Finite Group Databasebeta

← #27182574

Subgroups — #27182574

Order 12928.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265787C211yesmaximal, characteristic, complemented
2#20265780C211yesmaximal, characteristic
3#20265486C211yesmaximal, characteristic, complemented
4#20265483C211yesmaximal, characteristic, complemented
5#20265480C211yesmaximal, characteristic, complemented
6#20265475C211yesmaximal, characteristic, complemented
7#20265423C211yesmaximal, characteristic, complemented
8#20265408C211yesmaximal, characteristic, complemented
9#20265404C211yesmaximal, characteristic, complemented
10#20265381C211yesmaximal, characteristic, complemented
11#20265380C211yesmaximal, characteristic, complemented
12#20265376C211yesmaximal, characteristic, complemented
13#20265375C211yesmaximal, characteristic, complemented
14#20265205C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic, complemented
16C2×#13108063C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C22×#11717579C2211yescharacteristic
20#15692253C2211yescharacteristic
21#15692251C2211yescharacteristic
22#15692248—21no—
23#15692248—21no—
24#15692246—21no—
25#15692246C2211yescharacteristic
26#15692246C2211yescharacteristic
27#15692245—21no—
28#15692239—21no—
29#15692237—21no—
30#15692234C2211yescharacteristic
31#15692229C2211yescharacteristic
32#15692227—21no—
33#15692226—21no—
34Q8×Dic101C2211yescharacteristic
35#15692220C2211yescharacteristic
36D4×Dic101C2211yescharacteristic
37#15692217C2211yescharacteristic
38#15692214C2211yescharacteristic
39#15692189C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692184C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692177C2211yescharacteristic
50#15692176C2211yescharacteristic
51#15692175C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#107×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C2×D404C2311yescharacteristic
61C2×C22⋊D101—21no—
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C22×Dic101—21no—
65C2×C4×D101—21no—
66#13108063—41no—
67#13108062—41no—
68#13108061—41no—
69C2×#11717579—41no—
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101D411yescharacteristic
76C4⋊Dic101D411yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101D411yescharacteristic
88C22⋊Dic101D411yescharacteristic
89#13108051—21no—
90#13108051—21no—
91C4×Dic101—21no—
92C4×Dic101—21no—
93C4×Dic101C2311yescharacteristic
94C4×Dic101C2311yescharacteristic
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99SD16×C101—41no—
100M16×C101—41no—
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
110C22×D101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113D4×C101—21no—
114D4×C101—21no—
115D4×C101—81no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101C2×D411yescharacteristic
122C2×C4×C101C2411yescharacteristic
123C22⋊D101—81no—
124C22⋊D101—81no—
125C22⋊D101—81no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—81no—
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134D404—21no—
135D404—21no—
136C4×D101—81no—
137Dic202—21no—
138Dic202—21no—
139C8×C101—21no—
140C8×C101—21no—
141#11717579—21no—
142#11717579—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—21no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101C22×D411yescharacteristic
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C4×C101—21no—
154C4×C101—41no—
155C4×C101—41no—
156C4×C101—41no—
157C4×C101H(5,𝔽2)11yescharacteristic
158C4×C1012−1+411yescharacteristic
159Dic101—21no—
160Dic101—21no—
161Dic101—21no—
162Dic101—21no—
163Dic101—41no—
164Dic101—41no—
165Dic101—41no—
166C2×C101—41no—
167C2×C101—81no—
168C2×C101#53711yescharacteristic
169C2×C101#57711yescharacteristic
170C2×C101#57711yescharacteristic
171D101—81no—
172#3260—1011nomaximal, Sylow
173C101#326011yescharacteristic, complemented, Sylow
174#541—1011no—
175#540—1011no—
