Finite Group Databasebeta

← #27182976

Subgroups — #27182976

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#20265823C211yesmaximal, characteristic, complemented
2#20265534C211yesmaximal, characteristic, complemented
3#20265512C211yesmaximal, characteristic, complemented
4#20265501C211yesmaximal, characteristic, complemented
5#20265474C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265404C211yesmaximal, characteristic, complemented
8#20265392C211yesmaximal, characteristic, complemented
9#20265320C211yesmaximal, characteristic, complemented
10#20265310C211yesmaximal, characteristic, complemented
11C4×#13108059C211yesmaximal, characteristic, complemented
12D4×#11717579C211yesmaximal, characteristic, complemented
13#20265171C211yesmaximal, characteristic, complemented
14C2×#15692175C211yesmaximal, characteristic, complemented
15#553×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×#13108062C2211yescharacteristic, complemented
19C2×#13108059C2211yescharacteristic
20C2×#13108059C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22C22×#11717579C2211yescharacteristic, complemented
23#15692243C2211yescharacteristic
24#15692240—21no—
25#15692234C2211yescharacteristic
26#15692229C2211yescharacteristic
27#15692228—21no—
28#15692223—21no—
29C4×D404C2211yescharacteristic
30C22⋊C4×D101—21no—
31#15692188C2211yescharacteristic, complemented
32#15692188C2211yescharacteristic, complemented
33#15692186C2211yescharacteristic, complemented
34C4⋊#11717579C2211yescharacteristic, complemented
35#15692178C2211yescharacteristic, complemented
36#15692178C2211yescharacteristic, complemented
37#15692178C2211yescharacteristic, complemented
38#15692176C2211yescharacteristic, complemented
39#15692175C2211yescharacteristic
40#15692175C2211yescharacteristic
41#15692175C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44#15692175C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic, complemented
47C4×#11717579C2211yescharacteristic, complemented
48#126×C101—21no—
49#125×C101C2211yescharacteristic, complemented
50#123×C101—21no—
51#121×C101—21no—
52#121×C101C2211yescharacteristic
53#121×C101C2211yescharacteristic, complemented
54C4×D4×C101—21no—
55C4×D4×C101C2211yescharacteristic
56C4×D4×C101C2211yescharacteristic, complemented
57#117×C101C2211yescharacteristic, complemented
58C2×C4⋊C4×C101C2211yescharacteristic
59C23×D101—21no—
60#13108074—22no—
61#13108074—41no—
62C2×Dic202C2311yescharacteristic
63C2×D404—22no—
64C2×D404—21no—
65C2×D404—41no—
66C2×D404C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72#13108062—41no—
73#13108059—22no—
74#13108059—41no—
75C2×#11717579—41no—
76C2×#11717579—41no—
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C2×#11717579C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83C22⋊Dic101—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88C2×D4×C101—21no—
89C2×D4×C101—21no—
90C2×D4×C101D411yescharacteristic
91C2×D4×C101C2311yescharacteristic
92C22×C4×C101—21no—
93C22×C4×C101—21no—
94C22×C4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96C4⋊C4×C101—21no—
97C4⋊C4×C101—21no—
98C4⋊C4×C101—21no—
99C4⋊C4×C101D411yescharacteristic
100C4⋊C4×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C22⋊C4×C101D411yescharacteristic
108C22⋊C4×C101D411yescharacteristic
109C42×C101—21no—
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C23×C101C2×D411yescharacteristic
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—42no—
117C22×D101—42no—
118C22×D101—81no—
119D4×C101—22no—
120D4×C101—21no—
121D4×C101—21no—
122D4×C101—42no—
123D4×C101—81no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—81no—
132C2×C4×C101—81no—
133C2×C4×C101C2×D411yescharacteristic
134C2×C4×C101C2×D411yescharacteristic
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2411yescharacteristic
138C22⋊D101—42no—
139C22⋊D101—81no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142D404—21no—
143D404—21no—
144D404—21no—
145D404—42no—
146D404#4112yes—
147C4×D101—42no—
148C4×D101—81no—
149Dic202—21no—
150Dic202—21no—
151#11717579—21no—
152#11717579—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—21no—
158C22×C101—21no—
159C22×C101—21no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101—41no—
163C22×C101#13512yes—
164C22×C101C22×D411yescharacteristic
165C2×D101—22no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—84no—
172C4×C101—21no—
173C4×C101—21no—
174C4×C101—21no—
175C4×C101—21no—
176C4×C101—41no—
177C4×C101—41no—
178C4×C101—41no—
179C4×C101C2×#4111yescharacteristic
180C4×C101H(5,𝔽2)11yescharacteristic
181Dic101—41no—
182Dic101—41no—
183C2×C101—22no—
184C2×C101—41no—
185C2×C101—81no—
186C2×C101#54611yescharacteristic
187C2×C101C2×#13511yescharacteristic
