Finite Group Databasebeta

← #27183753

Subgroups — #27183753

Order 12928.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265891C211yesmaximal, characteristic
2#20265885C211yesmaximal, characteristic, complemented
3#20265581C211yesmaximal, characteristic, complemented
4#20265516C211yesmaximal, characteristic, complemented
5#20265515C211yesmaximal, characteristic, complemented
6#20265489C211yesmaximal, characteristic, complemented
7#20265488C211yesmaximal, characteristic, complemented
8#20265470C211yesmaximal, characteristic, complemented
9#20265465C211yesmaximal, characteristic, complemented
10#20265420C211yesmaximal, characteristic, complemented
11SD16×Dic101C211yesmaximal, characteristic, complemented
12#20265253C211yesmaximal, characteristic, complemented
13#20265219C211yesmaximal, characteristic, complemented
14#20265218C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic, complemented
16C2×#13108057C2211yescharacteristic
17#15692254C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692245—21no—
20#15692242—21no—
21#15692241—21no—
22#15692235—21no—
23#15692232—21no—
24#15692231C2211yescharacteristic
25#15692227—21no—
26#15692226C2211yescharacteristic
27Q8×Dic101C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29C4×C22⋊D101C2211yescharacteristic
30#15692215—21no—
31#15692214C2211yescharacteristic
32#15692213C2211yescharacteristic
33#15692211—21no—
34C2×C4⋊Dic101C2211yescharacteristic
35C4⋊C4×D101C2211yescharacteristic
36#15692189C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692187C2211yescharacteristic
42#15692187C2211yescharacteristic
43#15692186C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202—21no—
61C2×C22⋊D101—21no—
62C2×C22⋊D101D411yescharacteristic
63C22×Dic101—21no—
64C22×Dic101—21no—
65C2×C4×D101—21no—
66C2×C4×D101C2311yescharacteristic
67#13108057—22no—
68C2×#11717579C2311yescharacteristic
69C2×#11717579C2311yescharacteristic
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053D411yescharacteristic
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101D411yescharacteristic
85#13108051—21no—
86#13108051—21no—
87#13108051D411yescharacteristic
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C2×Q8×C101C2311yescharacteristic
93C2×D4×C101—21no—
94C2×D4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96SD16×C101—22no—
97M16×C101—22no—
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C23×C101—21no—
106C23×C101C2×D411yescharacteristic
107C22×D101C2×D411yescharacteristic
108Q8×C101—21no—
109Q8×C101—21no—
110D4×C101—21no—
111D4×C101—21no—
112D4×C101—81no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101C2×D411yescharacteristic
119C2×C4×C101C2411yescharacteristic
120C22⋊D101—22no—
121C22⋊D101—81no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—42no—
127C2×Dic101—81no—
128C2×Dic101C2×D411yescharacteristic
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C4×D101—21no—
132C4×D101—21no—
133C4×D101—81no—
134Dic202—81no—
135Dic202—81no—
136C8×C101—21no—
137C8×C101#4112yes—
138#11717579—21no—
139#11717579—21no—
140C22×C101—21no—
141C22×C101—21no—
142C22×C101—21no—
143C22×C101—41no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101C22×D411yescharacteristic
147C2×D101—21no—
148C2×D101—21no—
149C2×D101—21no—
150C4×C101—21no—
151C4×C101—41no—
152C4×C101—41no—
153C4×C101—41no—
154C4×C101C2×#4111yescharacteristic
155C4×C1012−1+411yescharacteristic
156Dic101—21no—
157Dic101—21no—
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161Dic101—41no—
162Dic101#13712yes—
163C2×C101—41no—
164C2×C101—81no—
165C2×C101#54711yescharacteristic
166C2×C101C2×#13711yescharacteristic
167C2×C101#57611yescharacteristic
168D101—41no—
169#3357—1011nomaximal, Sylow
170C101#335711yescharacteristic, complemented, Sylow
171#555—1011no—
172#554—1011no—
