Finite Group Databasebeta

← #27183818

Subgroups — #27183818

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#20265899C211yesmaximal, characteristic
3#20265533C211yesmaximal, characteristic
4#20265532C211yesmaximal, characteristic
5#20265526C211yesmaximal, characteristic, complemented
6#20265484C211yesmaximal, characteristic, complemented
7#20265483C211yesmaximal, characteristic, complemented
8#20265456C211yesmaximal, characteristic
9#20265401C211yesmaximal, characteristic
10#20265399C211yesmaximal, characteristic
11#20265385C211yesmaximal, characteristic, complemented
12C8×Dic202C211yesmaximal, characteristic, complemented
13#20265269C211yesmaximal, characteristic, complemented
14#20265262C211yesmaximal, characteristic
15C4×Q16×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108061C2211yescharacteristic
18#15692254—21no—
19#15692254C2211yescharacteristic
20#15692253C2211yescharacteristic
21#15692250—21no—
22#15692249—21no—
23#15692247—21no—
24#15692247C2211yescharacteristic
25#15692246—21no—
26#15692246C2211yescharacteristic
27#15692244C2211yescharacteristic
28#15692243C2211yescharacteristic
29#15692233—21no—
30Q8×Dic101C2211yescharacteristic
31Q8×Dic101C2211yescharacteristic
32C4×Dic202—21no—
33C4×Dic202C2211yescharacteristic
34#15692220C2211yescharacteristic
35#15692217C2211yescharacteristic
36#15692212—21no—
37#15692188C2211yescharacteristic
38#15692186C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692184C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692177C2211yescharacteristic
48#15692177C2211yescharacteristic
49#15692175C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×Q16×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54C4×Q8×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C4×C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202D411yescharacteristic
61C2×Dic202C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C4×D101—21no—
64#13108064—22no—
65#13108061—22no—
66C2×#11717579C2311yescharacteristic
67C2×#11717579C2311yescharacteristic
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101D411yescharacteristic
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
83#13108051—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87#13108051—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93C2×Q8×C101C2311yescharacteristic
94C2×Q8×C101C2311yescharacteristic
95Q16×C101—22no—
96C2×C8×C101C2311yescharacteristic
97C2×C8×C101C2311yescharacteristic
98C4⋊C4×C101—21no—
99C4⋊C4×C101—21no—
100C4⋊C4×C101C2311yescharacteristic
101C4⋊C4×C101C2311yescharacteristic
102C42×C101—21no—
103C42×C101—21no—
104C42×C101C2311yescharacteristic
105C22×D101—21no—
106Q8×C101—21no—
107Q8×C101—21no—
108Q8×C101—21no—
109Q8×C101#4112yes—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101C2×D411yescharacteristic
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2411yescharacteristic
117C2×Dic101—21no—
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101C2×D411yescharacteristic
121C2×Dic101C2×D411yescharacteristic
122C2×Dic101C2×D411yescharacteristic
123C2×Dic101C2×D411yescharacteristic
124D404—21no—
125D404—21no—
126C4×D101—42no—
127Dic202—22no—
128Dic202—21no—
129Dic202—21no—
130Dic202—42no—
131C8×C101—21no—
132C8×C101—21no—
133#11717579—21no—
134#11717579—21no—
135C22×C101C22×D411yescharacteristic
136C2×D101—41no—
137C2×D101—41no—
138C2×D101—41no—
139C4×C101—22no—
140C4×C101—21no—
141C4×C101—21no—
142C4×C101—41no—
