Finite Group Databasebeta

← #27183722

Subgroups — #27183722

Order 12928.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265891C211yesmaximal, characteristic, complemented
2#20265827C211yesmaximal, characteristic, complemented
3#20265574C211yesmaximal, characteristic, complemented
4#20265499C211yesmaximal, characteristic, complemented
5#20265498C211yesmaximal, characteristic, complemented
6#20265494C211yesmaximal, characteristic, complemented
7#20265493C211yesmaximal, characteristic, complemented
8#20265465C211yesmaximal, characteristic, complemented
9#20265455C211yesmaximal, characteristic, complemented
10#20265427C211yesmaximal, characteristic, complemented
11D8×Dic101C211yesmaximal, characteristic, complemented
12#20265253C211yesmaximal, characteristic, complemented
13#20265219C211yesmaximal, characteristic, complemented
14#20265216C211yesmaximal, characteristic, complemented
15#469×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C2×#13108057C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692254C2211yescharacteristic
20#15692242—21no—
21#15692241—21no—
22#15692237—21no—
23#15692236—21no—
24#15692232—21no—
25#15692227—21no—
26#15692226C2211yescharacteristic
27#15692226C2211yescharacteristic
28D4×Dic101C2211yescharacteristic
29D4×Dic101C2211yescharacteristic
30C4×C22⋊D101C2211yescharacteristic
31#15692214C2211yescharacteristic
32#15692213C2211yescharacteristic
33#15692211—21no—
34C2×C4⋊Dic101C2211yescharacteristic
35C4⋊C4×D101C2211yescharacteristic
36#15692189C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692186C2211yescharacteristic
42#15692186C2211yescharacteristic
43#15692186C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×D8×C101C2211yescharacteristic
53C2×M16×C101C2211yescharacteristic
54#121×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#106×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#13108073—44no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×C22⋊D101—21no—
63C2×C22⋊D101—21no—
64C2×C22⋊D101D411yescharacteristic
65C22×Dic101—21no—
66C22×Dic101—21no—
67C22×Dic101—21no—
68C2×C4×D101—21no—
69C2×C4×D101C2311yescharacteristic
70#13108057—22no—
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C4⋊Dic101—21no—
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053D411yescharacteristic
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101—21no—
86C22⋊Dic101—21no—
87C22⋊Dic101D411yescharacteristic
88#13108051—21no—
89#13108051D411yescharacteristic
90C4×Dic101—21no—
91C4×Dic101—21no—
92C4×Dic101C2311yescharacteristic
93C2×D4×C101—21no—
94C2×D4×C101—21no—
95C2×D4×C101C2311yescharacteristic
96C2×D4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98D8×C101—22no—
99M16×C101—22no—
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101C2×D411yescharacteristic
109C22×D101C2×D411yescharacteristic
110D4×C101—21no—
111D4×C101—21no—
112D4×C101—21no—
113D4×C101—21no—
114D4×C101—42no—
115D4×C101—81no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101C2×D411yescharacteristic
121C2×C4×C101C2411yescharacteristic
122C22⋊D101—22no—
123C22⋊D101—42no—
124C22⋊D101—81no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—42no—
130C2×Dic101—42no—
131C2×Dic101—42no—
132C2×Dic101C2×D411yescharacteristic
133C2×Dic101C2×D411yescharacteristic
134C2×Dic101C2×D411yescharacteristic
135C4×D101—21no—
136C4×D101—21no—
137C4×D101—42no—
138Dic202—42no—
139Dic202—81no—
140C8×C101—21no—
141C8×C101#4112yescomplemented
142#11717579—21no—
143#11717579—21no—
144C22×C101—21no—
145C22×C101—21no—
146C22×C101—21no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101—41no—
153C22×C101C22×D411yescharacteristic
154C2×D101—21no—
155C2×D101—21no—
156C2×D101—21no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101C2×#4111yescharacteristic
161C4×C1012−1+411yescharacteristic
162Dic101—21no—
163Dic101—21no—
164Dic101—41no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168Dic101AGL(1,ℤ/8)12yes—
169C2×C101—41no—
170C2×C101—81no—
171C2×C101—81no—
172C2×C101#54711yescharacteristic
