Finite Group Databasebeta

← #30650244

Subgroups — #30650244

Order 16256.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3#22550539C211yesmaximal, characteristic, complemented
4#22550520C211yesmaximal, characteristic
5#22550267C211yesmaximal, characteristic, complemented
6#22550121C211yesmaximal, characteristic, complemented
7#22550115C211yesmaximal, characteristic, complemented
8#22550097C211yesmaximal, characteristic, complemented
9#22550079C211yesmaximal, characteristic, complemented
10#22550039C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12#22550013C211yesmaximal, characteristic, complemented
13#22549973C211yesmaximal, characteristic, complemented
14#22549944C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic, complemented
19#16861238—24no—
20C2×Dic508C2211yescharacteristic, complemented
21C2×Dic508C2211yescharacteristic, complemented
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523236—24no—
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523231C2211yescharacteristic, complemented
29C22×C127⋊C8C2211yescharacteristic
30#16861208—21no—
31#16861207C2211yescharacteristic, complemented
32#16861196C2211yescharacteristic
33#16861190—21no—
34#16861189C2211yescharacteristic, complemented
35#16861188C2211yescharacteristic
36#16861187—21no—
37#16861186—21no—
38#16861181—21no—
39#16861180C2211yescharacteristic
40#16861178—21no—
41#16861174C2211yescharacteristic, complemented
42D4×Dic127C2211yescharacteristic
43C4×C22⋊D127C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C2×C22⋊Dic127—21no—
46#16861142C2211yescharacteristic, complemented
47Dic127⋊C8C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861133C2211yescharacteristic, complemented
51#16861132C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861119C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×SD16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254—22no—
65C2×Dic254—21no—
66C2×Dic254—41no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70Q8×D127—22no—
71Q8×D127—41no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C22×Dic127—21no—
76C22×Dic127—21no—
77C22×Dic127—21no—
78C2×C4×D127—21no—
79C2×C4×D127C2311yescharacteristic
80Dic508—22no—
81Dic508—41no—
82#14523238—22no—
83#14523236—22no—
84#14523236—22no—
85#14523236—22no—
86#14523236D414yescomplemented
87#14523231—22no—
88C2×C127⋊C8—22no—
89C2×C127⋊C8C2311yescharacteristic
90C2×C127⋊C8C2311yescharacteristic
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⋊Dic127—21no—
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108C24×C127—21no—
109C2×Q8×C127C2311yescharacteristic
110C2×D4×C127—22no—
111C2×D4×C127—21no—
112C2×D4×C127—41no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116SD16×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—
131Dic254C2×D412yes—
132C22⋊D127—22no—
133C22⋊D127—42no—
134C22⋊D127—81no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—42no—
140C2×Dic127—42no—
141C2×Dic127—81no—
142C2×Dic127C2×D411yescharacteristic
143C2×Dic127C2×D411yescharacteristic
144C2×Dic127C2×D411yescharacteristic
145C4×D127—21no—
146C4×D127—21no—
147C4×D127—42no—
148C127⋊C8—21no—
149C127⋊C8C2×D412yes—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—42no—
153C23×C127—42no—
154C23×C127—81no—
155C23×C127C2×D411yescharacteristic
156Q8×C127—21no—
157Q8×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—21no—
161D4×C127—42no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—22no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—84no—
180C22×C127#13712yescomplemented
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127C22×D411yescharacteristic
189C4×C127H(5,𝔽2)11yescharacteristic
190Dic127—21no—
191Dic127—22no—
192Dic127—21no—
193Dic127—21no—
194Dic127—41no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127—42no—
199C2×C127—81no—
200C2×C127D4211yescharacteristic
201C2×C127C2×#13711yescharacteristic
202C2×C127#57711yescharacteristic
203D127—41no—
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16—1271no—
208D4×Q8—1271no—
209#546—1271no—
210#497—1271no—
211#468—1271no—
212#467—1271no—
213#464—1271no—
214#461—1271no—
215#452—1271no—
216#451—1271no—
217#450D12711yescharacteristic, complemented
