Finite Group Databasebeta

← #27183812

Subgroups — #27183812

Order 12928.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265903C211yesmaximal, characteristic, complemented
2#20265896C211yesmaximal, characteristic
3#20265531C211yesmaximal, characteristic
4#20265530C211yesmaximal, characteristic
5#20265526C211yesmaximal, characteristic, complemented
6#20265485C211yesmaximal, characteristic, complemented
7#20265482C211yesmaximal, characteristic, complemented
8#20265455C211yesmaximal, characteristic
9#20265397C211yesmaximal, characteristic
10#20265385C211yesmaximal, characteristic, complemented
11#20265301C211yesmaximal, characteristic, complemented
12Q16×Dic101C211yesmaximal, characteristic, complemented
13C4×#13108064C211yesmaximal, characteristic, complemented
14SD16×Dic101C211yesmaximal, characteristic
15#464×C101C211yesmaximal, characteristic
16C2×#13108064C2211yescharacteristic
17#15692254—21no—
18#15692254C2211yescharacteristic
19#15692253C2211yescharacteristic
20#15692248—21no—
21#15692248—21no—
22#15692247—21no—
23#15692247C2211yescharacteristic
24#15692246—21no—
25#15692246C2211yescharacteristic
26#15692242C2211yescharacteristic
27#15692232—21no—
28Q8×Dic101C2211yescharacteristic
29Q8×Dic101C2211yescharacteristic
30Q8×Dic101C2211yescharacteristic
31C4×Dic202—21no—
32C4×Dic202C2211yescharacteristic
33C4×Dic202C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692211—21no—
36#15692189C2211yescharacteristic
37#15692188C2211yescharacteristic
38#15692187C2211yescharacteristic
39#15692184C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692180C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
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×Dic202C2311yescharacteristic
62C22×Dic101—21no—
63#13108064—22no—
64C2×#11717579C2311yescharacteristic
65C2×#11717579C2311yescharacteristic
66C4⋊Dic101—21no—
67C4⋊Dic101—21no—
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101D411yescharacteristic
71C4⋊Dic101C2311yescharacteristic
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74#13108053—21no—
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82#13108053D411yescharacteristic
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C4×Dic101—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—
106Q8×C101—21no—
107Q8×C101—21no—
108Q8×C101—21no—
109Q8×C101#4112yes—
110D4×C101—21no—
111D4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2411yescharacteristic
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—42no—
123C2×Dic101C2×D411yescharacteristic
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127Dic202—22no—
128Dic202—21no—
129Dic202—21no—
130Dic202—42no—
131C8×C101—21no—
132C8×C101—21no—
133#11717579—21no—
134#11717579—21no—
135C22×C101—41no—
136C22×C101—41no—
137C22×C101—41no—
138C22×C101C22×D411yescharacteristic
139C4×C101—22no—
140C4×C101—21no—
141C4×C101—21no—
142C4×C101—41no—
143C4×C101—41no—
144C4×C101C2×#4111yescharacteristic
145C4×C1012−1+411yescharacteristic
146Dic101—21no—
147Dic101—21no—
148Dic101—21no—
149Dic101—41no—
150Dic101—41no—
151Dic101—41no—
152Dic101—41no—
153Dic101#13512yes—
154C2×C101—81no—
155C2×C101#54811yescharacteristic
156C2×C101C2×#13511yescharacteristic
157C2×C101#57811yescharacteristic
158#3364—1011nomaximal, Sylow
159C101#336411yescharacteristic, complemented, Sylow
160#557—1011no—
161#556—1011no—
162#488—1011no—
163#488—1011no—
164#487—1011no—
165#479—1011no—
166#479—1011no—
167#474—1011no—
168#464—1011no—
