Finite Group Databasebeta

← #27181731

Subgroups — #27181731

Order 12928.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265729C211yesmaximal, characteristic, complemented
2#20265708C211yesmaximal, characteristic
3#20265585C211yesmaximal, characteristic, complemented
4#20265568C211yesmaximal, characteristic, complemented
5#20265544C211yesmaximal, characteristic, complemented
6#20265535C211yesmaximal, characteristic, complemented
7#20265435C211yesmaximal, characteristic, complemented
8#20265433C211yesmaximal, characteristic, complemented
9#20265421C211yesmaximal, characteristic
10#20265271C211yesmaximal, characteristic, complemented
11#20265269C211yesmaximal, characteristic, complemented
12#20265268C211yesmaximal, characteristic, complemented
13SD16×Dic101C211yesmaximal, characteristic, complemented
14M16×Dic101C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic
16C2×Dic404C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108058C2211yescharacteristic
20#15692254C2211yescharacteristic
21#15692253C2211yescharacteristic
22#15692248—21no—
23#15692246C2211yescharacteristic
24#15692245—21no—
25#15692242C2211yescharacteristic
26#15692236—21no—
27#15692234C2211yescharacteristic
28#15692234C2211yescharacteristic
29#15692232—21no—
30Q8×Dic101C2211yescharacteristic
31#15692220—21no—
32#15692220C2211yescharacteristic
33#15692220C2211yescharacteristic
34D4×Dic101—21no—
35D4×Dic101C2211yescharacteristic
36#15692211—21no—
37#15692211—21no—
38C2×C4×Dic101C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692165C2211yescharacteristic
49#15692165C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
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×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62C22×Dic101—21no—
63C22×Dic101D411yescharacteristic
64C22×Dic101D411yescharacteristic
65Dic404—22no—
66#13108064—22no—
67#13108062—22no—
68#13108058—22no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75#13108053—21no—
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C4×Dic101—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101D411yescharacteristic
92C4×Dic101D411yescharacteristic
93C4×Dic101C2311yescharacteristic
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99SD16×C101—22no—
100M16×C101—22no—
101C2×C8×C101C2311yescharacteristic
102C2×C8×C101C2311yescharacteristic
103C4⋊C4×C101—21no—
104C4⋊C4×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C22⋊C4×C101—21no—
107C22⋊C4×C101—21no—
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110Q8×C101—21no—
111Q8×C101—21no—
112D4×C101—21no—
113D4×C101—21no—
114D4×C101—42no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2411yescharacteristic
122C2×Dic101—22no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—41no—
128C2×Dic101—42no—
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133Dic202—21no—
134Dic202—21no—
135Dic202—21no—
136Dic202—21no—
137Dic202—42no—
138C8×C101—21no—
139C8×C101#4112yes—
140#11717579—21no—
141#11717579#4112yes—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101C22×D411yescharacteristic
149C4×C101—21no—
150C4×C101—41no—
151C4×C101—41no—
152C4×C101—41no—
153C4×C101C2×#4111yescharacteristic
154C4×C101C2×#4111yescharacteristic
155Dic101—21no—
156Dic101—21no—
157Dic101—21no—
158Dic101—41no—
159Dic101—41no—
160Dic101—41no—
161Dic101—41no—
162Dic101#13712yes—
163C2×C101—41no—
164C2×C101—81no—
165C2×C101#53211yescharacteristic
166C2×C101C2×#13711yescharacteristic
167C2×C101#57511yescharacteristic
168#3172—1011nomaximal, Sylow
169C101#317211yescharacteristic, complemented, Sylow
170#533—1011no—
