Finite Group Databasebeta

← #27181696

Subgroups — #27181696

Order 12928.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265711C211yesmaximal, characteristic, complemented
2#20265576C211yesmaximal, characteristic, complemented
3#20265559C211yesmaximal, characteristic, complemented
4#20265534C211yesmaximal, characteristic, complemented
5#20265531C211yesmaximal, characteristic, complemented
6#20265416C211yesmaximal, characteristic, complemented
7#20265411C211yesmaximal, characteristic
8#20265408C211yesmaximal, characteristic, complemented
9#20265404C211yesmaximal, characteristic, complemented
10C4×#13108064C211yesmaximal, characteristic, complemented
11C4×#13108061C211yesmaximal, characteristic, complemented
12C4×#13108062C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14C2×C4×#11717579C211yesmaximal, characteristic, complemented
15#529×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C22×#11717579C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22#15692249—21no—
23#15692247C2211yescharacteristic
24#15692243C2211yescharacteristic
25#15692238—21no—
26#15692234C2211yescharacteristic
27#15692229C2211yescharacteristic
28C4×Dic202C2211yescharacteristic
29C4×D404C2211yescharacteristic
30C4×C22⋊D101—21no—
31#15692212—21no—
32#15692188C2211yescharacteristic
33#15692188C2211yescharacteristic
34#15692186C2211yescharacteristic
35#15692186C2211yescharacteristic
36#15692179C2211yescharacteristic
37#15692179C2211yescharacteristic
38#15692178C2211yescharacteristic
39#15692178C2211yescharacteristic
40#15692176C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44C4×#11717579C2211yescharacteristic
45C4×#11717579C2211yescharacteristic
46C4×#11717579C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48#126×C101—21no—
49#125×C101C2211yescharacteristic
50#124×C101C2211yescharacteristic
51#123×C101—21no—
52C22⋊Q8×C101C2211yescharacteristic
53#121×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55C4×D4×C101—21no—
56C4×D4×C101C2211yescharacteristic
57#117×C101—21no—
58C2×C42×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C2×D404C2311yescharacteristic
61C2×C22⋊D101—21no—
62C2×C4×D101—21no—
63#13108064—22no—
64#13108062—22no—
65#13108061—22no—
66#13108059—22no—
67C2×#11717579—22no—
68C2×#11717579—22no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79C22⋊Dic101—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83#13108051—21no—
84C4×Dic101—21no—
85C2×Q8×C101C2311yescharacteristic
86C2×D4×C101—21no—
87C2×D4×C101C2311yescharacteristic
88C22×C4×C101—21no—
89C22×C4×C101D411yescharacteristic
90C22×C4×C101D411yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C4⋊C4×C101—21no—
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101C2311yescharacteristic
97C4⋊C4×C101C2311yescharacteristic
98C22⋊C4×C101—21no—
99C22⋊C4×C101—21no—
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101—21no—
103C42×C101—21no—
104C42×C101D411yescharacteristic
105C42×C101D411yescharacteristic
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113D4×C101—21no—
114D4×C101—21no—
115D4×C101—42no—
116C2×C4×C101—22no—
117C2×C4×C101—22no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—42no—
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2411yescharacteristic
130C22⋊D101—42no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134D404—21no—
135D404—21no—
136C4×D101—42no—
137Dic202—21no—
138Dic202—21no—
139#11717579—21no—
140#11717579—21no—
141#11717579#4112yes—
142#11717579#4112yes—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101#13512yes—
149C22×C101C22×D411yescharacteristic
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—21no—
156C4×C101—21no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101#13512yes—
161C4×C101C2×#4111yescharacteristic
162C4×C101C2×#4111yescharacteristic
163Dic101—41no—
164Dic101—41no—
165Dic101—41no—
166C2×C101—22no—
167C2×C101—81no—
168C2×C101#53211yescharacteristic
169C2×C101C2×#13511yescharacteristic
170C2×C101C2×#13511yescharacteristic
171D101—81no—