173C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C101#57711yescharacteristic
175D101—41no—
176#3354—1011nomaximal, Sylow
177C101#335411yescharacteristic, complemented, Sylow
178#555—1011no—
179#547—1011no—
180#498—1011no—
181#482—1011no—
182#482—1011no—
183#481—1011no—
184#481—1011no—
185#476—1011no—
186#474—1011no—
187#469—1011no—
188#469D10111yescharacteristic, complemented
189C4×D8—1011no—
190#436—1011no—
191#428—1011no—
192#428—1011no—
193C2×#41—2021no—
194C2×D8C2×D10111yescharacteristic
195C2×M16—1011no—
196C2×M16C2×D10111yescharacteristic
197C4⋊Q8—1011no—
198C4⋊Q8—1011no—
199#124—2021no—
200#124—2021no—
201#123—2021no—
202#123—2021no—
203C22⋊Q8—2021no—
204#121—1011no—
205#121—1011no—
206#121—2021no—
207#121C2×D10111yescharacteristic
208#121C2×D10111yescharacteristic
209C4×D4—1011no—
210C4×D4—1011no—
211C4×D4—1011no—
212#117—1011no—
213#117—1011no—
214#117—2021no—
215C2×C4⋊C4—1011no—
216C2×C4⋊C4—1011no—
217#107—1011no—
218#107—1011no—
219#107—1011no—
220#106—1011no—
221#106—1011no—
222#106—1011no—
223#106—1011no—
224#106—1011no—
225#106C2×D10111yescharacteristic
226C4⋊C8—1011no—
227#102—1011no—
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102C2×D10111yescharacteristic
232#102C2×D10111yescharacteristic
233C22⋊C8—1011no—
234C22⋊C8—1011no—
235C4×C8—1011no—
236#41—4044no—
237C2×Q8—2021no—
238C2×Q8—2021no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—1011no—
242C2×D4—2021no—
243C2×D4—2021no—
244C2×D4C22×D10111yescharacteristic
245C2×D4C22×D10111yescharacteristic
246C22×C4—1011no—
247C22×C4—2021no—
248C22×C4—2021no—
249C22×C4—2021no—
250C22×C4—2021no—
251C22×C4C22×D10111yescharacteristic
252D8—22no—
253M16—22no—
254M16—2022no—
255C2×C8—1011no—
256C2×C8—1011no—
257C2×C8C22×D10111yescharacteristic
258C2×C8C22×D10111yescharacteristic
259C4⋊C4—1011no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—1011no—
263C4⋊C4—1011no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4C22×D10111yescharacteristic
269C4⋊C4C22×D10111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1011no—
273C22⋊C4—1011no—
274C22⋊C4—2021no—
275C22⋊C4—2021no—
276C22⋊C4—2021no—
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C42—1011no—
281C42—2021no—
282C42—2021no—
283C23—21no—
284C23—21no—
285C23—1011no—
286C23D4×D10111yescharacteristic
287Q8—8081no—
288Q8—4042no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—8081no—
296D4—4042no—
297D4—2022no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1011no—
303C2×C4—1011no—
304C2×C4—1011no—
305C2×C4—4042no—
306C2×C4—4042no—
307C2×C4—4042no—
308C2×C4—4042no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4D4×D10111yescharacteristic
316C2×C4C23×D10111yescharacteristic
317C8—21no—
318C8—2021no—
319C8—2021no—
320C8#1310807312yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—2021no—
331C22—2021no—
332C22—2021no—
333C22C2×D4×D10111yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—1012no—
338C4—4041no—
339C4—4041no—
340C4—4041no—
341C4—4041no—
342C4—2021no—
343C4—2021no—
344C4C2×#1310807311yescharacteristic
345C4#1569230211yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—4041no—
350C2#2026582711yescharacteristic
351C2AGL(1,ℤ/8)×D10111yescharacteristic
352C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3354
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #20265891C2semidirect
2. #20265827C2semidirect
3. #20265574C2semidirect
4. #20265499C2semidirect
5. #20265498C2semidirect
6. #20265494C2semidirect
7. #20265493C2semidirect
8. #20265465C2semidirect
9. #20265455C2semidirect
10. #20265427C2semidirect
11. D8×Dic101C2semidirect
12. #20265253C2semidirect
13. #20265219C2semidirect
14. #20265216C2semidirect
15. #469×C101C2semidirect
141. C8×C101#41semidirect
177. C101#3354semidirect
188. #469D101semidirect
320. C8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658912
#202658272
#202655742
#202654992
#202654982
#202654942
#202654932
#202654652
#202654552
#202654272
D8×Dic1012
#202652532
#202652192
#202652162
#469×C1012
#3354101