Finite Group Databasebeta

← #27182647

Subgroups — #27182647

Order 12928.

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×#15692288C211yesmaximal, characteristic, complemented
2C22×#13108063C211yesmaximal, characteristic, complemented
3D4×Dic202C211yesmaximal, characteristic, complemented
4#20265823C211yesmaximal, characteristic
5#20265571C211yesmaximal, characteristic, complemented
6#20265423C211yesmaximal, characteristic, complemented
7#20265418C211yesmaximal, characteristic, complemented
8#20265400C211yesmaximal, characteristic, complemented
9#20265383C211yesmaximal, characteristic, complemented
10#20265341C211yesmaximal, characteristic, complemented
11#20265335C211yesmaximal, characteristic, complemented
12#20265330C211yesmaximal, characteristic, complemented
13#20265318C211yesmaximal, characteristic, complemented
14#20265279C211yesmaximal, characteristic, complemented
15#435×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18C22×Dic202C2211yescharacteristic, complemented
19C22×D404C2211yescharacteristic
20#15692288—24no—
21C2×Dic404C2211yescharacteristic, complemented
22C2×Dic404C2211yescharacteristic, complemented
23C2×Dic404C2211yescharacteristic, complemented
24C2×#13108063—24no—
25C2×#13108063C2211yescharacteristic
26C2×#13108063C2211yescharacteristic
27C2×#13108063C2211yescharacteristic
28C2×#13108063C2211yescharacteristic
29C2×#13108063C2211yescharacteristic
30C2×#13108063C2211yescharacteristic
31#15692255C2211yescharacteristic, complemented
32#15692254—21no—
33#15692243C2211yescharacteristic
34#15692240—21no—
35#15692234C2211yescharacteristic
36#15692234C2211yescharacteristic
37#15692234C2211yescharacteristic, complemented
38#15692232—21no—
39#15692232—21no—
40#15692229C2211yescharacteristic
41#15692228—21no—
42#15692223—21no—
43C4×Dic202C2211yescharacteristic, complemented
44C4×D404C2211yescharacteristic
45D4×Dic101—21no—
46C22⋊C4×D101—21no—
47#15692189C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692180C2211yescharacteristic, complemented
50#15692180C2211yescharacteristic, complemented
51#15692180C2211yescharacteristic, complemented
52#15692177C2211yescharacteristic
53#15692177C2211yescharacteristic
54C2×M16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic
56C4×D4×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic
59C22⋊C8×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202—22no—
65C2×Dic202—22no—
66C2×Dic202—21no—
67C2×Dic202—21no—
68C2×Dic202—41no—
69C2×Dic202—41no—
70C2×Dic202C2311yescharacteristic
71C2×Dic202C2311yescharacteristic
72C2×Dic202C2311yescharacteristic
73C2×Dic202C2311yescharacteristic
74C2×D404—22no—
75C2×D404—21no—
76C2×D404—41no—
77C2×D404C2311yescharacteristic
78C2×D404C2311yescharacteristic
79C2×C22⋊D101—21no—
80C2×C22⋊D101—21no—
81C22×Dic101—21no—
82C22×Dic101—21no—
83C2×C4×D101—21no—
84C2×C4×D101—21no—
85Dic404—22no—
86Dic404—22no—
87Dic404—41no—
88#13108063—22no—
89#13108063—22no—
90#13108063—22no—
91#13108063—22no—
92#13108063—41no—
93#13108063D414yescomplemented
94C4⋊Dic101—21no—
95C4⋊Dic101—21no—
96C4⋊Dic101C2311yescharacteristic
97C4⋊Dic101C2311yescharacteristic
98#13108053—21no—
99#13108053—21no—
100#13108053—21no—
101#13108053—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101—21no—
104C22⋊Dic101—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109C4×Dic101—21no—
