Finite Group Databasebeta

← #27181686

Subgroups — #27181686

Order 12928.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265729C211yesmaximal, characteristic, complemented
2#20265722C211yesmaximal, characteristic
3#20265574C211yesmaximal, characteristic, complemented
4#20265548C211yesmaximal, characteristic, complemented
5#20265526C211yesmaximal, characteristic, complemented
6#20265522C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265406C211yesmaximal, characteristic, complemented
9#20265404C211yesmaximal, characteristic, complemented
10#20265268C211yesmaximal, characteristic, complemented
11SD16×Dic101C211yesmaximal, characteristic, complemented
12#20265264C211yesmaximal, characteristic, complemented
13#20265262C211yesmaximal, characteristic, complemented
14C2×C8×Dic101C211yesmaximal, characteristic, complemented
15#465×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108063C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C22×#11717579C2211yescharacteristic
21#15692254C2211yescharacteristic
22#15692254C2211yescharacteristic
23#15692253C2211yescharacteristic
24#15692251C2211yescharacteristic
25#15692245—21no—
26#15692241—21no—
27#15692234C2211yescharacteristic
28#15692232—21no—
29#15692229C2211yescharacteristic
30#15692227—21no—
31Q8×Dic101C2211yescharacteristic
32#15692220—21no—
33#15692220C2211yescharacteristic
34D4×Dic101C2211yescharacteristic
35#15692217C2211yescharacteristic
36#15692215—21no—
37#15692211—21no—
38C2×C4×Dic101C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692178C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692175C2211yescharacteristic
48C8×Dic101C2211yescharacteristic
49C8×Dic101C2211yescharacteristic
50C8×Dic101C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×SD16×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×Dic202C2311yescharacteristic
63C2×D404C2311yescharacteristic
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C22×Dic101—21no—
68C22×Dic101D411yescharacteristic
69C22×Dic101D411yescharacteristic
70C2×C4×D101—21no—
71#13108063—22no—
72#13108062—22no—
73#13108061—22no—
74C2×#11717579—22no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C4⋊Dic101—21no—
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88#13108051—21no—
89#13108051—21no—
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101D411yescharacteristic
93C4×Dic101D411yescharacteristic
94C4×Dic101C2311yescharacteristic
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101C2311yescharacteristic
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100SD16×C101—22no—
101C2×C8×C101—22no—
102C2×C8×C101C2311yescharacteristic
103C2×C8×C101C2311yescharacteristic
104C4⋊C4×C101—21no—
105C4⋊C4×C101C2311yescharacteristic
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C22⋊C4×C101—21no—
109C23×C101—21no—
110C23×C101C2×D411yescharacteristic
111C22×D101—21no—
112Q8×C101—21no—
113Q8×C101—21no—
114D4×C101—21no—
115D4×C101—21no—
116D4×C101—42no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101—42no—
125C22⋊D101—42no—
126C22⋊D101—42no—
127C2×Dic101—22no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—42no—
132C2×Dic101SD1614yescomplemented
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137D404—21no—
138D404—21no—
139C4×D101—42no—
140Dic202—21no—
141Dic202—21no—
142Dic202—42no—
143Dic202—42no—
144C8×C101—21no—
145C8×C101#4112yescomplemented
146#11717579—21no—
147#11717579#4112yescomplemented
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101C2×SD1612yes—
154C22×C101C22×D411yescharacteristic
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C4×C101—21no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101C2×#4111yescharacteristic
163C4×C101C2×#4111yescharacteristic
164Dic101—21no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168Dic101C2×SD1612yes—
169Dic101C2×SD1612yes—
170Dic101#13512yes—
171C2×C101—22no—
172C2×C101—81no—
173C2×C101#53211yescharacteristic
174C2×C101C22×SD1611yescharacteristic
175C2×C101C2×#13511yescharacteristic
176D101—81no—
177#3167—1011nomaximal, Sylow
