Finite Group Databasebeta

← #27183823

Subgroups — #27183823

Order 12928.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265909C211yesmaximal, characteristic
2#20265903C211yesmaximal, characteristic, complemented
3#20265533C211yesmaximal, characteristic, complemented
4#20265531C211yesmaximal, characteristic, complemented
5#20265522C211yesmaximal, characteristic
6#20265478C211yesmaximal, characteristic, complemented
7#20265475C211yesmaximal, characteristic, complemented
8#20265460C211yesmaximal, characteristic, complemented
9#20265401C211yesmaximal, characteristic, complemented
10#20265375C211yesmaximal, characteristic
11#20265301C211yesmaximal, characteristic, complemented
12Q16×Dic101C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14C4×#13108061C211yesmaximal, characteristic, complemented
15#464×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17#15692254—21no—
18#15692254C2211yescharacteristic
19#15692251C2211yescharacteristic
20#15692247—21no—
21#15692247C2211yescharacteristic
22#15692246—21no—
23#15692246C2211yescharacteristic
24#15692244C2211yescharacteristic
25#15692241—21no—
26#15692239—21no—
27#15692237—21no—
28Q8×Dic101C2211yescharacteristic
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31C4×Dic202—21no—
32C4×Dic202C2211yescharacteristic
33C4×D404C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692215—21no—
36#15692189C2211yescharacteristic
37#15692187C2211yescharacteristic
38#15692186C2211yescharacteristic
39#15692184C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692179C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692175C2211yescharacteristic
49C4×#11717579C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54#124×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202D411yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C22×Dic101—21no—
65C2×C4×D101—21no—
66#13108061—22no—
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2311yescharacteristic
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101D411yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82#13108053D411yescharacteristic
83C22⋊Dic101—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87C4×Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101C2311yescharacteristic
91C4×Dic101C2311yescharacteristic
92C2×Q8×C101C2311yescharacteristic
93C2×Q8×C101C2311yescharacteristic
94C2×D4×C101C2311yescharacteristic
95Q16×C101—22no—
96SD16×C101—22no—
97C2×C8×C101C2311yescharacteristic
98C2×C8×C101C2311yescharacteristic
99C4⋊C4×C101—21no—
100C4⋊C4×C101C2311yescharacteristic
101C22⋊C4×C101—21no—
102C22⋊C4×C101—21no—
103C42×C101—21no—
104C42×C101C2311yescharacteristic
105C23×C101—21no—
106C22×D101—21no—
107Q8×C101—21no—
108Q8×C101—21no—
109Q8×C101—21no—
110Q8×C101#4112yes—
111D4×C101—21no—
112D4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2×D411yescharacteristic
118C2×C4×C101C2411yescharacteristic
119C22⋊D101—42no—
120C22⋊D101—81no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—81no—
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128C2×Dic101C2×D411yescharacteristic
129D404—21no—
130D404—21no—
131C4×D101—81no—
132Dic202—22no—
133Dic202—42no—
134C8×C101—21no—
135C8×C101—21no—
136#11717579—21no—
137#11717579—21no—
138C22×C101—41no—
139C22×C101—41no—
140C22×C101—41no—
141C22×C101C22×D411yescharacteristic
142C2×D101—41no—
143C2×D101—41no—
144C2×D101—41no—
145C4×C101—22no—
146C4×C101—21no—
147C4×C101—21no—
148C4×C101—41no—
149C4×C101—41no—
150C4×C101C2×#4111yescharacteristic
151C4×C1012−1+411yescharacteristic
152Dic101—21no—
153Dic101—21no—
154Dic101—21no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
158Dic101#13512yes—
159C2×C101—81no—
160C2×C101#54811yescharacteristic
161C2×C101C2×#13511yescharacteristic
162C2×C101#57711yescharacteristic
163D101—81no—