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41—1271no—
222C22×Q8—1271no—
223C22×Q8—1271no—
224C22×D4C2×D12711yescharacteristic
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16—1271no—
233C2×SD16—2544no—
234C2×SD16C2×D12711yescharacteristic
235C2×SD16C2×D12711yescharacteristic
236C2×M16—1271no—
237C22×C8—1271no—
238C4⋊Q8—1271no—
239C4⋊Q8—2541no—
240#124—1271no—
241#123—2541no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—2541no—
245C22⋊Q8—2541no—
246C22⋊Q8C2×D12711yescharacteristic
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8—1271no—
251C4×D4—1271no—
252C4×D4—1271no—
253C4×D4—2541no—
254C2×C22⋊C4—2541no—
255#107—1271no—
256C4⋊C8—1271no—
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103—1271no—
261#102C2×D12711yescharacteristic
262#102C2×D12711yescharacteristic
263C22⋊C8—1271no—
264C22⋊C8C2×D12711yescharacteristic
265#97—1271no—
266C24—21no—
267#41—5081no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—1271no—
272C2×Q8—5081no—
273C2×Q8—5081no—
274C2×Q8—2541no—
275C2×Q8—2542no—
276C2×Q8—2542no—
277C2×Q8—2541no—
278C2×Q8C22×D12711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1271no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4C22×D12711yescharacteristic
286C2×D4C22×D12711yescharacteristic
287C22×C4—1271no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4—2541no—
292C22×C4C22×D12711yescharacteristic
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—22no—
297SD16—41no—
298SD16—1274no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302M16—2542no—
303C2×C8—1271no—
304C2×C8—1271no—
305C2×C8—2542no—
306C2×C8C22×D12711yescharacteristic
307C2×C8C22×D12711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—1271no—
310C4⋊C4—1271no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4C22×D12711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1271no—
319C22⋊C4—1271no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C42—1271no—
327C42—2541no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1271no—
334C23D4×D12711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1272no—
338Q8—5082no—
339Q8—5082no—
340Q8—2541no—
341Q8—2541no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—10161no—
350D4—5082no—
351D4—2542no—
352D4D4×D12712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—1271no—
358C2×C4—1271no—
359C2×C4—1271no—
360C2×C4—10161no—
361C2×C4—5082no—
362C2×C4—5082no—
363C2×C4—5082no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4D4×D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—21no—
373C8—21no—
374C8—1272no—
375C8—2541no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2541no—
386C22—2541no—
387C22—2541no—
388C22#1686123812yescomplemented
389C22C2×D4×D12711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—5081no—
394C4—5081no—
395C4—5081no—
396C4—2541no—
397C4—2541no—
398C4—2541no—
399C4—2542no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—5081no—
406C2#2255050811yescharacteristic
407C2C2×#1686123811yescharacteristic
408C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radical#22550520C2
hypercenter#450D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C22×#14523236C2semidirect
3. #22550539C2semidirect
5. #22550267C2semidirect
6. #22550121C2semidirect
7. #22550115C2semidirect
8. #22550097C2semidirect
9. #22550079C2semidirect
10. #22550039C2semidirect
11. #22550034C2semidirect
12. #22550013C2semidirect
13. #22549973C2semidirect
14. #22549944C2semidirect
15. #450×C127C2semidirect
18. C2×Q8×D127C22semidirect
20. C2×Dic508C22semidirect
21. C2×Dic508C22semidirect
22. C2×#14523238C22semidirect
28. C2×#14523231C22semidirect
31. #16861207C22semidirect
34. #16861189C22semidirect
41. #16861174C22semidirect
46. #16861142C22semidirect
47. Dic127⋊C8C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861133C22semidirect
51. #16861132C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861119C22semidirect
86. #14523236D4semidirect
180. C22×C127#137semidirect
205. C127#3266semidirect
217. #450D127semidirect
388. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C22×#145232362
#225505392
#225505202
#225502672
#225501212
#225501152
#225500972
#225500792
#225500392
#225500342
#225500132
#225499732
#225499442
#450×C1272
#3266127