Finite Group Databasebeta

← #27182988

Subgroups — #27182988

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265879C211yesmaximal, characteristic, complemented
2#20265818C211yesmaximal, characteristic, complemented
3#20265533C211yesmaximal, characteristic, complemented
4#20265510C211yesmaximal, characteristic, complemented
5#20265503C211yesmaximal, characteristic, complemented
6#20265476C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic, complemented
8#20265394C211yesmaximal, characteristic, complemented
9#20265321C211yesmaximal, characteristic, complemented
10#20265310C211yesmaximal, characteristic, complemented
11#20265257C211yesmaximal, characteristic, complemented
12C8×C22⋊D101C211yesmaximal, characteristic, complemented
13#20265172C211yesmaximal, characteristic, complemented
14C2×#15692177C211yesmaximal, characteristic, complemented
15#465×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×#13108059C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic, complemented
21#15692248—21no—
22#15692246C2211yescharacteristic, complemented
23#15692244C2211yescharacteristic
24#15692240—21no—
25#15692239—21no—
26#15692230—21no—
27#15692230C2211yescharacteristic
28#15692227—21no—
29#15692226C2211yescharacteristic, complemented
30#15692225—21no—
31D4×Dic101—21no—
32D4×Dic101C2211yescharacteristic, complemented
33#15692217C2211yescharacteristic
34C4×C22⋊D101C2211yescharacteristic
35#15692213C2211yescharacteristic, complemented
36C2×C4⋊Dic101C2211yescharacteristic
37C22⋊C4×D101—21no—
38#15692188C2211yescharacteristic, complemented
39#15692188C2211yescharacteristic, complemented
40#15692184C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692177C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic, complemented
53C22×C8×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic, complemented
57#103×C101C2211yescharacteristic, complemented
58#102×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108075—22no—
61#13108075—41no—
62C2×D404—22no—
63C2×D404—21no—
64C2×D404—21no—
65C2×D404—21no—
66C2×D404—41no—
67C2×D404C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101D411yescharacteristic
71C22×Dic101—21no—
72C22×Dic101—21no—
73C2×C4×D101—21no—
74C2×C4×D101—21no—
75C2×C4×D101C2311yescharacteristic
76#13108059—22no—
77#13108059—41no—
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C8×D101—41no—
81C4⋊Dic101—21no—
82C4⋊Dic101—21no—
83C4⋊Dic101C2311yescharacteristic
84C4⋊Dic101C2311yescharacteristic
85#13108053—21no—
86#13108053D411yescharacteristic
87C22⋊Dic101—21no—
88C22⋊Dic101—21no—
89C22⋊Dic101D411yescharacteristic
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051D411yescharacteristic
96C4×Dic101—21no—
97C4×Dic101C2311yescharacteristic
98C2×Q8×C101C2311yescharacteristic
99C2×D4×C101—21no—
100C2×D4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102SD16×C101—41no—
103C2×C8×C101—41no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C22⋊C4×C101—21no—
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—81no—
118C22×D101C2×D411yescharacteristic
119Q8×C101—21no—
120Q8×C101—21no—
121D4×C101—21no—
122D4×C101—21no—
123D4×C101—42no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2411yescharacteristic
131C22⋊D101—22no—
132C22⋊D101—81no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—81no—
136C2×Dic101—81no—
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139C2×Dic101C2×D411yescharacteristic
140D404—21no—
141D404—21no—
142D404—21no—
143D404—42no—
144D404—42no—
145D404—81no—
146D404#4112yes—
147C4×D101—21no—
148C4×D101—21no—
149C4×D101—42no—
150C4×D101—81no—
151C8×C101—21no—
152C8×C101—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—41no—
158C22×C101—41no—
159C22×C101—41no—
160C22×C101#13512yes—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—22no—
164C2×D101—21no—
165C2×D101—21no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—84no—
172C4×C101—21no—
173C4×C101—41no—
174C4×C101—41no—
175C4×C101—41no—
176C4×C101C2×#4111yescharacteristic
177C4×C101H(5,𝔽2)11yescharacteristic
178Dic101—21no—
179Dic101—21no—
180Dic101—21no—
181Dic101—41no—
182Dic101—41no—
183C2×C101—22no—
184C2×C101—81no—
185C2×C101#54611yescharacteristic
186C2×C101C2×#13511yescharacteristic
187C2×C101#57611yescharacteristic
188D101—41no—
189D101—42no—
190D101—81no—
191#3297—1011nomaximal, Sylow
192C101#329711yescharacteristic, complemented, Sylow
193#553—1011no—
194#546—1011no—
195#488—1011no—
196#484—1011no—
197#482—1011no—
198#478—1011no—
199#466—1011no—
200#465D10111yescharacteristic, complemented
