Finite Group Databasebeta

← #27181361

Subgroups — #27181361

Order 12928.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265798C211yesmaximal, characteristic
2#20265518C211yesmaximal, characteristic, complemented
3#20265501C211yesmaximal, characteristic
4#20265472C211yesmaximal, characteristic, complemented
5#20265462C211yesmaximal, characteristic, complemented
6#20265399C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265333C211yesmaximal, characteristic, complemented
9#20265316C211yesmaximal, characteristic
10#20265282C211yesmaximal, characteristic, complemented
11#20265275C211yesmaximal, characteristic, complemented
12C4×#13108064C211yesmaximal, characteristic, complemented
13C4×#13108061C211yesmaximal, characteristic, complemented
14#20265238C211yesmaximal, characteristic, complemented
15#519×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20#15692255C2211yescharacteristic
21#15692250—21no—
22#15692243C2211yescharacteristic
23#15692241—21no—
24#15692237—21no—
25#15692234C2211yescharacteristic
26#15692233—21no—
27#15692229C2211yescharacteristic
28C4×Dic202C2211yescharacteristic
29C4×D404C2211yescharacteristic
30#15692188C2211yescharacteristic
31#15692188C2211yescharacteristic
32#15692186C2211yescharacteristic
33#15692186C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35C4⋊#11717579C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692179C2211yescharacteristic
39#15692179C2211yescharacteristic
40#15692178C2211yescharacteristic
41#15692178C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C22⋊#11717579C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46#15692166C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C22×Q8×C101C2211yescharacteristic
49C4×Q8×C101—21no—
50C4×Q8×C101—21no—
51C4×Q8×C101C2211yescharacteristic
52C4×Q8×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55#117×C101—21no—
56#117×C101—21no—
57#117×C101C2211yescharacteristic
58#117×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C4×D101—21no—
64#13108064—22no—
65#13108064—22no—
66#13108061—22no—
67#13108061—22no—
68C2×#11717579C2311yescharacteristic
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C4⋊Dic101—21no—
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77C22⋊Dic101—21no—
78#13108051—21no—
79#13108051—21no—
80#13108051—21no—
81#13108051—21no—
82C4×Dic101—21no—
83C2×Q8×C101—22no—
84C2×Q8×C101—22no—
85C2×Q8×C101—21no—
86C2×Q8×C101C2311yescharacteristic
87C2×Q8×C101C2311yescharacteristic
88C22×C4×C101—21no—
89C22×C4×C101C2311yescharacteristic
90C4⋊C4×C101—21no—
91C4⋊C4×C101—21no—
92C4⋊C4×C101—21no—
93C4⋊C4×C101—21no—
94C4⋊C4×C101D411yescharacteristic
95C4⋊C4×C101D411yescharacteristic
96C4⋊C4×C101C2311yescharacteristic
97C4⋊C4×C101C2311yescharacteristic
98C22⋊C4×C101—21no—
99C22⋊C4×C101D411yescharacteristic
100C22⋊C4×C101D411yescharacteristic
101C42×C101—21no—
102C42×C101—21no—
103C42×C101—21no—
104C42×C101—21no—
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101C2×D411yescharacteristic
109C22×D101—21no—
110Q8×C101—24no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114Q8×C101#4112yes—
115C2×C4×C101—24no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2411yescharacteristic
128C22⋊D101—81no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132D404—21no—
133D404—21no—
134C4×D101—81no—
135Dic202—21no—
136Dic202—21no—
137Dic202—81no—
138#11717579—21no—
139#11717579—21no—
140#11717579—21no—
141#11717579—21no—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101C22×D411yescharacteristic
146C2×D101—41no—
147C2×D101—41no—
148C2×D101—41no—
149C4×C101—22no—
150C4×C101—22no—
151C4×C101—21no—
152C4×C101—21no—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—21no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101C2×#4111yescharacteristic
159C4×C101C2×#4111yescharacteristic
160Dic101—41no—
161Dic101—41no—
162Dic101—41no—
163C2×C101—41no—
164C2×C101#52511yescharacteristic
165C2×C101#57711yescharacteristic