178C101#316711yescharacteristic, complemented, Sylow
179#533—1011no—
180#532—1011no—
181#498—1011no—
182#492—1011no—
183#487—1011no—
184#486—1011no—
185#465—1011no—
186#465—1011no—
187#465—1011no—
188#465D10111yescharacteristic, complemented
189C4×SD16—1011no—
190C4×SD16—1011no—
191C4×SD16—1011no—
192C4×SD16—1011no—
193C2×C4×C8—1011no—
194C2×#41—2021no—
195C2×SD16—1011no—
196C2×SD16—1011no—
197C2×SD16—1011no—
198C2×SD16C2×D10111yescharacteristic
199C22×C8—1011no—
200C22×C8C2×D10111yescharacteristic
201C4⋊Q8—1011no—
202C4⋊Q8—1011no—
203C4⋊Q8—1011no—
204#127—1011no—
205#125—2021no—
206#124—2021no—
207C22⋊Q8—1011no—
208C22⋊Q8—2021no—
209C22⋊Q8C2×D10111yescharacteristic
210#121—1011no—
211#121—2021no—
212#121C2×D10111yescharacteristic
213C4×Q8—1011no—
214C4×Q8—1011no—
215C4×Q8—2021no—
216C4×D4—1011no—
217C4×D4—1011no—
218C4×D4—2021no—
219#117—2021no—
220C2×C42—1011no—
221#106—1011no—
222#106—1011no—
223#106—1011no—
224#106C2×D10111yescharacteristic
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103C2×D10111yescharacteristic
229#102—1011no—
230#102—1011no—
231#102—1011no—
232#102C2×D10111yescharacteristic
233C4×C8—1011no—
234C4×C8—1011no—
235C4×C8—1011no—
236C4×C8—1011no—
237#41—4044no—
238C2×Q8—1011no—
239C2×Q8—2021no—
240C2×Q8—2021no—
241C2×Q8C22×D10111yescharacteristic
242C2×D4—21no—
243C2×D4—1011no—
244C2×D4—2021no—
245C2×D4—2021no—
246C2×D4—2021no—
247C2×D4C22×D10111yescharacteristic
248C22×C4—1011no—
249C22×C4—1011no—
250C22×C4—2021no—
251C22×C4—2021no—
252C22×C4C22×D10111yescharacteristic
253SD16—22no—
254SD16—2022no—
255SD16—2022no—
256SD16—2022no—
257C2×C8—22no—
258C2×C8—1011no—
259C2×C8—1011no—
260C2×C8—2022no—
261C2×C8C22×D10111yescharacteristic
262C2×C8C22×D10111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—1011no—
265C4⋊C4—1011no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4C22×D10111yescharacteristic
274C4⋊C4C22×D10111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C42—1011no—
282C42—1011no—
283C42—1011no—
284C42—1011no—
285C42—2021no—
286C42—2021no—
287C23—21no—
288C23—2021no—
289C23D4×D10111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—4042no—
293Q8—4042no—
294Q8—2021no—
295Q8—2021no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—4042no—
300D4—4042no—
301D4—4042no—
302D4—2021no—
303D4—2021no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—1011no—
312C2×C4—1014no—
313C2×C4—1011no—
314C2×C4—4042no—
315C2×C4—4042no—
316C2×C4—2022no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4D4×D10111yescharacteristic
321C2×C4C23×D10111yescharacteristic
322C8—21no—
323C8—1012no—
324C8—2021no—
325C8#1310807312yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—4041no—
332C22—4041no—
333C22—4041no—
334C22SD16×D10112yes—
335C22C2×D4×D10111yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1012no—
341C4—1012no—
342C4—1012no—
343C4—4041no—
344C4—4041no—
345C4—4041no—
346C4—2021no—
347C4#41×D10111yescharacteristic
348C4C2×#1310807311yescharacteristic
349C2—22no—
350C2—81no—
351C2—8081no—
352C2#2026572211yescharacteristic
353C2C2×SD16×D10111yescharacteristic
354C2#135×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#465×C101C2
socleC22×C101C22×D4
prime radical#20265722C2
hypercenter#465D101
hypocenterC101#3167
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20265729C2semidirect
3. #20265574C2semidirect
4. #20265548C2semidirect
5. #20265526C2semidirect
6. #20265522C2semidirect
7. #20265408C2semidirect
8. #20265406C2semidirect
9. #20265404C2semidirect
10. #20265268C2semidirect
11. SD16×Dic101C2semidirect
12. #20265264C2semidirect
13. #20265262C2semidirect
14. C2×C8×Dic101C2semidirect
15. #465×C101C2semidirect
132. C2×Dic101SD16semidirect
145. C8×C101#41semidirect
147. #11717579#41semidirect
178. C101#3167semidirect
188. #465D101semidirect
325. C8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202657292
#202657222
#202655742
#202655482
#202655262
#202655222
#202654082
#202654062
#202654042
#202652682
SD16×Dic1012
#202652642
#202652622
C2×C8×Dic1012
#465×C1012
#3167101