Finite Group Databasebeta

← #27181383

Subgroups — #27181383

Order 12928.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265785C211yesmaximal, characteristic, complemented
2#20265649C211yesmaximal, characteristic
3#20265520C211yesmaximal, characteristic, complemented
4#20265517C211yesmaximal, characteristic, complemented
5#20265480C211yesmaximal, characteristic, complemented
6#20265475C211yesmaximal, characteristic, complemented
7#20265381C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265329C211yesmaximal, characteristic
10#20265326C211yesmaximal, characteristic
11#20265279C211yesmaximal, characteristic, complemented
12C4×#13108061C211yesmaximal, characteristic, complemented
13C4×#13108062C211yesmaximal, characteristic, complemented
14#20265239C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108063C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20#15692253C2211yescharacteristic, complemented
21#15692250—21no—
22#15692249—21no—
23#15692247—21no—
24#15692246C2211yescharacteristic, complemented
25#15692245—21no—
26#15692235C2211yescharacteristic
27#15692231C2211yescharacteristic
28C4×Dic202C2211yescharacteristic, complemented
29C4×D404C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31#15692217C2211yescharacteristic
32C4×C22⋊D101—21no—
33#15692215—21no—
34#15692213C2211yescharacteristic
35#15692212C2211yescharacteristic
36C2×#13108051—21no—
37C22⋊C4×D101—21no—
38#15692189C2211yescharacteristic, complemented
39#15692186C2211yescharacteristic, complemented
40#15692186C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692184C2211yescharacteristic, complemented
43#15692180C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692178C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692167C2211yescharacteristic
50#15692167C2211yescharacteristic
51C4×#11717579C2211yescharacteristic, complemented
52C2×SD16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic, complemented
54C4×D4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63D4×D101—22no—
64D4×D101—22no—
65C22×Dic101—21no—
66C2×C4×D101—21no—
67C2×C4×D101—21no—
68C2×C4×D101C2311yescharacteristic
69#13108063—22no—
70#13108062—22no—
71#13108061—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053D411yescharacteristic
84#13108053D411yescharacteristic
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051D411yescharacteristic
92#13108051D411yescharacteristic
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101—21no—
99SD16×C101—22no—
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C22⋊C4×C101—21no—
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C22×D101—21no—
110C22×D101—24no—
111C22×D101—81no—
112C22×D101—81no—
113C22×D101C2×D411yescharacteristic
114Q8×C101—21no—
115Q8×C101—21no—
116D4×C101—21no—
117D4×C101#4112yes—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—81no—
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2411yescharacteristic
125C22⋊D101—24no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—81no—
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133D404—21no—
134D404#4112yes—
135C4×D101—21no—
136C4×D101—21no—
137C4×D101—81no—
138C4×D101—81no—
139Dic202—21no—
140Dic202—21no—
141C8×C101—21no—
142C8×C101—21no—
143#11717579—21no—
144#11717579—21no—
145C22×C101—22no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101C22×D411yescharacteristic
149C2×D101—21no—
150C2×D101—22no—
151C2×D101—21no—
152C2×D101—21no—
153C2×D101—41no—
154C2×D101—41no—
155C2×D101—84no—
156C4×C101—21no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101C2×#4111yescharacteristic
162C4×C101C2×#4111yescharacteristic
163Dic101—21no—
164Dic101—21no—
165Dic101—21no—
166Dic101—41no—
167Dic101—41no—
168Dic101—41no—
169C2×C101—42no—
170C2×C101#52511yescharacteristic
171C2×C101#57711yescharacteristic
172C2×C101#57711yescharacteristic
173D101—41no—
174D101—42no—
175#3123—1011nomaximal, Sylow
176C101#312311yescharacteristic, complemented, Sylow
177#541—1011no—
178#518—1011no—
179#485—1011no—
180#485—1011no—
181#478—1011no—
182#478—1011no—
183#461—1011no—
184#461—1011no—
185#450—1011no—
