Finite Group Databasebeta

← #27182983

Subgroups — #27182983

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265871C211yesmaximal, characteristic, complemented
2#20265815C211yesmaximal, characteristic, complemented
3#20265533C211yesmaximal, characteristic, complemented
4#20265511C211yesmaximal, characteristic, complemented
5#20265501C211yesmaximal, characteristic, complemented
6#20265475C211yesmaximal, characteristic, complemented
7#20265413C211yesmaximal, characteristic, complemented
8#20265409C211yesmaximal, characteristic, complemented
9#20265395C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265312C211yesmaximal, characteristic, complemented
12#20265257C211yesmaximal, characteristic, complemented
13C8×C22⋊D101C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#416×C101C211yesmaximal, characteristic, complemented
16C2×D4×D101C2211yescharacteristic
17C2×#13108061C2211yescharacteristic, complemented
18C2×D808C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×C8×D101C2211yescharacteristic
21#15692249—21no—
22#15692246C2211yescharacteristic, complemented
23#15692244C2211yescharacteristic
24#15692238—21no—
25#15692237—21no—
26#15692231C2211yescharacteristic
27#15692230C2211yescharacteristic
28#15692229C2211yescharacteristic
29#15692229C2211yescharacteristic, complemented
30#15692228—21no—
31#15692227—21no—
32#15692223—21no—
33C4×D404—21no—
34D4×Dic101C2211yescharacteristic
35#15692217C2211yescharacteristic, complemented
36C4×C22⋊D101C2211yescharacteristic
37C4⋊C4×D101C2211yescharacteristic
38C2×#13108051—21no—
39#15692189C2211yescharacteristic, complemented
40#15692188C2211yescharacteristic, complemented
41#15692186C2211yescharacteristic, complemented
42#15692184C2211yescharacteristic, complemented
43#15692179C2211yescharacteristic, complemented
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic, complemented
46#15692176C2211yescharacteristic
47#15692175C2211yescharacteristic
48#15692175C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic, complemented
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic, complemented
54#117×C101C2211yescharacteristic, complemented
55#103×C101C2211yescharacteristic, complemented
56#103×C101C2211yescharacteristic, complemented
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C23×D101—21no—
60C2×D404—21no—
61C2×D404C2311yescharacteristic
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101D411yescharacteristic
67D4×D101—22no—
68D4×D101—41no—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101C2311yescharacteristic
73#13108061—41no—
74D808—41no—
75#13108059—22no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C8×D101—41no—
79C4⋊Dic101—21no—
80C4⋊Dic101C2311yescharacteristic
81C4⋊Dic101C2311yescharacteristic
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101D411yescharacteristic
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051D411yescharacteristic
96C4×Dic101C2311yescharacteristic
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—21no—
101C2×D4×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104C2×C8×C101—41no—
105C2×C8×C101C2311yescharacteristic
106C2×C8×C101C2311yescharacteristic
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C22⋊C4×C101—21no—
110C42×C101—21no—
111C23×C101—21no—
112C23×C101C2×D411yescharacteristic
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—42no—
116C22×D101—42no—
117C22×D101—81no—
118C22×D101C2×D411yescharacteristic
119Q8×C101—21no—
120Q8×C101—21no—
121D4×C101—21no—
122D4×C101—42no—
123D4×C101#4112yes—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—42no—
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C22⋊D101—22no—
133C22⋊D101—42no—
134C22⋊D101—81no—
135C22⋊D101—81no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—81no—
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141C2×Dic101C2×D411yescharacteristic
142D404—21no—
143D404—21no—
144D404—21no—
145D404—21no—
146D404—42no—
147C4×D101—21no—
148C4×D101—21no—
149C4×D101—81no—
150C4×D101—81no—
151C8×C101—21no—
152C8×C101—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—22no—
156C22×C101—21no—
157C22×C101—21no—
158C22×C101—21no—
159C22×C101—41no—
160C22×C101—41no—
161C22×C101C22×D411yescharacteristic
162C2×D101—21no—
163C2×D101—21no—
164C2×D101—41no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—84no—
171C2×D101#13512yes—
172C4×C101—21no—
173C4×C101—41no—
174C4×C101—41no—
175C4×C101—41no—
176C4×C101C2×#4111yescharacteristic
177C4×C101H(5,𝔽2)11yescharacteristic
178Dic101—21no—
179Dic101—21no—
180Dic101—21no—
181Dic101—41no—
182Dic101—41no—
183C2×C101—42no—
184C2×C101—41no—
185C2×C101#54611yescharacteristic
186C2×C101C2×#13511yescharacteristic
187C2×C101#57611yescharacteristic
188D101—22no—
189D101—81no—
190D101—81no—
191#3297—1011nomaximal, Sylow