164#3365—1011nomaximal, Sylow
165C101#336511yescharacteristic, complemented, Sylow
166#559—1011no—
167#557—1011no—
168#488—1011no—
169#488—1011no—
170#486—1011no—
171#478—1011no—
172#478—1011no—
173#475—1011no—
174#464—1011no—
175#464D10111yescharacteristic, complemented
176#460—1011no—
177C8×Q8—1011no—
178C4×Q16—1011no—
179C4×SD16—1011no—
180C4×SD16—1011no—
181C2×Q16C2×D10111yescharacteristic
182C2×SD16—1011no—
183C2×SD16C2×D10111yescharacteristic
184C4⋊Q8—1011no—
185C4⋊Q8—2021no—
186#127—1011no—
187#125—1011no—
188#125—1011no—
189#125—2021no—
190#125—2021no—
191#124—1011no—
192#124—2021no—
193#124C2×D10111yescharacteristic
194#123—2021no—
195#123—2021no—
196C4×Q8—1011no—
197C4×Q8—1011no—
198C4×Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—2021no—
201C4×Q8C2×D10111yescharacteristic
202C4×D4—1011no—
203C4×D4—1011no—
204C4×D4—2021no—
205#107—1011no—
206#106—1011no—
207#106—1011no—
208C4⋊C8—1011no—
209C4⋊C8—1011no—
210C4⋊C8C2×D10111yescharacteristic
211#103—1011no—
212#103—1011no—
213#103—1011no—
214#103C2×D10111yescharacteristic
215#102—1011no—
216#102—1011no—
217#102—1011no—
218#102—1011no—
219#102—1011no—
220#102C2×D10111yescharacteristic
221C4×C8—1011no—
222C4×C8—1011no—
223C4×C8—1011no—
224C2×Q8—1011no—
225C2×Q8—2021no—
226C2×Q8C22×D10111yescharacteristic
227C2×Q8C22×D10111yescharacteristic
228C2×D4—1011no—
229C2×D4—2021no—
230C2×D4—2021no—
231C2×D4C22×D10111yescharacteristic
232C22×C4—2021no—
233C22×C4—2021no—
234Q16—22no—
235SD16—22no—
236SD16—2022no—
237C2×C8—1011no—
238C2×C8—1011no—
239C2×C8C22×D10111yescharacteristic
240C2×C8C22×D10111yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—1011no—
243C4⋊C4—1011no—
244C4⋊C4—1011no—
245C4⋊C4—1011no—
246C4⋊C4—1011no—
247C4⋊C4—1011no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4C22×D10111yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C42—21no—
264C42—1011no—
265C42—1011no—
266C42—2021no—
267C42—2021no—
268C42—2021no—
269C42C22×D10111yescharacteristic
270C23—21no—
271C23—2021no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—4042no—
276Q8—2022no—
277Q8#1310807312yes—
278D4—21no—
279D4—21no—
280D4—8081no—
281D4—4042no—
282D4—2021no—
283D4—2021no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—8081no—
288C2×C4—8081no—
289C2×C4—1011no—
290C2×C4—1011no—
291C2×C4—1011no—
292C2×C4—1011no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4D4×D10111yescharacteristic
297C2×C4D4×D10111yescharacteristic
298C2×C4C23×D10111yescharacteristic
299C8—21no—
300C8—21no—
301C8—2021no—
302C8—2021no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—4041no—
307C22—4041no—
308C22—4041no—
309C22C2×D4×D10111yescharacteristic
310C4—21no—
311C4—22no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—1012no—
316C4—4041no—
317C4—4041no—
318C4—4041no—
319C4—2021no—
320C4—2021no—
321C4—2021no—
322C4C2×#1310807311yescharacteristic
323C4#1569230211yescharacteristic
324C2—81no—
325C2—8081no—
326C2#2026583311yescharacteristic
327C2#135×D10111yescharacteristic
328C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radical#15692251C22
hypercenter#464D101
hypocenterC101#3365
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #20265903C2semidirect
3. #20265533C2semidirect
4. #20265531C2semidirect
6. #20265478C2semidirect
7. #20265475C2semidirect
8. #20265460C2semidirect
9. #20265401C2semidirect
11. #20265301C2semidirect
12. Q16×Dic101C2semidirect
13. SD16×Dic101C2semidirect
14. C4×#13108061C2semidirect
15. #464×C101C2semidirect
165. C101#3365semidirect
175. #464D101semidirect

Maximal subgroups

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

subgroupindex
#202659092
#202659032
#202655332
#202655312
#202655222
#202654782
#202654752
#202654602
#202654012
#202653752
#202653012
Q16×Dic1012
SD16×Dic1012
C4×#131080612
#464×C1012
#3365101