110C2×D4×C101D411yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113M16×C101—22no—
114C2×C8×C101—22no—
115C2×C8×C101C2311yescharacteristic
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C4⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C42×C101C2311yescharacteristic
123C23×C101C2×D411yescharacteristic
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—42no—
128C22×D101—42no—
129C22×D101—81no—
130D4×C101—22no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—42no—
141C22⋊D101—81no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—42no—
149C2×Dic101—42no—
150D404—21no—
151D404—21no—
152D404—21no—
153D404—42no—
154D404C2×D412yes—
155C4×D101—42no—
156C4×D101—81no—
157Dic202—21no—
158Dic202—21no—
159Dic202—21no—
160Dic202—21no—
161Dic202—21no—
162Dic202—21no—
163Dic202—21no—
164Dic202—42no—
165Dic202—42no—
166Dic202C2×D412yes—
167C8×C101—21no—
168C8×C101—21no—
169C8×C101—21no—
170C8×C101C2×D412yes—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101#13712yescomplemented
177C22×C101C22×D411yescharacteristic
178C2×D101—22no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C2×D101—84no—
185C4×C101—21no—
186C4×C101—21no—
187C4×C101—21no—
188C4×C101—21no—
189C4×C101C22×D411yescharacteristic
190C4×C101H(5,𝔽2)11yescharacteristic
191Dic101—22no—
192Dic101—41no—
193Dic101—41no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—22no—
198C2×C101—41no—
199C2×C101D4211yescharacteristic
200C2×C101C2×#13711yescharacteristic
201C2×C101#57711yescharacteristic
202D101—42no—
203D101—81no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137—1011no—
207C22×SD16—1011no—
208D4×Q8—1011no—
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467—1011no—
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450—1011no—
218#448—1011no—
219#440—1011no—
220#435D10111yescharacteristic, complemented
221C2×#41—1011no—
222C22×Q8—1011no—
223C22×Q8—1011no—
224C22×D4—1011no—
225#137—2024no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×Q16—1011no—
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—2024no—
236C2×M16C2×D10111yescharacteristic, complemented
237C22×C8C2×D10111yescharacteristic
238C4⋊Q8—1011no—
239C4⋊Q8—2021no—
240#124—1011no—
241#123—2021no—
242C22⋊Q8—1011no—
243C22⋊Q8—1011no—
244C22⋊Q8—1011no—
245C22⋊Q8—2021no—
246C22⋊Q8—2021no—
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8—1011no—
251C4×D4—1011no—
252C4×D4—2021no—
253C4×D4C2×D10111yescharacteristic
254C2×C22⋊C4—2021no—
255#107—1011no—
256C4⋊C8C2×D10111yescharacteristic, complemented
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103—1011no—
261#102—1011no—
262#102—1011no—
263C22⋊C8C2×D10111yescharacteristic
264C22⋊C8C2×D10111yescharacteristic, complemented
265#97C2×D10111yescharacteristic, complemented
266C24—2021no—
267#41—4041no—
268#41—2022no—
269C2×Q8—1011no—
270C2×Q8—1011no—
271C2×Q8—1011no—
272C2×Q8—1011no—
273C2×Q8—4041no—
274C2×Q8—4041no—
275C2×Q8—2022no—
276C2×Q8—2022no—
277C2×Q8—2021no—
278C2×Q8—2021no—
279C2×D4—1011no—
280C2×D4—1011no—
281C2×D4—4041no—
282C2×D4—2022no—
283C2×D4—2021no—
284C2×D4—2021no—
285C2×D4—2021no—
286C2×D4D40411yescharacteristic
287C22×C4—2021no—
288C22×C4—2021no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4C22×D10111yescharacteristic
292C22×C4C22×D10111yescharacteristic
293Q16—4041no—
294Q16—2022no—
295Q16—2022no—
296SD16—1014no—
297SD16—4041no—
298SD16—2022no—
299SD16—2022no—
300SD16—2022no—
