Finite Group Databasebeta

← #30650298

Subgroups — #30650298

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
2C2×Q16×D127C211yesmaximal, characteristic, complemented
3#22550539C211yesmaximal, characteristic, complemented
4#22550533C211yesmaximal, characteristic
5#22550267C211yesmaximal, characteristic, complemented
6#22550132C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550097C211yesmaximal, characteristic, complemented
9#22550084C211yesmaximal, characteristic, complemented
10#22550034C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550013C211yesmaximal, characteristic, complemented
13#22549980C211yesmaximal, characteristic, complemented
14#22549950C211yesmaximal, characteristic, complemented
15#452×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861239—24no—
20C2×Dic508C2211yescharacteristic
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25Q16×D127—24no—
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523232C2211yescharacteristic
29C2×C8×D127C2211yescharacteristic
30#16861208—21no—
31#16861207C2211yescharacteristic
32#16861196—21no—
33#16861196C2211yescharacteristic
34#16861189C2211yescharacteristic
35#16861188—21no—
36#16861188C2211yescharacteristic
37#16861187—21no—
38#16861186—21no—
39#16861180—21no—
40#16861180C2211yescharacteristic
41Q8×Dic127C2211yescharacteristic
42#16861174C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861140C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861134C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861132C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861119C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61C2×Dic254—21no—
62C2×Dic254—21no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—22no—
71Q8×D127—41no—
72Q8×D127—41no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C22×Dic127—21no—
76C2×C4×D127—21no—
77C2×C4×D127—21no—
78C2×C4×D127C2311yescharacteristic
79Dic508—22no—
80Dic508—41no—
81#14523238—22no—
82#14523238—22no—
83#14523238D414yescomplemented
84#14523236—22no—
85#14523236—22no—
86#14523232—22no—
87C2×C127⋊C8C2311yescharacteristic
88C2×C127⋊C8C2311yescharacteristic
89C8×D127—22no—
90C4⋊Dic127—21no—
91C4⋊Dic127—21no—
92C4⋊Dic127C2311yescharacteristic
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4D411yescharacteristic
98C22⋊Dic127—21no—
99C22⋊Dic127—21no—
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—22no—
117SD16×C127—41no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130Dic254—42no—
131Dic254—42no—
132Dic254C2×D412yes—
133C22⋊D127—22no—
134C22⋊D127—42no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C4×D127—21no—
144C4×D127—21no—
145C4×D127—42no—
146C4×D127—42no—
147C127⋊C8—21no—
148C127⋊C8C2×D412yes—
149C23×C127—21no—
150C23×C127C2×D411yescharacteristic
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127C2×D412yes—
155D4×C127—21no—
156D4×C127—21no—
157D4×C127—42no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—21no—
177C2×D127#13712yescomplemented
178C4×C127—21no—
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C22×D411yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—21no—
185Dic127—22no—
186Dic127—21no—
187Dic127—21no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—41no—
192C2×C127—81no—
193C2×C127D4211yescharacteristic
194C2×C127C2×#13711yescharacteristic
195C2×C127#57811yescharacteristic
196D127—22no—
197#3270—1271nomaximal, Sylow
198C127#327011yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C22×Q16—1271no—
201D4×Q8—1271no—
202#548—1271no—
203#497—1271no—
204#470—1271no—
205#465—1271no—
206#464—1271no—
207#462—1271no—
208#452D12711yescharacteristic, complemented
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×Q8—1271no—
218#137—2544no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16—2544no—
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16—1271no—
228C2×SD16C2×D12711yescharacteristic
229C2×M16—1271no—
230C22×C8—1271no—
231C4⋊Q8—1271no—
232C4⋊Q8—2541no—
233#124—1271no—
234#124—2541no—
235C22⋊Q8—1271no—
236C22⋊Q8—1271no—
237C22⋊Q8—2541no—
238C22⋊Q8—2541no—
239C22⋊Q8—2541no—
240C22⋊Q8C2×D12711yescharacteristic
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#103C2×D12711yescharacteristic
255#102C2×D12711yescharacteristic
256C22⋊C8—1271no—
257C22⋊C8C2×D12711yescharacteristic
258#97—1271no—
259#41—22no—
260#41—41no—
261C2×Q8—1271no—
262C2×Q8—1271no—
263C2×Q8—1271no—
264C2×Q8—5081no—
265C2×Q8—5081no—
266C2×Q8—5081no—
267C2×Q8—2541no—
268C2×Q8—2541no—
269C2×Q8—2542no—
270C2×Q8—2542no—
271C2×Q8—2542no—
272C2×Q8—2541no—
273C2×Q8C22×D12711yescharacteristic
274C2×Q8C22×D12711yescharacteristic
275C2×D4—21no—
276C2×D4—1271no—
277C2×D4—2541no—
278C2×D4C22×D12711yescharacteristic
279C22×C4—21no—
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—22no—
286Q16—1274no—
287Q16—5081no—
288Q16—2542no—
289Q16—2542no—
290Q16—2542no—
291SD16—41no—
292SD16—2542no—
293SD16—2542no—
294M16—2542no—
295C2×C8—1271no—
296C2×C8—1271no—
297C2×C8—2542no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—1271no—
302C4⋊C4—1271no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4C22×D12711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—1271no—
312C22⋊C4—1271no—
313C22⋊C4—2541no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C42—1271no—
319C42—2541no—
320C42—2541no—
321C23—21no—
322C23—1271no—
323C23D4×D12711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—1272no—
328Q8—5082no—
329Q8—5082no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336Q8D4×D12712yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—5082no—
341D4—2542no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—1271no—
349C2×C4—1271no—
350C2×C4—1271no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—5082no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—21no—
364C8—1272no—
365C8—2541no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1272no—
373C22—2541no—
374C22—2541no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—2541no—
384C4—2541no—
385C4—2541no—
386C4—2542no—
387C4C2×D4×D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—41no—
390C2—81no—
391C2—2542no—
392C2#2255050811yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#452×C127C2
socleC22×C127C22×D4
prime radical#22550533C2
hypercenter#452D127
hypocenterC127#3270
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×Q16×D127C2semidirect
3. #22550539C2semidirect
5. #22550267C2semidirect
6. #22550132C2semidirect
7. #22550104C2semidirect
8. #22550097C2semidirect
9. #22550084C2semidirect
10. #22550034C2semidirect
11. #22550033C2semidirect
12. #22550013C2semidirect
13. #22549980C2semidirect
14. #22549950C2semidirect
15. #452×C127C2semidirect
83. #14523238D4semidirect
177. C2×D127#137semidirect
198. C127#3270semidirect
208. #452D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×Q16×D1272
#225505392
#225505332
#225502672
#225501322
#225501042
#225500972
#225500842
#225500342
#225500332
#225500132
#225499802
#225499502
#452×C1272
#3270127