Finite Group Databasebeta

← #30650245

Subgroups — #30650245

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×#16861242C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic, complemented
3D4×Dic254C211yesmaximal, characteristic, complemented
4#22550522C211yesmaximal, characteristic
5#22550270C211yesmaximal, characteristic, complemented
6#22550122C211yesmaximal, characteristic, complemented
7#22550117C211yesmaximal, characteristic, complemented
8#22550099C211yesmaximal, characteristic, complemented
9#22550082C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550034C211yesmaximal, characteristic, complemented
12#22550029C211yesmaximal, characteristic, complemented
13#22550017C211yesmaximal, characteristic, complemented
14#22549978C211yesmaximal, characteristic, complemented
15#435×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C22×Dic254C2211yescharacteristic
18C22×Dic254C2211yescharacteristic, complemented
19C22×D508C2211yescharacteristic
20#16861242—24no—
21C2×Dic508C2211yescharacteristic, complemented
22C2×Dic508C2211yescharacteristic, complemented
23C2×Dic508C2211yescharacteristic, complemented
24C2×#14523237—24no—
25C2×#14523237C2211yescharacteristic
26C2×#14523237C2211yescharacteristic
27C2×#14523237C2211yescharacteristic
28C2×#14523237C2211yescharacteristic
29C2×#14523237C2211yescharacteristic
30C2×#14523237C2211yescharacteristic
31#16861209C2211yescharacteristic, complemented
32#16861208—21no—
33#16861197C2211yescharacteristic
34#16861194—21no—
35#16861188C2211yescharacteristic
36#16861188C2211yescharacteristic
37#16861188C2211yescharacteristic, complemented
38#16861186—21no—
39#16861186—21no—
40#16861183C2211yescharacteristic
41#16861182—21no—
42#16861177—21no—
43C4×Dic254C2211yescharacteristic, complemented
44C4×D508C2211yescharacteristic
45D4×Dic127—21no—
46C22⋊C4×D127—21no—
47#16861143C2211yescharacteristic, complemented
48#16861134C2211yescharacteristic, complemented
49#16861134C2211yescharacteristic, complemented
50#16861134C2211yescharacteristic, complemented
51#16861134C2211yescharacteristic, complemented
52#16861131C2211yescharacteristic
53#16861131C2211yescharacteristic
54C2×M16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic
56C4×D4×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic
59C22⋊C8×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62#14523248—22no—
63#14523248—41no—
64C2×Dic254—22no—
65C2×Dic254—22no—
66C2×Dic254—21no—
67C2×Dic254—21no—
68C2×Dic254—41no—
69C2×Dic254—41no—
70C2×Dic254C2311yescharacteristic
71C2×Dic254C2311yescharacteristic
72C2×Dic254C2311yescharacteristic
73C2×Dic254C2311yescharacteristic
74C2×D508—22no—
75C2×D508—21no—
76C2×D508—41no—
77C2×D508C2311yescharacteristic
78C2×D508C2311yescharacteristic
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81C22×Dic127—21no—
82C22×Dic127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85Dic508—22no—
86Dic508—22no—
87Dic508—41no—
88#14523237—22no—
89#14523237—22no—
90#14523237—22no—
91#14523237—22no—
92#14523237—41no—
93#14523237D414yescomplemented
94C4⋊Dic127—21no—
95C4⋊Dic127—21no—
96C4⋊Dic127C2311yescharacteristic
97C4⋊Dic127C2311yescharacteristic
98Dic127⋊C4—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104C22⋊Dic127—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109C4×Dic127—21no—
110C2×D4×C127D411yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113M16×C127—22no—
114C2×C8×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127D411yescharacteristic
120C22⋊C4×C127D411yescharacteristic
121C22⋊C4×C127D411yescharacteristic
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—42no—
126C22×D127—42no—
127C22×D127—81no—
128Dic254—21no—
129Dic254—21no—
130Dic254—21no—
131Dic254—21no—
132Dic254—21no—
133Dic254—21no—
134Dic254—21no—
135Dic254—42no—
136Dic254—42no—
137Dic254C2×D412yes—
138D508—21no—
139D508—21no—
140D508—21no—
141D508—42no—
142D508C2×D412yes—
143C22⋊D127—42no—
144C22⋊D127—81no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C2×Dic127—42no—
152C2×Dic127—42no—
153C4×D127—42no—
154C4×D127—81no—
155C23×C127C2×D411yescharacteristic
156C23×C127C2×D411yescharacteristic
157D4×C127—22no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127C2×D411yescharacteristic
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2411yescharacteristic
167C8×C127—21no—
168C8×C127—21no—
169C8×C127—21no—
170C8×C127C2×D412yes—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127#13712yescomplemented
177C22×C127C22×D411yescharacteristic
178C2×D127—22no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—84no—
185C4×C127—21no—
186C4×C127—21no—
187C4×C127—21no—
188C4×C127—21no—
189C4×C127C22×D411yescharacteristic
190C4×C127H(5,𝔽2)11yescharacteristic
191Dic127—22no—
192Dic127—41no—
193Dic127—41no—
194Dic127—41no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127—41no—
199C2×C127D4211yescharacteristic