171#529—1011no—
172#501—1011no—
173#497—1011no—
174#491—1011no—
175#489—1011no—
176#470—1011no—
177#470—1011no—
178#468—1011no—
179#468D10111yescharacteristic
180C4×Q16—1011no—
181C4×Q16—1011no—
182C4×SD16—1011no—
183C4×SD16—1011no—
184C4×M16—1011no—
185C2×Q16—1011no—
186C2×Q16—1011no—
187C2×SD16—1011no—
188C2×SD16C2×D10111yescharacteristic
189C2×M16—1011no—
190C2×M16C2×D10111yescharacteristic
191C4⋊Q8—1011no—
192C4⋊Q8—1011no—
193#126—2021no—
194#125—1011no—
195#125—2021no—
196#124—1011no—
197#123—2021no—
198C22⋊Q8—1011no—
199C22⋊Q8—1011no—
200C22⋊Q8—2021no—
201C22⋊Q8C2×D10111yescharacteristic
202#121C2×D10111yescharacteristic
203C4×Q8—1011no—
204C4×Q8—1011no—
205C4×Q8—1011no—
206C4×Q8—2021no—
207C4×D4—1011no—
208C4×D4—2021no—
209#117—2021no—
210#117—2021no—
211C2×C42—1011no—
212#107—1011no—
213#107C2×D10111yescharacteristic
214#106—1011no—
215#106—1011no—
216#103—1011no—
217#103—1011no—
218#103—1011no—
219#103—1011no—
220#103—1011no—
221#103C2×D10111yescharacteristic
222#102—1011no—
223#102C2×D10111yescharacteristic
224#97—1011no—
225#97—1011no—
226C4×C8—1011no—
227C4×C8—1011no—
228C2×Q8—1011no—
229C2×Q8—1011no—
230C2×Q8—2021no—
231C2×Q8C22×D10111yescharacteristic
232C2×D4—21no—
233C2×D4C22×D10111yescharacteristic
234C22×C4—1011no—
235C22×C4—1011no—
236C22×C4—2021no—
237C22×C4C22×D10111yescharacteristic
238Q16—2022no—
239Q16—2022no—
240SD16—22no—
241SD16—2022no—
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—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4—2021no—
257C4⋊C4—2021no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4C22×D10111yescharacteristic
262C4⋊C4C22×D10111yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—2021no—
266C22⋊C4—2021no—
267C22⋊C4—2021no—
268C22⋊C4—2021no—
269C22⋊C4—2021no—
270C42—1011no—
271C42—1011no—
272C42—1011no—
273C42—1011no—
274C42—2021no—
275C42—2021no—
276C42—2021no—
277C23—21no—
278C23D4×D10111yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—4042no—
282Q8—2021no—
283Q8—2021no—
284Q8—2021no—
285Q8—2021no—
286D4—21no—
287D4—21no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—1011no—
295C2×C4—1011no—
296C2×C4—1011no—
297C2×C4—1011no—
298C2×C4—4042no—
299C2×C4—4041no—
300C2×C4—2022no—
301C2×C4—2021no—
302C2×C4—2021no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4D4×D10111yescharacteristic
306C2×C4C23×D10111yescharacteristic
307C8—21no—
308C8—1012no—
309C8—2021no—
310C8#1310807312yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C2×D4×D10111yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—1012no—
323C4—4041no—
324C4—4041no—
325C4—4041no—
326C4—4041no—
327C4—2021no—
328C4—2021no—
329C4—2021no—
330C4#41×D10111yescharacteristic
331C4C2×#1310807311yescharacteristic
332C2—41no—
333C2—81no—
334C2#2026572211yescharacteristic
335C2#137×D10111yescharacteristic
336C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
prime radical#121×C101C22
hypercenter#468D101
hypocenterC101#3172
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
1. #20265729C2semidirect
3. #20265585C2semidirect
4. #20265568C2semidirect
5. #20265544C2semidirect
6. #20265535C2semidirect
7. #20265435C2semidirect
8. #20265433C2semidirect
10. #20265271C2semidirect
11. #20265269C2semidirect
12. #20265268C2semidirect
13. SD16×Dic101C2semidirect
14. M16×Dic101C2semidirect
169. C101#3172semidirect

Maximal subgroups

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

subgroupindex
#202657292
#202657082
#202655852
#202655682
#202655442
#202655352
#202654352
#202654332
#202654212
#202652712
#202652692
#202652682
SD16×Dic1012
M16×Dic1012
#468×C1012
#3172101