—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25Q16×D127—24no—
26C2×#14523236C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861208—21no—
30#16861195—21no—
31#16861187—21no—
32#16861187—21no—
33#16861186—21no—
34#16861185C2211yescharacteristic
35#16861185C2211yescharacteristic
36#16861185C2211yescharacteristic
37#16861182—21no—
38#16861180C2211yescharacteristic
39C4×Dic254C2211yescharacteristic
40C4×Dic254C2211yescharacteristic
41C4×D508C2211yescharacteristic
42#16861171—21no—
43(C22⋊D127)⋊C4—21no—
44#16861141C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861133C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861130C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53(C2×D127)⋊C8C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic
58C4⋊Q8×C127C2211yescharacteristic
59#124×C127C2211yescharacteristic
60#97×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254—21no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68Q8×D127—22no—
69Q8×D127—22no—
70Q8×D127—22no—
71Q8×D127—41no—
72Q8×D127—41no—
73Q8×D127—41no—
74C2×D508D411yescharacteristic
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C22×Dic127—21no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80C2×C4×D127—21no—
81C2×C4×D127C2311yescharacteristic
82C2×C4×D127C2311yescharacteristic
83Dic508D414yescomplemented
84#14523238—22no—
85#14523238—22no—
86#14523238—22no—
87#14523236—22no—
88#14523231—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
91C8×D127—22no—
92C4⋊Dic127D411yescharacteristic
93C4⋊Dic127C2311yescharacteristic
94C4⋊Dic127C2311yescharacteristic
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100C22⋊Dic127—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4D411yescharacteristic
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127—21no—
108C2×Q8×C127C2311yescharacteristic
109C2×Q8×C127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127C2311yescharacteristic
112Q16×C127—22no—
113Q16×C127—41no—
114SD16×C127—22no—
115SD16×C127—41no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127C2×D411yescharacteristic
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—42no—
128Dic254—42no—
129Dic254—42no—
130Dic254C2×D412yes—
131Dic254C2×D412yes—
132D508—22no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—21no—
143C4×D127—21no—
144C4×D127—21no—
145C4×D127—21no—
146C4×D127—42no—
147C4×D127—42no—
148C4×D127—42no—
149C127⋊C8—21no—
150C127⋊C8—21no—
151C23×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—21no—
155Q8×C127—21no—
156Q8×C127—21no—
157Q8×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127C2×D412yes—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127C22×D411yescharacteristic
172C2×D127—21no—
173C2×D127—21no—
174C2×D127—21no—
175C2×D127—21no—
176C2×D127—21no—
177C2×D127#13712yescomplemented
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127C22×D411yescharacteristic
184C4×C127H(5,𝔽2)11yescharacteristic
185Dic127—21no—
186Dic127—21no—
187Dic127—22no—
188Dic127—22no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—81no—
192C2×C127D4211yescharacteristic
193C2×C127C2×#13711yescharacteristic
194C2×C127#57811yescharacteristic
195D127—22no—
196D127—41no—
197#3270—1271nomaximal, Sylow
198C127#327011yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C22×Q16—1271no—
201D4×Q8—1271no—
202#548—1271no—
203#497D12711yescharacteristic, complemented
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×#41—1271no—
215C22×Q8—1271no—
216C22×Q8—1271no—
217C22×Q8—1271no—
218#137—2544no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—2544no—
224C2×Q16C2×D12711yescharacteristic
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16C2×D12711yescharacteristic
228C2×SD16C2×D12711yescharacteristic
229C2×M16—1271no—
230C22×C8—1271no—
231C4⋊Q8—2541no—
232C4⋊Q8C2×D12711yescharacteristic
233#124—2541no—
234#124C2×D12711yescharacteristic
235C22⋊Q8—1271no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—2541no—
239C22⋊Q8—2541no—
240C22⋊Q8—2541no—
241#121—1271no—
242#121—2541no—
243C4×Q8—1271no—
244C4×Q8—1271no—
245C4×D4—1271no—
246C4×D4—2541no—
247C4×D4—2541no—
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#97C2×D12711yescharacteristic
259#41—5081no—
260#41—2542no—
261C2×Q8—1271no—
262C2×Q8—1271no—
263C2×Q8—5081no—
264C2×Q8—5081no—
265C2×Q8—5081no—
266C2×Q8—2541no—
267C2×Q8—2541no—
268C2×Q8—2541no—
269C2×Q8—2542no—
270C2×Q8—2542no—
271C2×Q8—2542no—
272C2×Q8C22×D12711yescharacteristic
273C2×Q8C22×D12711yescharacteristic
274C2×Q8C22×D12711yescharacteristic
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4—2541no—
278C2×D4C22×D12711yescharacteristic
279C22×C4—1271no—
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4—2541no—
285Q16—22no—
286Q16—41no—
287Q16—1274no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—22no—
292SD16—41no—
293SD16—2542no—
294M16—2542no—
295C2×C8—1271no—
296C2×C8—1271no—
297C2×C8—2542no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—1271no—
303C4⋊C4—1271no—
304C4⋊C4—1271no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1271no—
313C22⋊C4—1271no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C42—2541no—
319C42—2541no—
320C42C22×D12711yescharacteristic
321C23—21no—
322C23—1271no—
323C23—1271no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—1272no—
331Q8—1272no—
332Q8—5082no—
333Q8—5082no—
334Q8—5082no—
335Q8—2541no—
336Q8—2541no—
337D4—21no—
338D4—21no—
339D4—5082no—
340D4—5082no—
341D4—2542no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—1271no—
346C2×C4—1271no—
347C2×C4—5082no—
348C2×C4—5082no—
349C2×C4—5082no—
350C2×C4—5082no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4D4×D12711yescharacteristic
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—2541no—
364C8—2541no—
365C8D4×D12712yes—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—1272no—
370C22—2541no—
371C22—2541no—
372C22—2541no—
373C22—2541no—
374C22—2541no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4—2542no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—81no—
390C2—5081no—
391C2—2542no—
392C2#2255051111yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#497×C127C2
socleC22×C127C22×D4
prime radical#22550534C2
hypercenter#497D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C2×Q16×D127C2semidirect
3. #22550541C2semidirect
5. #22550131C2semidirect
6. #22550102C2semidirect
7. #22550095C2semidirect
8. #22550083C2semidirect
9. #22550035C2semidirect
10. #22550030C2semidirect
11. #22550030C2semidirect
12. #22550013C2semidirect
13. #22549984C2semidirect
14. #22549949C2semidirect
15. #497×C127C2semidirect
83. Dic508D4semidirect
177. C2×D127#137semidirect
198. C127#3270semidirect
203. #497D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×Q16×D1272
#225505412
#225505342
#225501312
#225501022
#225500952
#225500832
#225500352
#225500302
#225500132
#225499842
#225499492
#497×C1272
#3270127