172#3168—1011nomaximal, Sylow
173C101#316811yescharacteristic, complemented, Sylow
174#529—1011no—
175#529D10111yescharacteristic, complemented
176#499—1011no—
177#495—1011no—
178#488—1011no—
179#488—1011no—
180#467—1011no—
181#466—1011no—
182#465—1011no—
183#465—1011no—
184C4×Q16—1011no—
185C4×SD16—1011no—
186C4×SD16—1011no—
187C4×D8—1011no—
188C2×C4×C8—1011no—
189C2×Q16—1011no—
190C2×SD16—1011no—
191C2×SD16—1011no—
192C2×D8—1011no—
193C22×C8—1011no—
194C22×C8—1011no—
195#126—21no—
196#126—2021no—
197#125—1011no—
198#125C2×D10111yescharacteristic
199#124—1011no—
200#124C2×D10111yescharacteristic
201#123—21no—
202#123—2021no—
203C22⋊Q8—1011no—
204C22⋊Q8C2×D10111yescharacteristic
205#121—1011no—
206#121C2×D10111yescharacteristic
207C4×Q8—1011no—
208C4×Q8C2×D10111yescharacteristic
209C4×D4—21no—
210C4×D4—1011no—
211C4×D4—2021no—
212C4×D4C2×D10111yescharacteristic
213#117—21no—
214#117—2021no—
215C2×C42C2×D10111yescharacteristic
216#107—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220#103—1011no—
221#103—1011no—
222#103—1011no—
223#103—1011no—
224#102—1011no—
225#102—1011no—
226#102—1011no—
227#102—1011no—
228C4×C8—1011no—
229C4×C8—1011no—
230C4×C8—1011no—
231C4×C8—1011no—
232C2×Q8—1011no—
233C2×Q8C22×D10111yescharacteristic
234C2×D4—21no—
235C2×D4—1011no—
236C2×D4—2021no—
237C2×D4C22×D10111yescharacteristic
238C22×C4—21no—
239C22×C4—2021no—
240C22×C4C22⋊D10111yescharacteristic
241C22×C4C22⋊D10111yescharacteristic
242C22×C4C22×D10111yescharacteristic
243Q16—2022no—
244SD16—2022no—
245SD16—2022no—
246D8—2022no—
247C2×C8—1011no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8—1011no—
251C2×C8—2022no—
252C2×C8—2022no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—1011no—
258C4⋊C4—1011no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4C22×D10111yescharacteristic
264C4⋊C4C22×D10111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2021no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C42—21no—
276C42—2021no—
277C42C22⋊D10111yescharacteristic
278C42C22⋊D10111yescharacteristic
279C42C22×D10111yescharacteristic
280C42C22×D10111yescharacteristic
281C23—21no—
282C23—2021no—
283C23C2×C22⋊D10111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—2021no—
287Q8—2021no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—4042no—
292D4—2021no—
293D4—2021no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—4042no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4C2×C22⋊D10111yescharacteristic
307C2×C4C2×C22⋊D10111yescharacteristic
308C2×C4C2×C22⋊D10111yescharacteristic
309C2×C4C2×C22⋊D10111yescharacteristic
310C2×C4C2×C22⋊D10111yescharacteristic
311C2×C4C23×D10111yescharacteristic
312C8—1012no—
313C8—1012no—
314C8—2021no—
315C8—2021no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4041no—
322C22—4041no—
323C22—4041no—
324C22#1569227512yes—
325C22C22×C22⋊D10111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4041no—
334C4—4041no—
335C4—4041no—
336C4#1569227512yes—
337C4#41×D10111yescharacteristic
338C4#41×D10111yescharacteristic
339C2—22no—
340C2—81no—
341C2—8081no—
342C2#2026572311yescharacteristic
343C2C2×#1569227511yescharacteristic
344C2C2×#1569227511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#529×C101C2
socleC22×C101C22×D4
prime radical#20265411C2
hypercenter#529D101
hypocenterC101#3168
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20265711C2semidirect
2. #20265576C2semidirect
3. #20265559C2semidirect
4. #20265534C2semidirect
5. #20265531C2semidirect
6. #20265416C2semidirect
8. #20265408C2semidirect
9. #20265404C2semidirect
10. C4×#13108064C2semidirect
11. C4×#13108061C2semidirect
12. C4×#13108062C2semidirect
13. C4×#13108059C2semidirect
14. C2×C4×#11717579C2semidirect
15. #529×C101C2semidirect
173. C101#3168semidirect
175. #529D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202655762
#202655592
#202655342
#202655312
#202654162
#202654112
#202654082
#202654042
C4×#131080642
C4×#131080612
C4×#131080622
C4×#131080592
C2×C4×#117175792
#529×C1012
#3168101