Finite Group Databasebeta

← #27181701

Subgroups — #27181701

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#20265712C211yesmaximal, characteristic, complemented
2#20265706C211yesmaximal, characteristic, complemented
3#20265576C211yesmaximal, characteristic, complemented
4#20265559C211yesmaximal, characteristic, complemented
5#20265532C211yesmaximal, characteristic, complemented
6#20265417C211yesmaximal, characteristic, complemented
7#20265412C211yesmaximal, characteristic
8#20265409C211yesmaximal, characteristic, complemented
9#20265405C211yesmaximal, characteristic, complemented
10#20265269C211yesmaximal, characteristic, complemented
11#20265268C211yesmaximal, characteristic, complemented
12#20265262C211yesmaximal, characteristic, complemented
13#20265257C211yesmaximal, characteristic, complemented
14C4×C8×D101C211yesmaximal, characteristic, complemented
15#488×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21C2×C8×D101C2211yescharacteristic
22#15692250—21no—
23#15692248—21no—
24#15692247C2211yescharacteristic
25#15692243C2211yescharacteristic
26#15692240—21no—
27#15692237—21no—
28#15692235C2211yescharacteristic
29#15692231C2211yescharacteristic
30#15692230C2211yescharacteristic
31#15692226C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33#15692220C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692217—21no—
36#15692217C2211yescharacteristic
37#15692215—21no—
38#15692213—21no—
39#15692211—21no—
40C42×D101C2211yescharacteristic
41#15692188C2211yescharacteristic
42#15692188C2211yescharacteristic
43#15692186C2211yescharacteristic
44#15692186C2211yescharacteristic
45#15692180C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692177C2211yescharacteristic
49C4×#11717579C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52#125×C101C2211yescharacteristic
53#124×C101C2211yescharacteristic
54#103×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C4×C8×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C2×D404—21no—
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C22×Dic101—21no—
64C2×C4×D101—21no—
65C2×C4×D101D411yescharacteristic
66C2×C4×D101D411yescharacteristic
67C2×C4×D101C2311yescharacteristic
68#13108064—22no—
69#13108062—22no—
70#13108061—22no—
71#13108059—22no—
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C8×D101—22no—
75C8×D101—22no—
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101D411yescharacteristic
95C4×Dic101D411yescharacteristic
96C4×Dic101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98C2×D4×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C2×C8×C101C2311yescharacteristic
101C4⋊C4×C101—21no—
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C22⋊C4×C101—21no—
106C22⋊C4×C101—21no—
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C22×D101—21no—
110C22×D101C2×D411yescharacteristic
111Q8×C101—21no—
112Q8×C101—21no—
113D4×C101—21no—
114D4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2×D411yescharacteristic
120C2×C4×C101C2411yescharacteristic
121C22⋊D101—42no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—42no—
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129D404—21no—
130D404—21no—
131D404—42no—
132C4×D101—22no—
133C4×D101—21no—
134C4×D101—21no—
135C4×D101—22no—
136C4×D101—42no—
137Dic202—21no—
138Dic202—21no—
139C8×C101—21no—
140C8×C101—21no—
141#11717579#4112yes—
142#11717579#4112yes—
143C22×C101—41no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101C22×D411yescharacteristic
147C2×D101—21no—
148C2×D101—21no—
149C2×D101—41no—
150C2×D101—41no—
151C2×D101—41no—
152C2×D101#13512yes—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101—41no—
158C4×C101C2×#4111yescharacteristic
159C4×C101C2×#4111yescharacteristic
160Dic101—21no—
161Dic101—21no—
162Dic101—41no—
163Dic101—41no—
164Dic101—41no—
165Dic101#13512yes—
166C2×C101—81no—