143C4×C101—41no—
144C4×C101—41no—
145C4×C101C2×#4111yescharacteristic
146C4×C1012−1+411yescharacteristic
147Dic101—21no—
148Dic101—21no—
149Dic101—21no—
150Dic101—41no—
151Dic101—41no—
152Dic101—41no—
153Dic101#13512yes—
154C2×C101#54811yescharacteristic
155C2×C101C2×#13511yescharacteristic
156C2×C101#57811yescharacteristic
157D101—81no—
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#464—1011no—
170#462—1011no—
171C8×Q8—1011no—
172C4×Q16—1011no—
173C4×Q16D10111yescharacteristic, complemented
174C4×SD16—1011no—
175C2×Q16—1011no—
176C2×Q16C2×D10111yescharacteristic
177C2×SD16—1011no—
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#124—1011no—
189C22⋊Q8—2021no—
190C4×Q8—1011no—
191C4×Q8—1011no—
192C4×Q8—1011no—
193C4×Q8—1011no—
194C4×Q8—2021no—
195C4×Q8C2×D10111yescharacteristic
196C4×Q8C2×D10111yescharacteristic
197C4×D4—1011no—
198#117—2021no—
199#107—1011no—
200#107C2×D10111yescharacteristic
201#106—1011no—
202C4⋊C8—1011no—
203C4⋊C8—1011no—
204C4⋊C8—1011no—
205#103—1011no—
206#103—1011no—
207#103—1011no—
208#103—1011no—
209#103—1011no—
210#103C2×D10111yescharacteristic
211#103C2×D10111yescharacteristic
212#102—1011no—
213#102—1011no—
214#102—1011no—
215C4×C8—1011no—
216C4×C8—1011no—
217C4×C8C2×D10111yescharacteristic
218C2×Q8—1011no—
219C2×Q8—1011no—
220C2×Q8—2021no—
221C2×Q8C22×D10111yescharacteristic
222C2×Q8C22×D10111yescharacteristic
223C2×D4—1011no—
224C22×C4—2021no—
225Q16—22no—
226Q16—2022no—
227SD16—2022no—
228C2×C8—1011no—
229C2×C8—1011no—
230C2×C8C22×D10111yescharacteristic
231C2×C8C22×D10111yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
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⋊C4C22×D10111yescharacteristic
250C4⋊C4C22×D10111yescharacteristic
251C22⋊C4—2021no—
252C22⋊C4—2021no—
253C22⋊C4—2021no—
254C22⋊C4—2021no—
255C22⋊C4—2021no—
256C42—21no—
257C42—21no—
258C42—1011no—
259C42—1011no—
260C42—2021no—
261C42—2021no—
262C42—2021no—
263C42C22×D10111yescharacteristic
264C23—2021no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—4042no—
269Q8—2022no—
270Q8—2021no—
271Q8—2021no—
272Q8#1310807412yes—
273D4—2021no—
274D4—2021no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—1011no—
280C2×C4—1011no—
281C2×C4—1011no—
282C2×C4—1011no—
283C2×C4—4042no—
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—4041no—
295C22—4041no—
296C22—4041no—
297C22C2×D4×D10111yescharacteristic
298C4—21no—
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—1012no—
305C4—4041no—
306C4—4041no—
307C4—4041no—
308C4—2021no—
309C4—2021no—
310C4—2021no—
311C4C2×#1310807411yescharacteristic
312C4#1569230211yescharacteristic
313C2—8081no—
314C2#2026583611yescharacteristic
315C2#135×D10111yescharacteristic
316C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×Q16×C101C2
socleC22×C101C22×D4
prime radicalC2×D404C23
hypercenterC4×Q16D101
hypocenterC101#3364
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #20265903C2semidirect
5. #20265526C2semidirect
6. #20265484C2semidirect
7. #20265483C2semidirect
11. #20265385C2semidirect
12. C8×Dic202C2semidirect
13. #20265269C2semidirect
15. C4×Q16×C101C2semidirect
159. C101#3364semidirect
173. C4×Q16D101semidirect

Maximal subgroups

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

subgroupindex
#202659032
#202658992
#202655332
#202655322
#202655262
#202654842
#202654832
#202654562
#202654012
#202653992
#202653852
C8×Dic2022
#202652692
#202652622
C4×Q16×C1012
#3364101