188C2×C101#57611yescharacteristic
189D101—42no—
190D101—81no—
191#3297—1011nomaximal, Sylow
192C101#329711yescharacteristic, complemented, Sylow
193#553D10111yescharacteristic, complemented
194#546—1011no—
195#488—1011no—
196#484—1011no—
197#482—1011no—
198#478—1011no—
199#466—1011no—
200#465—1011no—
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×SD16—1011no—
211C2×D8—1011no—
212C2×D8—1011no—
213C22×C8—1011no—
214C22×C8—1011no—
215#126—21no—
216#125C2×D10111yescharacteristic, complemented
217#124—1011no—
218#123—21no—
219#123—2021no—
220C22⋊Q8—1011no—
221#121—21no—
222#121—1011no—
223#121—2021no—
224#121C2×D10111yescharacteristic
225#121C2×D10111yescharacteristic, complemented
226#120—2021no—
227C4×D4—21no—
228C4×D4—1011no—
229C4×D4C2×D10111yescharacteristic
230C4×D4C2×D10111yescharacteristic, complemented
231#117C2×D10111yescharacteristic, complemented
232C2×C4⋊C4C2×D10111yescharacteristic
233C2×C22⋊C4—2021no—
234#107—1011no—
235#107—1011no—
236#106—1011no—
237C4⋊C8—1011no—
238#103—1011no—
239#103—1011no—
240#103—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102—1011no—
247#102—1011no—
248C22⋊C8—1011no—
249C22⋊C8—1011no—
250C4×C8—1011no—
251C24—2021no—
252#41—4041no—
253#41—2022no—
254C2×Q8—1011no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—4041no—
260C2×D4—2022no—
261C2×D4—2021no—
262C2×D4—2021no—
263C2×D4—2021no—
264C2×D4C22⋊D10111yescharacteristic
265C2×D4C22×D10111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2021no—
269C22×C4—2021no—
270C22×C4C22×D10111yescharacteristic
271C22×C4C22×D10111yescharacteristic
272SD16—4041no—
273D8—4041no—
274D8—2022no—
275C2×C8—1011no—
276C2×C8—1011no—
277C2×C8—1011no—
278C2×C8—1011no—
279C2×C8—4041no—
280C2×C8—4041no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—1011no—
285C4⋊C4—2021no—
286C4⋊C4C22⋊D10111yescharacteristic
287C4⋊C4C22×D10111yescharacteristic
288C4⋊C4C22×D10111yescharacteristic
289C4⋊C4C22×D10111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C22⋊C4C22⋊D10111yescharacteristic
300C22⋊C4C22⋊D10111yescharacteristic
301C42—21no—
302C42C22×D10111yescharacteristic
303C23—21no—
304C23—8081no—
305C23—4042no—
306C23—4042no—
307C23—2021no—
308C23—2021no—
309C23C2×C22⋊D10111yescharacteristic
310C23C2×C22⋊D10111yescharacteristic
311Q8—2021no—
312Q8—2021no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—8081no—
319D4—1012no—
320D4—4042no—
321D4—4042no—
322D4—2021no—
323D4—2021no—
324D4—2021no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—8081no—
335C2×C4—4042no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4C2×C22⋊D10111yescharacteristic
339C2×C4C2×C22⋊D10111yescharacteristic
340C2×C4C2×C22⋊D10111yescharacteristic
341C2×C4C2×C22⋊D10111yescharacteristic
342C2×C4C23×D10111yescharacteristic
343C8—2021no—
344C8—2021no—
345C8—2021no—
346C8—2021no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—8084no—
356C22—4041no—
357C22—4041no—
358C22—4041no—
359C22—4041no—
360C22—4041no—
361C22—2022no—
362C22#1569227512yes—
363C22C22×C22⋊D10111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—4041no—
372C4—4041no—
373C4#41×D10111yescharacteristic
374C4#1569230111yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—8081no—
379C2—4042no—
380C2#2026581511yescharacteristic
381C2C2×#1569227511yescharacteristic
382C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#553×C101C2
socleC22×C101C22×D4
hypercenter#553D101
hypocenterC101#3297
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20265823C2semidirect
2. #20265534C2semidirect
3. #20265512C2semidirect
4. #20265501C2semidirect
5. #20265474C2semidirect
6. #20265411C2semidirect
7. #20265404C2semidirect
8. #20265392C2semidirect
9. #20265320C2semidirect
10. #20265310C2semidirect
11. C4×#13108059C2semidirect
12. D4×#11717579C2semidirect
13. #20265171C2semidirect
14. C2×#15692175C2semidirect
15. #553×C101C2semidirect
18. C2×#13108062C22semidirect
22. C22×#11717579C22semidirect
31. #15692188C22semidirect
32. #15692188C22semidirect
33. #15692186C22semidirect
34. C4⋊#11717579C22semidirect
35. #15692178C22semidirect
36. #15692178C22semidirect
37. #15692178C22semidirect
38. #15692176C22semidirect
46. C22⋊#11717579C22semidirect
47. C4×#11717579C22semidirect
49. #125×C101C22semidirect
53. #121×C101C22semidirect
56. C4×D4×C101C22semidirect
57. #117×C101C22semidirect
192. C101#3297semidirect
193. #553D101semidirect
216. #125C2×D101semidirect
225. #121C2×D101semidirect
230. C4×D4C2×D101semidirect
231. #117C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658232
#202655342
#202655122
#202655012
#202654742
#202654112
#202654042
#202653922
#202653202
#202653102
C4×#131080592
D4×#117175792
#202651712
C2×#156921752
#553×C1012
#3297101