167C2×C101#53211yescharacteristic
168C2×C101C2×#13511yescharacteristic
169C2×C101C2×#13511yescharacteristic
170D101—22no—
171D101—81no—
172#3168—1011nomaximal, Sylow
173C101#316811yescharacteristic, complemented, Sylow
174#529—1011no—
175#529—1011no—
176#499—1011no—
177#495—1011no—
178#488—1011no—
179#488D10111yescharacteristic, complemented
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—2021no—
196#126—2021no—
197#125—1011no—
198#125C2×D10111yescharacteristic
199#124—1011no—
200#124C2×D10111yescharacteristic
201#123—2021no—
202#123—2021no—
203C22⋊Q8—1011no—
204C22⋊Q8—1011no—
205#121—1011no—
206#121—1011no—
207C4×Q8—1011no—
208C4×Q8—1011no—
209C4×D4—1011no—
210C4×D4—1011no—
211C4×D4—2021no—
212C4×D4—2021no—
213#117—2021no—
214#117—2021no—
215C2×C42—1011no—
216#107—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220#103—1011no—
221#103—1011no—
222#103C2×D10111yescharacteristic
223#103C2×D10111yescharacteristic
224#102—1011no—
225#102—1011no—
226#102C2×D10111yescharacteristic
227#102C2×D10111yescharacteristic
228C4×C8—1011no—
229C4×C8—1011no—
230C4×C8—1011no—
231C4×C8C2×D10111yescharacteristic
232C2×Q8—1011no—
233C2×Q8C22×D10111yescharacteristic
234C2×D4—1011no—
235C2×D4—2021no—
236C2×D4—2021no—
237C2×D4C22×D10111yescharacteristic
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—2021no—
242C22×C4—2021no—
243Q16—2022no—
244SD16—2022no—
245SD16—2022no—
246D8—2022no—
247C2×C8—1011no—
248C2×C8—1011no—
249C2×C8—2022no—
250C2×C8—2022no—
251C2×C8C22×D10111yescharacteristic
252C2×C8C22×D10111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—1011no—
256C4⋊C4—1011no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
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—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C22⋊C4—2021no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C42—1011no—
276C42—1011no—
277C42—1011no—
278C42—2021no—
279C42—2021no—
280C42C22×D10111yescharacteristic
281C23—21no—
282C23—1011no—
283C23—2021no—
284Q8—21no—
285Q8—21no—
286Q8—2021no—
287Q8—2021no—
288D4—21no—
289D4—21no—
290D4—4042no—
291D4—4042no—
292D4—2021no—
293D4—2021no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—1011no—
298C2×C4—1011no—
299C2×C4—1011no—
300C2×C4—4042no—
301C2×C4—4042no—
302C2×C4—2022no—
303C2×C4—2022no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4D4×D10111yescharacteristic
310C2×C4D4×D10111yescharacteristic
311C2×C4C23×D10111yescharacteristic
312C8—21no—
313C8—21no—
314C8—1012no—
315C8—1012no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—1012no—
320C22—4041no—
321C22—4041no—
322C22—4041no—
323C22—2021no—
324C22—2021no—
325C22C2×D4×D10111yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—1012no—
332C4—4041no—
333C4—4041no—
334C4—4041no—
335C4—2021no—
336C4—2021no—
337C4#41×D10111yescharacteristic
338C4#41×D10111yescharacteristic
339C2—81no—
340C2—8081no—
341C2—2022no—
342C2#2026572611yescharacteristic
343C2#135×D10111yescharacteristic
344C2#135×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#488×C101C2
socleC22×C101C22×D4
prime radical#20265412C2
hypercenter#488D101
hypocenterC101#3168
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20265712C2semidirect
2. #20265706C2semidirect
3. #20265576C2semidirect
4. #20265559C2semidirect
5. #20265532C2semidirect
6. #20265417C2semidirect
8. #20265409C2semidirect
9. #20265405C2semidirect
10. #20265269C2semidirect
11. #20265268C2semidirect
12. #20265262C2semidirect
13. #20265257C2semidirect
14. C4×C8×D101C2semidirect
15. #488×C101C2semidirect
173. C101#3168semidirect
179. #488D101semidirect

Maximal subgroups

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

subgroupindex
#202657122
#202657062
#202655762
#202655592
#202655322
#202654172
#202654122
#202654092
#202654052
#202652692
#202652682
#202652622
#202652572
C4×C8×D1012
#488×C1012
#3168101