Finite Group Databasebeta

← #27183862

Subgroups — #27183862

Order 12928.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265903C211yesmaximal, characteristic, complemented
2#20265848C211yesmaximal, characteristic
3#20265542C211yesmaximal, characteristic, complemented
4#20265540C211yesmaximal, characteristic, complemented
5#20265522C211yesmaximal, characteristic, complemented
6#20265480C211yesmaximal, characteristic, complemented
7#20265477C211yesmaximal, characteristic, complemented
8#20265456C211yesmaximal, characteristic, complemented
9#20265377C211yesmaximal, characteristic, complemented
10#20265375C211yesmaximal, characteristic, complemented
11#20265375C211yesmaximal, characteristic, complemented
12#20265305C211yesmaximal, characteristic, complemented
13#20265280C211yesmaximal, characteristic, complemented
14#20265262C211yesmaximal, characteristic, complemented
15#440×C101C211yesmaximal, characteristic, complemented
16C2×#13108074—21no—
17C2×#13108061C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19#15692254—21no—
20#15692254C2211yescharacteristic
21#15692252C2211yescharacteristic
22#15692251C2211yescharacteristic
23#15692251C2211yescharacteristic
24#15692247—21no—
25#15692247C2211yescharacteristic
26#15692246—21no—
27#15692246C2211yescharacteristic
28#15692229—21no—
29#15692228—21no—
30#15692228—21no—
31Q8×Dic101C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33C4×Dic202—21no—
34C4×Dic202C2211yescharacteristic
35#15692217C2211yescharacteristic
36#15692217C2211yescharacteristic
37#15692188C2211yescharacteristic
38#15692186C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692177C2211yescharacteristic
43#15692177C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49#15692175C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×D4×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#97×C101C2211yescharacteristic
59#13108074—44no—
60C2×Dic202—21no—
61C2×Dic202D411yescharacteristic
62C2×D404—21no—
63C2×D404—21no—
64C2×D404C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72#13108061—22no—
73#13108061—22no—
74C2×#11717579C2311yescharacteristic
75C2×#11717579C2311yescharacteristic
76C4⋊Dic101—21no—
77C4⋊Dic101—21no—
78C4⋊Dic101—21no—
79C4⋊Dic101D411yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86#13108053D411yescharacteristic
87#13108053D411yescharacteristic
88#13108051—21no—
89#13108051—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101C2311yescharacteristic
93C4×Dic101C2311yescharacteristic
94C2×Q8×C101C2311yescharacteristic
95C2×D4×C101C2311yescharacteristic
96C22×C4×C101—21no—
97SD16×C101—22no—
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C42×C101—21no—
105C42×C101C2311yescharacteristic
106C23×C101—21no—
107C22×D101—21no—
108C22×D101—21no—
109Q8×C101—21no—
110Q8×C101#4112yescomplemented
111D4×C101—21no—
112D4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—42no—
117C2×C4×C101C2×D411yescharacteristic
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2411yescharacteristic
120C22⋊D101—42no—
121C22⋊D101—42no—
122C22⋊D101—81no—
123C22⋊D101—81no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130D404—21no—
131D404—21no—
132D404—21no—
133D404—21no—
134D404—42no—
135D404—81no—
136C4×D101—42no—
137C4×D101—42no—
138Dic202—22no—
139Dic202—42no—
140C8×C101—21no—
141C8×C101—21no—
142#11717579—21no—
143#11717579—21no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101C22×D411yescharacteristic
148C2×D101—41no—
149C2×D101—41no—
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C2×D101—41no—
154C4×C101—22no—
155C4×C101—21no—
156C4×C101—21no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101C2×#4111yescharacteristic
160C4×C1012−1+411yescharacteristic
161Dic101—21no—
162Dic101—21no—
163Dic101—21no—
164Dic101—41no—
165Dic101—41no—
166Dic101AGL(1,ℤ/8)12yes—
167C2×C101—81no—
168C2×C101#54811yescharacteristic
169C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C101#57711yescharacteristic