301SD16—2022no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D10111yescharacteristic
305C2×C8C22×D10111yescharacteristic
306C2×C8C22×D10111yescharacteristic
307C2×C8C22×D10111yescharacteristic
308C4⋊C4—1011no—
309C4⋊C4—1011no—
310C4⋊C4—2021no—
311C4⋊C4—2021no—
312C4⋊C4—2021no—
313C4⋊C4—2021no—
314C4⋊C4—2021no—
315C4⋊C4—2021no—
316C4⋊C4D40411yescharacteristic
317C22⋊C4—2021no—
318C22⋊C4—2021no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4D40411yescharacteristic
325C22⋊C4D40411yescharacteristic
326C42—2021no—
327C42C22×D10111yescharacteristic
328C23—8081no—
329C23—4042no—
330C23—4042no—
331C23—2021no—
332C23—2021no—
333C23C2×D40411yescharacteristic
334C23C2×D40411yescharacteristic
335Q8—1012no—
336Q8—4042no—
337Q8—4042no—
338Q8—2021no—
339Q8—2021no—
340Q8—2021no—
341Q8—2021no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345D4—22no—
346D4—8081no—
347D4—1012no—
348D4—4042no—
349D4—4042no—
350D4—2021no—
351D4—2021no—
352D4—2021no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8081no—
358C2×C4—4042no—
359C2×C4—4042no—
360C2×C4—4042no—
361C2×C4—2021no—
362C2×C4—2021no—
363C2×C4—2021no—
364C2×C4—2021no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4C2×D40411yescharacteristic
368C2×C4C2×D40411yescharacteristic
369C2×C4C2×D40411yescharacteristic
370C2×C4C2×D40411yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D10112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8084no—
382C22—4041no—
383C22—4041no—
384C22—4041no—
385C22—4041no—
386C22—4041no—
387C22—2022no—
388C22#1569228812yescomplemented
389C22C22×D40411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—4041no—
395C4—4041no—
396C4—4041no—
397C4—4041no—
398C4—4041no—
399C4—2022no—
400C4C2×D4×D10111yescharacteristic
401C4#1569230111yescharacteristic
402C2—22no—
403C2—41no—
404C2—8081no—
405C2—4042no—
406C2D4×D40411yescharacteristic
407C2C2×#1569228811yescharacteristic
408C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#435×C101C2
socleC22×C101C22×D4
prime radical#20265823C2
hypercenter#435D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15692288C2semidirect
2. C22×#13108063C2semidirect
3. D4×Dic202C2semidirect
5. #20265571C2semidirect
6. #20265423C2semidirect
7. #20265418C2semidirect
8. #20265400C2semidirect
9. #20265383C2semidirect
10. #20265341C2semidirect
11. #20265335C2semidirect
12. #20265330C2semidirect
13. #20265318C2semidirect
14. #20265279C2semidirect
15. #435×C101C2semidirect
18. C22×Dic202C22semidirect
21. C2×Dic404C22semidirect
22. C2×Dic404C22semidirect
23. C2×Dic404C22semidirect
31. #15692255C22semidirect
37. #15692234C22semidirect
43. C4×Dic202C22semidirect
47. #15692189C22semidirect
48. #15692180C22semidirect
49. #15692180C22semidirect
50. #15692180C22semidirect
51. #15692180C22semidirect
54. C2×M16×C101C22semidirect
57. C4⋊C8×C101C22semidirect
59. C22⋊C8×C101C22semidirect
60. #97×C101C22semidirect
93. #13108063D4semidirect
176. C22×C101#137semidirect
205. C101#3266semidirect
220. #435D101semidirect
236. C2×M16C2×D101semidirect
256. C4⋊C8C2×D101semidirect
264. C22⋊C8C2×D101semidirect
265. #97C2×D101semidirect
388. C22#15692288semidirect

Maximal subgroups

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

subgroupindex
C2×#156922882
C22×#131080632
D4×Dic2022
#202658232
#202655712
#202654232
#202654182
#202654002
#202653832
#202653412
#202653352
#202653302
#202653182
#202652792
#435×C1012
#3266101