Finite Group Databasebeta

← #27180623

Subgroups — #27180623

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692287C211yesmaximal, characteristic, complemented
2C2×#15692282C211yesmaximal, characteristic
3#124×D101C211yesmaximal, characteristic, complemented
4C4×#13108075C211yesmaximal, characteristic, complemented
5#20265401C211yesmaximal, characteristic, complemented
6#20265399C211yesmaximal, characteristic, complemented
7#20265397C211yesmaximal, characteristic, complemented
8#20265395C211yesmaximal, characteristic, complemented
9#20265393C211yesmaximal, characteristic, complemented
10#20265391C211yesmaximal, characteristic, complemented
11#20265389C211yesmaximal, characteristic, complemented
12#20265199C211yesmaximal, characteristic, complemented
13#20265184C211yesmaximal, characteristic, complemented
14#20265180C211yesmaximal, characteristic, complemented
15#464×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692287—24no—
20#15692282—24no—
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108063C2211yescharacteristic, complemented
23C2×#13108061C2211yescharacteristic
24C2×#13108061C2211yescharacteristic, complemented
25C2×D808C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108057C2211yescharacteristic
28C2×#13108057C2211yescharacteristic
29#15692244C2211yescharacteristic
30#15692243C2211yescharacteristic
31#15692242C2211yescharacteristic
32#15692241—21no—
33#15692241—21no—
34Q8×Dic101C2211yescharacteristic, complemented
35C4×D404—21no—
36C4×D404C2211yescharacteristic, complemented
37#15692217—21no—
38#15692217C2211yescharacteristic, complemented
39#15692213—21no—
40#15692213C2211yescharacteristic
41C22⋊C4×D101—21no—
42C22⋊C4×D101—21no—
43C42×D101—21no—
44C42×D101C2211yescharacteristic
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692184C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692178C2211yescharacteristic, complemented
51#15692177C2211yescharacteristic
52#15692176C2211yescharacteristic
53#15692175C2211yescharacteristic
54C2×Q16×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic
56#124×C101C2211yescharacteristic
57C4×Q8×C101C2211yescharacteristic, complemented
58C4⋊C8×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic
61C23×D101—21no—
62#13108075—22no—
63#13108075D414yes—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66Q8×D101—22no—
67Q8×D101—22no—
68C2×D404—21no—
69C2×D404C2311yescharacteristic
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72D4×D101—22no—
73D4×D101—22no—
74C2×C4×D101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101D411yescharacteristic
79C2×C4×D101D411yescharacteristic
80C2×C4×D101C2311yescharacteristic
81#13108064—22no—
82#13108063—22no—
83#13108061—22no—
84#13108061—22no—
85D808—22no—
86#13108059—22no—
87#13108057—22no—
88#13108057—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C4⋊Dic101—21no—
92C4⋊Dic101C2311yescharacteristic
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102C4×Dic101—21no—
103C4×Dic101—21no—
104C4×Dic101D411yescharacteristic
105C4×Dic101D411yescharacteristic
106C4×Dic101C2311yescharacteristic
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110Q16×C101—22no—
111SD16×C101—22no—
112C2×C8×C101C2311yescharacteristic
113C2×C8×C101C2311yescharacteristic
114C4⋊C4×C101—21no—
115C4⋊C4×C101C2311yescharacteristic
116C22⋊C4×C101—21no—
117C22⋊C4×C101—21no—
118C42×C101—21no—
119C42×C101C2311yescharacteristic
120C23×C101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—42no—
124C22×D101—42no—
125C22×D101—42no—
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101—21no—
129Q8×C101—21no—
130Q8×C101C2×D412yes—
131D4×C101—21no—
132D4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2411yescharacteristic
139C22⋊D101—42no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101C2×D411yescharacteristic
143C2×Dic101C2×D411yescharacteristic
144C2×Dic101C2×D411yescharacteristic
145D404—24no—
146D404—21no—
147D404—21no—
148D404—21no—
149D404C2×D412yes—
150C4×D101—24no—
151C4×D101—41no—
152C4×D101—41no—
153C4×D101—42no—
154C4×D101—42no—
155C4×D101—42no—
156C4×D101C2×D412yes—
157C4×D101#4112yes—
158Dic202—21no—
159Dic202—21no—
160Dic202—42no—
161C8×C101—21no—
162C8×C101—21no—
163#11717579—21no—
164#11717579—21no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101C22×D411yescharacteristic
169C2×D101—21no—
170C2×D101—21no—
171C2×D101—22no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—41no—
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—84no—
178C2×D101#12112yes—
179C4×C101—22no—
180C4×C101—21no—
181C4×C101—21no—
182C4×C101—41no—
183C4×C101—41no—
184C4×C101C22×D411yescharacteristic
185C4×C101C2×#4111yescharacteristic
186Dic101—21no—
187Dic101—21no—
188Dic101—41no—
189Dic101—41no—
190Dic101#12112yes—
191C2×C101—81no—
192C2×C101C2×#12111yescharacteristic
193C2×C101#57511yescharacteristic
194C2×C101#57511yescharacteristic
195D101—22no—
196D101—42no—
197D101—81no—
198#3040—1011nomaximal, Sylow
199C101#304011yescharacteristic, complemented, Sylow
200C2×#137—1011no—