201#463—1011no—
202#449—1011no—
203#447—1011no—
204C4×D8—1011no—
205C8×D4—1011no—
206#416—1011no—
207C2×#102—1011no—
208C2×#41—1011no—
209C22×D4—1011no—
210C2×SD16C2×D10111yescharacteristic, complemented
211C2×D8—1011no—
212C2×D8—1011no—
213C22×C8—1011no—
214C22×C8C2×D10111yescharacteristic
215#126—2021no—
216#125—1011no—
217#124—1011no—
218#123—2021no—
219#123—2021no—
220C22⋊Q8C2×D10111yescharacteristic
221#121—1011no—
222#121—1011no—
223#121—2021no—
224#121—2021no—
225#121C2×D10111yescharacteristic
226#120—2021no—
227C4×D4—1011no—
228C4×D4—1011no—
229C4×D4—1011no—
230C4×D4—2021no—
231#117—1011no—
232C2×C4⋊C4—1011no—
233C2×C22⋊C4—2021no—
234#107—1011no—
235#107—1011no—
236#106C2×D10111yescharacteristic, complemented
237C4⋊C8—1011no—
238#103—1011no—
239#103—1011no—
240#103C2×D10111yescharacteristic, complemented
241#102—1011no—
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102C2×D10111yescharacteristic, complemented
248C22⋊C8—1011no—
249C22⋊C8—1011no—
250C4×C8—1011no—
251C24—2021no—
252#41—4041no—
253#41—2022no—
254C2×Q8C22×D10111yescharacteristic
255C2×D4—21no—
256C2×D4—1011no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—4041no—
260C2×D4—2021no—
261C2×D4—2022no—
262C2×D4—2021no—
263C2×D4—2021no—
264C2×D4—2021no—
265C2×D4C22×D10111yescharacteristic
266C22×C4—1011no—
267C22×C4—2021no—
268C22×C4—2021no—
269C22×C4—2021no—
270C22×C4—2021no—
271C22×C4C22×D10111yescharacteristic
272SD16—41no—
273D8—4041no—
274D8—2022no—
275C2×C8—41no—
276C2×C8—1011no—
277C2×C8—1011no—
278C2×C8—4041no—
279C2×C8C22×D10111yescharacteristic
280C2×C8C22×D10111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—1011no—
283C4⋊C4—1011no—
284C4⋊C4—1011no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4—2021no—
288C4⋊C4C22×D10111yescharacteristic
289C4⋊C4C22×D10111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1011no—
293C22⋊C4—1011no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C22⋊C4—2021no—
300C22⋊C4—2021no—
301C42—1011no—
302C42—2021no—
303C23—21no—
304C23—8081no—
305C23—1011no—
306C23—4042no—
307C23—4042no—
308C23—2021no—
309C23—2021no—
310C23D4×D10111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—8081no—
317D4—8081no—
318D4—1012no—
319D4—4042no—
320D4—4042no—
321D4—2022no—
322D4—2021no—
323D4—2021no—
324D4—2021no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—8081no—
331C2×C4—8081no—
332C2×C4—8081no—
333C2×C4—1011no—
334C2×C4—1011no—
335C2×C4—1011no—
336C2×C4—4042no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4D4×D10111yescharacteristic
342C2×C4C23×D10111yescharacteristic
343C8—21no—
344C8—21no—
345C8—2021no—
346C8—2021no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—8084no—
353C22—4041no—
354C22—4041no—
355C22—4041no—
356C22—4041no—
357C22—4041no—
358C22—2022no—
359C22—2021no—
360C22—2021no—
361C22—2021no—
362C22#1569227612yes—
363C22C2×D4×D10111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—4041no—
369C4—4041no—
370C4—2021no—
371C4—2021no—
372C4—2021no—
373C4#41×D10111yescharacteristic
374C4#1569230011yescharacteristic
375C2—22no—
376C2—81no—
377C2—8081no—
378C2—4042no—
379C2—4041no—
380C2#2026581611yescharacteristic
381C2C2×#1569227611yescharacteristic
382C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#465×C101C2
socleC22×C101C22×D4
hypercenter#465D101
hypocenterC101#3297
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20265879C2semidirect
2. #20265818C2semidirect
3. #20265533C2semidirect
4. #20265510C2semidirect
5. #20265503C2semidirect
6. #20265476C2semidirect
7. #20265412C2semidirect
8. #20265394C2semidirect
9. #20265321C2semidirect
10. #20265310C2semidirect
11. #20265257C2semidirect
12. C8×C22⋊D101C2semidirect
13. #20265172C2semidirect
14. C2×#15692177C2semidirect
15. #465×C101C2semidirect
20. C2×C8×D101C22semidirect
22. #15692246C22semidirect
29. #15692226C22semidirect
32. D4×Dic101C22semidirect
35. #15692213C22semidirect
38. #15692188C22semidirect
39. #15692188C22semidirect
40. #15692184C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
50. #15692167C22semidirect
51. C8×Dic101C22semidirect
52. C2×SD16×C101C22semidirect
56. #106×C101C22semidirect
57. #103×C101C22semidirect
58. #102×C101C22semidirect
192. C101#3297semidirect
200. #465D101semidirect
210. C2×SD16C2×D101semidirect
236. #106C2×D101semidirect
240. #103C2×D101semidirect
247. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658792
#202658182
#202655332
#202655102
#202655032
#202654762
#202654122
#202653942
#202653212
#202653102
#202652572
C8×C22⋊D1012
#202651722
C2×#156921772
#465×C1012
#3297101