169#464D10111yescharacteristic
170#462—1011no—
171C8×Q8—1011no—
172C4×Q16—1011no—
173C4×Q16—1011no—
174C4×SD16—1011no—
175C2×Q16—1011no—
176C2×Q16C2×D10111yescharacteristic
177C2×SD16C2×D10111yescharacteristic
178C4⋊Q8—1011no—
179C4⋊Q8—1011no—
180C4⋊Q8—2021no—
181#126—2021no—
182#126—2021no—
183#125—1011no—
184#125—1011no—
185#125—2021no—
186#125—2021no—
187#124—1011no—
188#124C2×D10111yescharacteristic
189C22⋊Q8—2021no—
190C4×Q8—1011no—
191C4×Q8—1011no—
192C4×Q8—1011no—
193C4×Q8—1011no—
194C4×Q8—1011no—
195C4×Q8—2021no—
196C4×Q8C2×D10111yescharacteristic
197C4×D4—1011no—
198#117—2021no—
199#107—1011no—
200#107—1011no—
201#106—1011no—
202C4⋊C8—1011no—
203C4⋊C8—1011no—
204C4⋊C8C2×D10111yescharacteristic
205#103—1011no—
206#103—1011no—
207#103—1011no—
208#103—1011no—
209#103—1011no—
210#103—1011no—
211#103C2×D10111yescharacteristic
212#102—1011no—
213#102—1011no—
214#102C2×D10111yescharacteristic
215C4×C8—1011no—
216C4×C8—1011no—
217C4×C8—1011no—
218C2×Q8—1011no—
219C2×Q8—1011no—
220C2×Q8—2021no—
221C2×Q8C22×D10111yescharacteristic
222C2×Q8C22×D10111yescharacteristic
223C2×D4C22×D10111yescharacteristic
224C22×C4—2021no—
225Q16—22no—
226Q16—2022no—
227SD16—22no—
228C2×C8—1011no—
229C2×C8—1011no—
230C2×C8C22×D10111yescharacteristic
231C2×C8C22×D10111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—1011no—
234C4⋊C4—1011no—
235C4⋊C4—1011no—
236C4⋊C4—1011no—
237C4⋊C4—1011no—
238C4⋊C4—1011no—
239C4⋊C4—2021no—
240C4⋊C4—2021no—
241C4⋊C4—2021no—
242C4⋊C4—2021no—
243C4⋊C4—2021no—
244C4⋊C4—2021no—
245C4⋊C4—2021no—
246C4⋊C4—2021no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4C22×D10111yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—2021no—
254C22⋊C4—2021no—
255C22⋊C4—2021no—
256C42—21no—
257C42—1011no—
258C42—1011no—
259C42—2021no—
260C42—2021no—
261C42—2021no—
262C42—2021no—
263C42C22×D10111yescharacteristic
264C23—21no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—4042no—
269Q8—2022no—
270Q8—2021no—
271Q8—2021no—
272Q8#1310807312yes—
273D4—21no—
274D4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—1011no—
279C2×C4—1011no—
280C2×C4—1011no—
281C2×C4—1011no—
282C2×C4—4042no—
283C2×C4—2021no—
284C2×C4—2021no—
285C2×C4—2021no—
286C2×C4—2021no—
287C2×C4D4×D10111yescharacteristic
288C2×C4D4×D10111yescharacteristic
289C2×C4C23×D10111yescharacteristic
290C8—21no—
291C8—21no—
292C8—2021no—
293C8—2021no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22C2×D4×D10111yescharacteristic
298C4—21no—
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—1012no—
304C4—4041no—
305C4—4041no—
306C4—4041no—
307C4—4041no—
308C4—2021no—
309C4—2021no—
310C4—2021no—
311C4C2×#1310807311yescharacteristic
312C4#1569230211yescharacteristic
313C2—81no—
314C2#2026583311yescharacteristic
315C2#135×D10111yescharacteristic
316C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×C101C23
hypercenter#464D101
hypocenterC101#3364
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
1. #20265903C2semidirect
5. #20265526C2semidirect
6. #20265485C2semidirect
7. #20265482C2semidirect
10. #20265385C2semidirect
11. #20265301C2semidirect
12. Q16×Dic101C2semidirect
13. C4×#13108064C2semidirect
159. C101#3364semidirect

Maximal subgroups

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

subgroupindex
#202659032
#202658962
#202655312
#202655302
#202655262
#202654852
#202654822
#202654552
#202653972
#202653852
#202653012
Q16×Dic1012
C4×#131080642
SD16×Dic1012
#464×C1012
#3364101