200C2×C127C2×#13711yescharacteristic
201C2×C127#57711yescharacteristic
202D127—42no—
203D127—81no—
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#450—1271no—
218#448—1271no—
219#440—1271no—
220#435D12711yescharacteristic, complemented
221C2×#41—1271no—
222C22×Q8—1271no—
223C22×Q8—1271no—
224C22×D4—1271no—
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—1271no—
234C2×SD16—1271no—
235C2×SD16—2544no—
236C2×M16C2×D12711yescharacteristic, complemented
237C22×C8C2×D12711yescharacteristic
238C4⋊Q8—1271no—
239C4⋊Q8—2541no—
240#124—1271no—
241#123—2541no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—1271no—
245C22⋊Q8—2541no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8—1271no—
251C4×D4—1271no—
252C4×D4—2541no—
253C4×D4C2×D12711yescharacteristic
254C2×C22⋊C4—2541no—
255#107—1271no—
256C4⋊C8C2×D12711yescharacteristic, complemented
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103—1271no—
261#102—1271no—
262#102—1271no—
263C22⋊C8C2×D12711yescharacteristic
264C22⋊C8C2×D12711yescharacteristic, complemented
265#97C2×D12711yescharacteristic, complemented
266C24—2541no—
267#41—5081no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—1271no—
272C2×Q8—1271no—
273C2×Q8—5081no—
274C2×Q8—5081no—
275C2×Q8—2542no—
276C2×Q8—2542no—
277C2×Q8—2541no—
278C2×Q8—2541no—
279C2×D4—1271no—
280C2×D4—1271no—
281C2×D4—5081no—
282C2×D4—2542no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4D50811yescharacteristic
287C22×C4—2541no—
288C22×C4—2541no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4C22×D12711yescharacteristic
292C22×C4C22×D12711yescharacteristic
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—1274no—
297SD16—5081no—
298SD16—2542no—
299SD16—2542no—
300SD16—2542no—
301SD16—2542no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D12711yescharacteristic
305C2×C8C22×D12711yescharacteristic
306C2×C8C22×D12711yescharacteristic
307C2×C8C22×D12711yescharacteristic
308C4⋊C4—1271no—
309C4⋊C4—1271no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4D50811yescharacteristic
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4D50811yescharacteristic
325C22⋊C4D50811yescharacteristic
326C42—2541no—
327C42C22×D12711yescharacteristic
328C23—10161no—
329C23—5082no—
330C23—5082no—
331C23—2541no—
332C23—2541no—
333C23C2×D50811yescharacteristic
334C23C2×D50811yescharacteristic
335Q8—1272no—
336Q8—5082no—
337Q8—5082no—
338Q8—2541no—
339Q8—2541no—
340Q8—2541no—
341Q8—2541no—
342Q8—2541no—
343Q8—2541no—
344Q8—2541no—
345D4—22no—
346D4—1272no—
347D4—10161no—
348D4—5082no—
349D4—5082no—
350D4—2541no—
351D4—2541no—
352D4—2541no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—10161no—
358C2×C4—5082no—
359C2×C4—5082no—
360C2×C4—5082no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4C2×D50811yescharacteristic
368C2×C4C2×D50811yescharacteristic
369C2×C4C2×D50811yescharacteristic
370C2×C4C2×D50811yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D12712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—10164no—
382C22—5081no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—5081no—
387C22—2542no—
388C22#1686124212yescomplemented
389C22C22×D50811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—5081no—
395C4—5081no—
396C4—5081no—
397C4—5081no—
398C4—5081no—
399C4—2542no—
400C4C2×D4×D12711yescharacteristic
401C4#1686125511yescharacteristic
402C2—22no—
403C2—41no—
404C2—10161no—
405C2—5082no—
406C2D4×D50811yescharacteristic
407C2C2×#1686124211yescharacteristic
408C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#435×C127C2
socleC22×C127C22×D4
prime radical#22550522C2
hypercenter#435D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C22×#14523237C2semidirect
3. D4×Dic254C2semidirect
5. #22550270C2semidirect
6. #22550122C2semidirect
7. #22550117C2semidirect
8. #22550099C2semidirect
9. #22550082C2semidirect
10. #22550040C2semidirect
11. #22550034C2semidirect
12. #22550029C2semidirect
13. #22550017C2semidirect
14. #22549978C2semidirect
15. #435×C127C2semidirect
18. C22×Dic254C22semidirect
21. C2×Dic508C22semidirect
22. C2×Dic508C22semidirect
23. C2×Dic508C22semidirect
31. #16861209C22semidirect
37. #16861188C22semidirect
43. C4×Dic254C22semidirect
47. #16861143C22semidirect
48. #16861134C22semidirect
49. #16861134C22semidirect
50. #16861134C22semidirect
51. #16861134C22semidirect
54. C2×M16×C127C22semidirect
57. C4⋊C8×C127C22semidirect
59. C22⋊C8×C127C22semidirect
60. #97×C127C22semidirect
93. #14523237D4semidirect
176. C22×C127#137semidirect
205. C127#3266semidirect
220. #435D127semidirect
236. C2×M16C2×D127semidirect
256. C4⋊C8C2×D127semidirect
264. C22⋊C8C2×D127semidirect
265. #97C2×D127semidirect
388. C22#16861242semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C22×#145232372
D4×Dic2542
#225505222
#225502702
#225501222
#225501172
#225500992
#225500822
#225500402
#225500342
#225500292
#225500172
#225499782
#435×C1272
#3266127