176#479—1011no—
177#479—1011no—
178#478—1011no—
179#478—1011no—
180#468—1011no—
181#468D10111yescharacteristic, complemented
182#465—1011no—
183#465—1011no—
184#461—1011no—
185#461—1011no—
186#460—1011no—
187#460—1011no—
188#424—1011no—
189C2×SD16—1011no—
190C2×SD16—1011no—
191C2×SD16—1011no—
192C2×SD16C2×D10111yescharacteristic
193C2×M16C2×D10111yescharacteristic
194C22×C8—1011no—
195C4⋊Q8—1011no—
196#127—1011no—
197#126—2021no—
198#126—2021no—
199#125—1011no—
200#125—1011no—
201#125—2021no—
202#125—2021no—
203#123—2021no—
204#123—2021no—
205C22⋊Q8—1011no—
206C22⋊Q8C2×D10111yescharacteristic
207#121—1011no—
208#121—2021no—
209#121—2021no—
210#121C2×D10111yescharacteristic
211C4×Q8—1011no—
212C4×Q8—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215#117—1011no—
216#107—1011no—
217#107C2×D10111yescharacteristic
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223C4⋊C8—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103C2×D10111yescharacteristic
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102C2×D10111yescharacteristic
232C2×Q8—1011no—
233C2×Q8C22×D10111yescharacteristic
234C2×D4—21no—
235C2×D4—1011no—
236C2×D4—2021no—
237C2×D4—2021no—
238C2×D4—2021no—
239C2×D4C22×D10111yescharacteristic
240C22×C4—2021no—
241C22×C4—2021no—
242C22×C4C22×D10111yescharacteristic
243SD16—41no—
244SD16—4041no—
245SD16—4041no—
246SD16—4041no—
247M16—41no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8—4041no—
251C2×C8C22×D10111yescharacteristic
252C2×C8C22×D10111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1011no—
255C4⋊C4—1011no—
256C4⋊C4—1011no—
257C4⋊C4—1011no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4—2021no—
262C4⋊C4—2021no—
263C4⋊C4—2021no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4C22×D10111yescharacteristic
268C4⋊C4C22×D10111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1011no—
272C22⋊C4—1011no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C42—1011no—
278C42—1011no—
279C42—2021no—
280C42—2021no—
281C23—21no—
282C23—2021no—
283C23D4×D10111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—2021no—
287Q8—2021no—
288D4—21no—
289D4—21no—
290D4—81no—
291D4—8081no—
292D4—8081no—
293D4—8081no—
294D4—2021no—
295D4—2021no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—8081no—
302C2×C4—8081no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—1011no—
306C2×C4—1011no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4D4×D10111yescharacteristic
311C2×C4C23×D10111yescharacteristic
312C8—21no—
313C8—21no—
314C8—2021no—
315C8—2021no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—4041no—
323C22—4041no—
324C22—4041no—
325C22C2×D4×D10111yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—4041no—
331C4—4041no—
332C4—4041no—
333C4—2021no—
334C4—2021no—
335C4—2021no—
336C4—2021no—
337C4#1569230011yescharacteristic
338C4#1569230211yescharacteristic
339C2—41no—
340C2—81no—
341C2—8081no—
342C2#2026575411yescharacteristic
343C2#2026598611yescharacteristic
344C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
prime radical#20265780C2
hypercenter#468D101
hypocenterC101#3260
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. #20265787C2semidirect
3. #20265486C2semidirect
4. #20265483C2semidirect
5. #20265480C2semidirect
6. #20265475C2semidirect
7. #20265423C2semidirect
8. #20265408C2semidirect
9. #20265404C2semidirect
10. #20265381C2semidirect
11. #20265380C2semidirect
12. #20265376C2semidirect
13. #20265375C2semidirect
14. #20265205C2semidirect
15. #468×C101C2semidirect
173. C101#3260semidirect
181. #468D101semidirect

Maximal subgroups

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

subgroupindex
#202657872
#202657802
#202654862
#202654832
#202654802
#202654752
#202654232
#202654082
#202654042
#202653812
#202653802
#202653762
#202653752
#202652052
#468×C1012
#3260101