171D101—81no—
172D101—81no—
173#3368—1011nomaximal, Sylow
174C101#336811yescharacteristic, complemented, Sylow
175#557—1011no—
176#550—1011no—
177#490—1011no—
178#490—1011no—
179#486—1011no—
180#478—1011no—
181#478—1011no—
182#474—1011no—
183#460—1011no—
184#460—1011no—
185#460—1011no—
186#446—1011no—
187#440—1011no—
188#440D10111yescharacteristic, complemented
189C4×SD16—1011no—
190C2×#41—2021no—
191C2×SD16—1011no—
192C2×SD16—1011no—
193C2×SD16C2×D10111yescharacteristic
194C4⋊Q8—1011no—
195C4⋊Q8—2021no—
196#127—1011no—
197#127—1011no—
198#127—1011no—
199#125—1011no—
200#125—1011no—
201#125—2021no—
202#125—2021no—
203#121—2021no—
204#121—2021no—
205#121—2021no—
206C4×Q8—1011no—
207C4×Q8—1011no—
208C4×Q8—1011no—
209C4×Q8—2021no—
210C4×Q8C2×D10111yescharacteristic
211C4×D4—1011no—
212C4×D4—1011no—
213C4×D4C2×D10111yescharacteristic
214#107—1011no—
215#107C2×D10111yescharacteristic
216#106—1011no—
217C4⋊C8—1011no—
218C4⋊C8—1011no—
219C4⋊C8—1011no—
220#103C2×D10111yescharacteristic
221#102—1011no—
222#102—1011no—
223#102—1011no—
224#102—1011no—
225#102—1011no—
226#102—1011no—
227#102—1011no—
228#102—1011no—
229#102C2×D10111yescharacteristic
230#97—1011no—
231#97C2×D10111yescharacteristic
232C4×C8—1011no—
233#41—4044no—
234C2×Q8—1011no—
235C2×Q8—2021no—
236C2×Q8C22×D10111yescharacteristic
237C2×D4—1011no—
238C2×D4—1011no—
239C2×D4—2021no—
240C2×D4—2021no—
241C2×D4—2021no—
242C2×D4—2021no—
243C2×D4—2021no—
244C2×D4—2021no—
245C2×D4C22×D10111yescharacteristic
246C22×C4—21no—
247C22×C4—2021no—
248C22×C4—2021no—
249SD16—22no—
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—1011no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4C22×D10111yescharacteristic
270C4⋊C4C22×D10111yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C42—21no—
275C42—1011no—
276C42—1011no—
277C42—2021no—
278C42—2021no—
279C42C22×D10111yescharacteristic
280C23—21no—
281C23—2021no—
282C23—2021no—
283Q8—21no—
284Q8—4042no—
285Q8—2022no—
286Q8#1310807412yescomplemented
287D4—21no—
288D4—21no—
289D4—8081no—
290D4—8081no—
291D4—8081no—
292D4—4042no—
293D4—4042no—
294D4—4042no—
295D4—2021no—
296D4—2021no—
297D4—2021no—
298D4—2021no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—1011no—
306C2×C4—1011no—
307C2×C4—4042no—
308C2×C4—4042no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4D4×D10111yescharacteristic
312C2×C4D4×D10111yescharacteristic
313C2×C4C23×D10111yescharacteristic
314C8—21no—
315C8—21no—
316C8—2021no—
317C8—2021no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4041no—
322C22—4041no—
323C22—4041no—
324C22—4041no—
325C22—4041no—
326C22—4041no—
327C22C2×D4×D10111yescharacteristic
328C4—21no—
329C4—22no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—1012no—
334C4—4041no—
335C4—4041no—
336C4—2021no—
337C4—2021no—
338C4—2021no—
339C4C2×#1310807411yescharacteristic
340C4#1569230211yescharacteristic
341C2—81no—
342C2—8081no—
343C2—8081no—
344C2#2026583611yescharacteristic
345C2AGL(1,ℤ/8)×D10111yescharacteristic
346C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#440×C101C2
socleC22×C101C22×D4
prime radical#20265848C2
hypercenter#440D101
hypocenterC101#3368
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. #20265903C2semidirect
3. #20265542C2semidirect
4. #20265540C2semidirect
5. #20265522C2semidirect
6. #20265480C2semidirect
7. #20265477C2semidirect
8. #20265456C2semidirect
9. #20265377C2semidirect
10. #20265375C2semidirect
11. #20265375C2semidirect
12. #20265305C2semidirect
13. #20265280C2semidirect
14. #20265262C2semidirect
15. #440×C101C2semidirect
110. Q8×C101#41semidirect
174. C101#3368semidirect
188. #440D101semidirect
286. Q8#13108074semidirect

Maximal subgroups

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

subgroupindex
#202659032
#202658482
#202655422
#202655402
#202655222
#202654802
#202654772
#202654562
#202653772
#202653752
#202653052
#202652802
#202652622
#440×C1012
#3368101