186#450—1011no—
187#440—1011no—
188#440D10111yescharacteristic, complemented
189C4×SD16—1011no—
190C4×SD16—1011no—
191#433—1011no—
192C22×D4—1011no—
193C2×SD16—1011no—
194C2×SD16—1011no—
195C2×SD16—1011no—
196C2×SD16C2×D10111yescharacteristic
197C4⋊Q8—1011no—
198#126—2021no—
199#126—2021no—
200#125—1011no—
201#125—2021no—
202#125—2021no—
203C22⋊Q8—1011no—
204C22⋊Q8—1011no—
205C4×Q8—1011no—
206C4×Q8C2×D10111yescharacteristic, complemented
207C4×D4—1011no—
208C4×D4—1011no—
209C4×D4—1011no—
210C4×D4—2021no—
211C4×D4—2021no—
212C4×D4C2×D10111yescharacteristic
213#117—1011no—
214#117—1011no—
215C2×C22⋊C4—2021no—
216C2×C22⋊C4—2021no—
217#107—1011no—
218#107C2×D10111yescharacteristic, complemented
219#106—1011no—
220#106—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#103C2×D10111yescharacteristic, complemented
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102C2×D10111yescharacteristic
231C22⋊C8—1011no—
232C22⋊C8—1011no—
233#97C2×D10111yescharacteristic, complemented
234C4×C8—1011no—
235C24—2021no—
236C2×Q8—1011no—
237C2×Q8C22×D10111yescharacteristic
238C2×D4—1011no—
239C2×D4—2021no—
240C2×D4—2022no—
241C2×D4—2022no—
242C2×D4C22×D10111yescharacteristic
243C22×C4—21no—
244C22×C4—1011no—
245C22×C4—2021no—
246C22×C4—2021no—
247C22×C4—2021no—
248SD16—22no—
249SD16—2022no—
250SD16—2022no—
251SD16—2022no—
252C2×C8—1011no—
253C2×C8—1011no—
254C2×C8C22×D10111yescharacteristic
255C2×C8C22×D10111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—1011no—
260C4⋊C4—1011no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C4⋊C4C22×D10111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1011no—
272C22⋊C4—1011no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C42—21no—
280C42—1011no—
281C42—2021no—
282C42—2021no—
283C42C22×D10111yescharacteristic
284C23—21no—
285C23—8081no—
286C23—8081no—
287C23—1011no—
288C23—2024no—
289C23—2021no—
290Q8—21no—
291Q8—21no—
292Q8—2021no—
293Q8—2021no—
294D4—21no—
295D4—1012no—
296D4—2024no—
297D4—2021no—
298D4#1310807412yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—8081no—
304C2×C4—8081no—
305C2×C4—8081no—
306C2×C4—1011no—
307C2×C4—1011no—
308C2×C4—1011no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4D4×D10111yescharacteristic
315C2×C4D4×D10111yescharacteristic
316C2×C4C23×D10111yescharacteristic
317C8—21no—
318C8—21no—
319C8—2021no—
320C8—2021no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—8084no—
325C22—4041no—
326C22—4041no—
327C22—2021no—
328C22—2021no—
329C22—2021no—
330C22—2022no—
331C22C2×D4×D10111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4041no—
338C4—4041no—
339C4—4041no—
340C4—2021no—
341C4—2021no—
342C4—2021no—
343C4#41×D10111yescharacteristic
344C4C2×#1310807411yescharacteristic
345C2—42no—
346C2—4041no—
347C2—4042no—
348C2#2026568611yescharacteristic
349C2#2026598611yescharacteristic
350C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#440D101
hypocenterC101#3123
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20265785C2semidirect
3. #20265520C2semidirect
4. #20265517C2semidirect
5. #20265480C2semidirect
6. #20265475C2semidirect
7. #20265381C2semidirect
8. #20265380C2semidirect
11. #20265279C2semidirect
12. C4×#13108061C2semidirect
13. C4×#13108062C2semidirect
14. #20265239C2semidirect
15. #440×C101C2semidirect
20. #15692253C22semidirect
24. #15692246C22semidirect
28. C4×Dic202C22semidirect
38. #15692189C22semidirect
39. #15692186C22semidirect
40. #15692186C22semidirect
41. #15692184C22semidirect
42. #15692184C22semidirect
43. #15692180C22semidirect
44. #15692179C22semidirect
45. #15692178C22semidirect
51. C4×#11717579C22semidirect
53. C4×Q8×C101C22semidirect
55. #107×C101C22semidirect
56. #103×C101C22semidirect
58. #97×C101C22semidirect
176. C101#3123semidirect
188. #440D101semidirect
206. C4×Q8C2×D101semidirect
218. #107C2×D101semidirect
226. #103C2×D101semidirect
233. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657852
#202656492
#202655202
#202655172
#202654802
#202654752
#202653812
#202653802
#202653292
#202653262
#202652792
C4×#131080612
C4×#131080622
#202652392
#440×C1012
#3123101