173#500—1011no—
174#485—1011no—
175#485—1011no—
176#480—1011no—
177#480—1011no—
178#477—1011no—
179#476—1011no—
180#468—1011no—
181#468D10111yescharacteristic, complemented
182C4×SD16—1011no—
183#436—1011no—
184#428—1011no—
185#428—1011no—
186C2×SD16C2×D10111yescharacteristic
187C2×M16—1011no—
188C2×M16C2×D10111yescharacteristic
189C4⋊Q8—1011no—
190C4⋊Q8—1011no—
191#125—2021no—
192#124—2021no—
193#124—2021no—
194C22⋊Q8—1011no—
195C22⋊Q8—2021no—
196C22⋊Q8—2021no—
197C22⋊Q8C2×D10111yescharacteristic
198#121—1011no—
199#121—2021no—
200#121C2×D10111yescharacteristic
201C4×Q8—1011no—
202C4×D4—1011no—
203C4×D4—1011no—
204C4×D4—2021no—
205#117—1011no—
206#117—1011no—
207#117—2021no—
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—1011no—
210#107—1011no—
211#107—1011no—
212#107—1011no—
213#107—1011no—
214#107—1011no—
215#107C2×D10111yescharacteristic
216#106—1011no—
217#106—1011no—
218#106—1011no—
219C4⋊C8—1011no—
220#103—1011no—
221#103—1011no—
222#103C2×D10111yescharacteristic
223#102—1011no—
224#102—1011no—
225#102C2×D10111yescharacteristic
226C22⋊C8—1011no—
227C22⋊C8—1011no—
228C4×C8—1011no—
229C2×Q8—2021no—
230C2×Q8—2021no—
231C2×Q8C22×D10111yescharacteristic
232C2×D4—21no—
233C2×D4—1011no—
234C2×D4—2021no—
235C2×D4C22×D10111yescharacteristic
236C22×C4—1011no—
237C22×C4—2021no—
238C22×C4—2021no—
239C22×C4—2021no—
240C22×C4C22×D10111yescharacteristic
241SD16—22no—
242M16—22no—
243M16—2022no—
244C2×C8—1011no—
245C2×C8—1011no—
246C2×C8C22×D10111yescharacteristic
247C2×C8C22×D10111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—1011no—
250C4⋊C4—1011no—
251C4⋊C4—1011no—
252C4⋊C4—1011no—
253C4⋊C4—1011no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4C22×D10111yescharacteristic
261C4⋊C4C22×D10111yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—1011no—
265C22⋊C4—1011no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C22⋊C4—2021no—
271C42—1011no—
272C42—2021no—
273C42—2021no—
274C42—2021no—
275C23—21no—
276C23—1011no—
277C23D4×D10111yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—8081no—
281Q8—8081no—
282D4—21no—
283D4—21no—
284D4—81no—
285D4—8081no—
286D4—2022no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—8081no—
293C2×C4—8081no—
294C2×C4—1011no—
295C2×C4—1011no—
296C2×C4—1011no—
297C2×C4—4042no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4—2021no—
303C2×C4—2021no—
304C2×C4D4×D10111yescharacteristic
305C2×C4C23×D10111yescharacteristic
306C8—21no—
307C8—2021no—
308C8—2021no—
309C8#1310807312yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—2021no—
317C22—2021no—
318C22—2021no—
319C22C2×D4×D10111yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—1012no—
325C4—4041no—
326C4—4041no—
327C4—4041no—
328C4—4041no—
329C4—2021no—
330C4—2021no—
331C4C2×#1310807311yescharacteristic
332C4#1569230211yescharacteristic
333C2—41no—
334C2—81no—
335C2—4041no—
336C2#2026582711yescharacteristic
337C2#137×D10111yescharacteristic
338C2#2026598111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
prime radical#20265891C2
hypercenter#468D101
hypocenterC101#3357
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20265885C2semidirect
3. #20265581C2semidirect
4. #20265516C2semidirect
5. #20265515C2semidirect
6. #20265489C2semidirect
7. #20265488C2semidirect
8. #20265470C2semidirect
9. #20265465C2semidirect
10. #20265420C2semidirect
11. SD16×Dic101C2semidirect
12. #20265253C2semidirect
13. #20265219C2semidirect
14. #20265218C2semidirect
15. #468×C101C2semidirect
170. C101#3357semidirect
181. #468D101semidirect

Maximal subgroups

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

subgroupindex
#202658912
#202658852
#202655812
#202655162
#202655152
#202654892
#202654882
#202654702
#202654652
#202654202
SD16×Dic1012
#202652532
#202652192
#202652182
#468×C1012
#3357101