192C101#329711yescharacteristic, complemented, Sylow
193#553—1011no—
194#546—1011no—
195#488—1011no—
196#484—1011no—
197#482—1011no—
198#478—1011no—
199#466—1011no—
200#465—1011no—
201#463—1011no—
202#449—1011no—
203#447—1011no—
204C4×D8—1011no—
205C8×D4—1011no—
206#416D10111yescharacteristic, complemented
207C2×#102—1011no—
208C2×#41C2×D10111yescharacteristic
209C22×D4—1011no—
210C2×SD16—1011no—
211C2×D8—1011no—
212C2×D8—1011no—
213C22×C8—1011no—
214C22×C8C2×D10111yescharacteristic, complemented
215#126—2021no—
216#125—1011no—
217#124—1011no—
218#123—2021no—
219#123—2021no—
220C22⋊Q8—1011no—
221#121—1011no—
222#121—1011no—
223#121—1011no—
224#121—2021no—
225#121—2021no—
226#120—2021no—
227C4×D4—1011no—
228C4×D4—1011no—
229C4×D4—1011no—
230C4×D4—2021no—
231#117C2×D10111yescharacteristic, complemented
232C2×C4⋊C4—1011no—
233C2×C22⋊C4—2021no—
234#107—1011no—
235#107—1011no—
236#106—1011no—
237C4⋊C8—1011no—
238#103—1011no—
239#103C2×D10111yescharacteristic, complemented
240#103C2×D10111yescharacteristic, complemented
241#102—1011no—
242#102—1011no—
243#102—1011no—
244#102—1011no—
245#102—1011no—
246#102C2×D10111yescharacteristic
247#102C2×D10111yescharacteristic
248C22⋊C8—1011no—
249C22⋊C8—1011no—
250C4×C8—1011no—
251C24—2021no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10111yescharacteristic
255C2×D4—21no—
256C2×D4—1011no—
257C2×D4—1011no—
258C2×D4—1011no—
259C2×D4—4041no—
260C2×D4—2021no—
261C2×D4—2021no—
262C2×D4—2021no—
263C2×D4—2021no—
264C2×D4—2022no—
265C2×D4C22×D10111yescharacteristic
266C22×C4—21no—
267C22×C4—1011no—
268C22×C4—2021no—
269C22×C4—2021no—
270C22×C4—2021no—
271C22×C4C22×D10111yescharacteristic
272SD16—4041no—
273D8—4041no—
274D8—2022no—
275C2×C8—41no—
276C2×C8—1011no—
277C2×C8—1011no—
278C2×C8—4041no—
279C2×C8C22×D10111yescharacteristic
280C2×C8C22×D10111yescharacteristic
281C4⋊C4—1011no—
282C4⋊C4—1011no—
283C4⋊C4—1011no—
284C4⋊C4—2021no—
285C4⋊C4—2021no—
286C4⋊C4—2021no—
287C4⋊C4—2021no—
288C4⋊C4C22×D10111yescharacteristic
289C4⋊C4C22×D10111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—1011no—
292C22⋊C4—1011no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4—2021no—
297C22⋊C4—2021no—
298C22⋊C4—2021no—
299C22⋊C4—2021no—
300C22⋊C4—2021no—
301C42—21no—
302C42—1011no—
303C23—21no—
304C23—8081no—
305C23—1011no—
306C23—4042no—
307C23—4042no—
308C23—2021no—
309C23—2021no—
310C23D4×D10111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—8081no—
316D4—8081no—
317D4—4042no—
318D4—4042no—
319D4—2022no—
320D4—2021no—
321D4—2021no—
322D4—2021no—
323D4—2021no—
324D4#1310807412yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—8081no—
332C2×C4—8081no—
333C2×C4—8081no—
334C2×C4—1011no—
335C2×C4—1011no—
336C2×C4—1011no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4D4×D10111yescharacteristic
342C2×C4C23×D10111yescharacteristic
343C8—21no—
344C8—21no—
345C8—2021no—
346C8—2021no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—8084no—
354C22—1012no—
355C22—4041no—
356C22—4041no—
357C22—4041no—
358C22—4041no—
359C22—4041no—
360C22—4041no—
361C22—2021no—
362C22—2021no—
363C22C2×D4×D10111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—4041no—
369C4—4041no—
370C4—2021no—
371C4—2021no—
372C4—2021no—
373C4C2×#1310807411yescharacteristic
374C4#1569230011yescharacteristic
375C2—41no—
376C2—42no—
377C2—8081no—
378C2—8081no—
379C2—2022no—
380C2#2026581711yescharacteristic
381C2#135×D10111yescharacteristic
382C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
hypercenter#416D101
hypocenterC101#3297
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20265871C2semidirect
2. #20265815C2semidirect
3. #20265533C2semidirect
4. #20265511C2semidirect
5. #20265501C2semidirect
6. #20265475C2semidirect
7. #20265413C2semidirect
8. #20265409C2semidirect
9. #20265395C2semidirect
10. #20265322C2semidirect
11. #20265312C2semidirect
12. #20265257C2semidirect
13. C8×C22⋊D101C2semidirect
14. #102×D101C2semidirect
15. #416×C101C2semidirect
17. C2×#13108061C22semidirect
22. #15692246C22semidirect
29. #15692229C22semidirect
35. #15692217C22semidirect
39. #15692189C22semidirect
40. #15692188C22semidirect
41. #15692186C22semidirect
42. #15692184C22semidirect
43. #15692179C22semidirect
45. #15692177C22semidirect
49. C22⋊#11717579C22semidirect
51. C8×Dic101C22semidirect
53. C22×C8×C101C22semidirect
54. #117×C101C22semidirect
55. #103×C101C22semidirect
56. #103×C101C22semidirect
192. C101#3297semidirect
206. #416D101semidirect
214. C22×C8C2×D101semidirect
231. #117C2×D101semidirect
239. #103C2×D101semidirect
240. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658712
#202658152
#202655332
#202655112
#202655012
#202654752
#202654132
#202654092
#202653952
#202653222
#202653122
#202652572
C8×C22⋊D1012
#102×D1012
#416×C1012
#3297101