166C2×C101#57811yescharacteristic
167D101—81no—
168#3121—1011nomaximal, Sylow
169C101#312111yescharacteristic, complemented, Sylow
170#542—1011no—
171#519D10111yescharacteristic, complemented
172#485—1011no—
173#482—1011no—
174#477—1011no—
175#475—1011no—
176#464—1011no—
177#464—1011no—
178#451—1011no—
179#448—1011no—
180#441—1011no—
181#440—1011no—
182C4×Q16—1011no—
183C4×SD16—1011no—
184#433—1011no—
185C22×Q8C2×D10111yescharacteristic
186C2×Q16—1011no—
187C2×Q16—1011no—
188C2×SD16—1011no—
189C2×SD16—1011no—
190C4⋊Q8—1011no—
191#126—2021no—
192#124—1011no—
193#124—2021no—
194#123—2021no—
195C22⋊Q8—1011no—
196C22⋊Q8—2021no—
197#121—1011no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—1011no—
201C4×Q8C2×D10111yescharacteristic
202C4×Q8C2×D10111yescharacteristic
203C4×Q8C2×D10111yescharacteristic
204C4×Q8C2×D10111yescharacteristic
205C4×D4—1011no—
206#117—21no—
207#117—21no—
208#117C2×D10111yescharacteristic
209#117C2×D10111yescharacteristic
210#107—1011no—
211#107—1011no—
212#106—1011no—
213#106—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216#103—1011no—
217#103—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103—1011no—
222#102—1011no—
223#102—1011no—
224C22⋊C8—1011no—
225C22⋊C8—1011no—
226#97—1011no—
227C4×C8—1011no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—1011no—
232C2×Q8—2021no—
233C2×Q8C22×D10111yescharacteristic
234C2×Q8C22×D10111yescharacteristic
235C2×D4—1011no—
236C2×D4—2021no—
237C22×C4—21no—
238C22×C4—2021no—
239C22×C4C22×D10111yescharacteristic
240Q16—2022no—
241Q16—2022no—
242SD16—2022no—
243SD16—2022no—
244C2×C8—1011no—
245C2×C8—1011no—
246C2×C8—1011no—
247C2×C8—1011no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1011no—
253C4⋊C4—1011no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4C22⋊D10111yescharacteristic
258C4⋊C4C22⋊D10111yescharacteristic
259C4⋊C4C22×D10111yescharacteristic
260C4⋊C4C22×D10111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2021no—
263C22⋊C4—2021no—
264C22⋊C4—2021no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4C22⋊D10111yescharacteristic
268C22⋊C4C22⋊D10111yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—2021no—
275C42C22×D10111yescharacteristic
276C42C22×D10111yescharacteristic
277C23—2021no—
278C23C2×C22⋊D10111yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—8081no—
283Q8—2021no—
284Q8—2021no—
285Q8#1310807412yes—
286Q8#1310807412yes—
287D4—8081no—
288D4—2021no—
289D4—2021no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—8081no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4C2×C22⋊D10111yescharacteristic
302C2×C4C2×C22⋊D10111yescharacteristic
303C2×C4C2×C22⋊D10111yescharacteristic
304C2×C4C2×C22⋊D10111yescharacteristic
305C2×C4C2×C22⋊D10111yescharacteristic
306C2×C4C23×D10111yescharacteristic
307C8—2021no—
308C8—2021no—
309C8—2021no—
310C8—2021no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—4041no—
315C22—4041no—
316C22—4041no—
317C22C22×C22⋊D10111yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—4041no—
328C4—4041no—
329C4—4041no—
330C4C2×#1310807411yescharacteristic
331C4C2×#1310807411yescharacteristic
332C2—41no—
333C2—8081no—
334C2#2026568811yescharacteristic
335C2#2026598511yescharacteristic
336C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#519×C101C2
socleC22×C101C22×D4
prime radical#15692229C22
hypercenter#519D101
hypocenterC101#3121
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20265518C2semidirect
4. #20265472C2semidirect
5. #20265462C2semidirect
6. #20265399C2semidirect
7. #20265399C2semidirect
8. #20265333C2semidirect
10. #20265282C2semidirect
11. #20265275C2semidirect
12. C4×#13108064C2semidirect
13. C4×#13108061C2semidirect
14. #20265238C2semidirect
15. #519×C101C2semidirect
169. C101#3121semidirect
171. #519D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202655182
#202655012
#202654722
#202654622
#202653992
#202653332
#202653162
#202652822
#202652752
C4×#131080642
C4×#131080612
#202652382
#519×C1012
#3121101