201C2×AGL(1,ℤ/8)—1011no—
202C2×#124—1011no—
203C4×#41—1011no—
204#464—1011no—
205#464—1011no—
206#464—1011no—
207#464D10111yescharacteristic, complemented
208#463—1011no—
209#463—1011no—
210#463—1011no—
211#463—1011no—
212#423—1011no—
213#419—1011no—
214#418—1011no—
215C2×#41—1011no—
216C22×Q8—1011no—
217C22×D4—1011no—
218#137—2024no—
219AGL(1,ℤ/8)—2024no—
220C2×Q16—1011no—
221C2×Q16C2×D10111yescharacteristic, complemented
222C2×SD16—1011no—
223C2×SD16—1011no—
224C2×SD16—1011no—
225C2×SD16C2×D10111yescharacteristic
226C2×D8—1011no—
227C2×D8—1011no—
228C2×M16—1011no—
229C2×M16—1011no—
230#124—1011no—
231#124—1011no—
232#124—1011no—
233#124—2021no—
234#124—2021no—
235#124C2×D10111yescharacteristic
236C4×Q8—1011no—
237C4×Q8C2×D10111yescharacteristic, complemented
238C4×D4—1011no—
239C4×D4—1011no—
240C4×D4—2021no—
241C4×D4—2021no—
242#117—1011no—
243#117—2021no—
244C2×C22⋊C4—2021no—
245C2×C22⋊C4—2021no—
246C2×C42—1011no—
247C2×C42—2021no—
248C4⋊C8—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251C4⋊C8C2×D10111yescharacteristic, complemented
252#103—1011no—
253#103—1011no—
254#103—1011no—
255#103C2×D10111yescharacteristic, complemented
256#102—1011no—
257#102—1011no—
258#102—1011no—
259#102C2×D10111yescharacteristic
260C24—2021no—
261#41—1014no—
262#41—2022no—
263C2×Q8—1011no—
264C2×Q8—2021no—
265C2×Q8—2022no—
266C2×Q8—2022no—
267C2×Q8C22×D10111yescharacteristic
268C2×Q8C22×D10111yescharacteristic
269C2×D4—1011no—
270C2×D4—1011no—
271C2×D4—2021no—
272C2×D4—2022no—
273C2×D4—2022no—
274C2×D4—2021no—
275C2×D4C22×D10111yescharacteristic
276C22×C4—1011no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—2021no—
280C22×C4—2021no—
281C22×C4—2021no—
282C22×C4—2021no—
283Q16—22no—
284Q16—2022no—
285SD16—22no—
286SD16—2022no—
287SD16—2022no—
288SD16—2022no—
289D8—2022no—
290D8—2022no—
291M16—2022no—
292M16—2022no—
293C2×C8—1011no—
294C2×C8—1011no—
295C2×C8C22×D10111yescharacteristic
296C2×C8C22×D10111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1011no—
299C4⋊C4—2021no—
300C4⋊C4C22×D10111yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2021no—
304C22⋊C4—2021no—
305C22⋊C4—2021no—
306C22⋊C4—2021no—
307C22⋊C4—2021no—
308C22⋊C4—2021no—
309C22⋊C4—2021no—
310C22⋊C4—2021no—
311C22⋊C4—2021no—
312C42—21no—
313C42—1011no—
314C42—1011no—
315C42—1011no—
316C42—2021no—
317C42—2021no—
318C42C22×D10111yescharacteristic
319C23—21no—
320C23—1011no—
321C23—4042no—
322C23—4042no—
323C23—4042no—
324C23—2021no—
325C23—2021no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—4042no—
330Q8—2021no—
331Q8—2021no—
332Q8D4×D10112yes—
333D4—21no—
334D4—21no—
335D4—1012no—
336D4—4042no—
337D4—2024no—
338D4—2021no—
339D4—2021no—
340D4—2021no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1011no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1011no—
348C2×C4—1011no—
349C2×C4—4042no—
350C2×C4—4041no—
351C2×C4—4041no—
352C2×C4—4042no—
353C2×C4—4042no—
354C2×C4—2024no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4D4×D10111yescharacteristic
358C2×C4D4×D10111yescharacteristic
359C2×C4C23×D10111yescharacteristic
360C8—21no—
361C8—21no—
362C8—2021no—
363C8—2021no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—8084no—
368C22—1012no—
369C22—4041no—
370C22—4041no—
371C22—4041no—
372C22—4041no—
373C22—4041no—
374C22—2022no—
375C22—2021no—
376C22—2021no—
377C22C2×D4×D10111yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—1012no—
384C4—4041no—
385C4—4041no—
386C4—2021no—
387C4—2021no—
388C4C2×D4×D10111yescharacteristic
389C4#41×D10111yescharacteristic
390C2—81no—
391C2—8081no—
392C2—4042no—
393C2—2022no—
394C2#121×D10111yescharacteristic
395C2#2026597411yescharacteristic
396C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#464×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692282C2
hypercenter#464D101
hypocenterC101#3040
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#15692287C2semidirect
3. #124×D101C2semidirect
4. C4×#13108075C2semidirect
5. #20265401C2semidirect
6. #20265399C2semidirect
7. #20265397C2semidirect
8. #20265395C2semidirect
9. #20265393C2semidirect
10. #20265391C2semidirect
11. #20265389C2semidirect
12. #20265199C2semidirect
13. #20265184C2semidirect
14. #20265180C2semidirect
15. #464×C101C2semidirect
21. C2×#13108064C22semidirect
22. C2×#13108063C22semidirect
24. C2×#13108061C22semidirect
34. Q8×Dic101C22semidirect
36. C4×D404C22semidirect
38. #15692217C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692184C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692178C22semidirect
54. C2×Q16×C101C22semidirect
57. C4×Q8×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #103×C101C22semidirect
199. C101#3040semidirect
207. #464D101semidirect
221. C2×Q16C2×D101semidirect
237. C4×Q8C2×D101semidirect
251. C4⋊C8C2×D101semidirect
255. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922872
C2×#156922822
#124×D1012
C4×#131080752
#202654012
#202653992
#202653972
#202653952
#202653932
#202653912
#202653892
#202651992
#202651842
#202651802
#464×C1012
#3040101