Finite Group Databasebeta

← #27180639

Subgroups — #27180639

Order 12928.

494 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×#15692282C211yesmaximal, characteristic
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265752C211yesmaximal, characteristic, complemented
4#20265649C211yesmaximal, characteristic, complemented
5#20265395C211yesmaximal, characteristic, complemented
6#20265394C211yesmaximal, characteristic, complemented
7#20265393C211yesmaximal, characteristic, complemented
8#20265392C211yesmaximal, characteristic, complemented
9#20265381C211yesmaximal, characteristic, complemented
10#20265380C211yesmaximal, characteristic, complemented
11#20265379C211yesmaximal, characteristic, complemented
12#20265202C211yesmaximal, characteristic, complemented
13#20265184C211yesmaximal, characteristic, complemented
14#20265172C211yesmaximal, characteristic, complemented
15#461×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692282—24no—
20C2×#13108063C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108061C2211yescharacteristic
23C2×#13108061C2211yescharacteristic
24SD16×D101—24no—
25C2×D808C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108057C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692253C2211yescharacteristic, complemented
30#15692244C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic, complemented
32#15692240—21no—
33#15692240—21no—
34#15692233—21no—
35#15692233—21no—
36C4×D404C2211yescharacteristic
37D4×Dic101C2211yescharacteristic
38#15692217C2211yescharacteristic
39C4×C22⋊D101—21no—
40#15692215—21no—
41#15692213C2211yescharacteristic
42#15692212C2211yescharacteristic
43C2×#13108051—21no—
44C22⋊C4×D101—21no—
45C4⋊#11717579C2211yescharacteristic, complemented
46#15692184C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692179C2211yescharacteristic, complemented
51#15692178C2211yescharacteristic, complemented
52#15692177C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic
55C2×SD16×C101C2211yescharacteristic
56C4⋊Q8×C101C2211yescharacteristic, complemented
57C4×D4×C101C2211yescharacteristic
58C4⋊C8×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#103×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108075—22no—
63#13108075—22no—
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×D101D414yes—
74C22×Dic101—21no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101—21no—
79C2×C4×D101C2311yescharacteristic
80#13108063—22no—
81#13108062—22no—
82#13108061—22no—
83#13108061—22no—
84D808—22no—
85#13108059—22no—
86#13108057—22no—
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C8×D101—22no—
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053—21no—
93#13108053D411yescharacteristic
94#13108053D411yescharacteristic
95C22⋊Dic101—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051D411yescharacteristic
104#13108051D411yescharacteristic
105C4×Dic101—21no—
106C4×Dic101C2311yescharacteristic
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101—21no—
111SD16×C101—22no—
112SD16×C101—22no—
113C2×C8×C101C2311yescharacteristic
114C2×C8×C101C2311yescharacteristic
115C4⋊C4×C101—21no—
116C4⋊C4×C101—21no—
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C42×C101C2311yescharacteristic
120C23×C101—21no—
121C22×D101—21no—
122C22×D101—24no—
123C22×D101—21no—
124C22×D101—81no—
125C22×D101—81no—
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101—21no—
129Q8×C101—21no—
130Q8×C101—21no—
131D4×C101—21no—
132D4×C101C2×D412yes—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—81no—
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2×D411yescharacteristic
139C2×C4×C101C2411yescharacteristic
140C22⋊D101—24no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—81no—
144C2×Dic101C2×D411yescharacteristic
145C2×Dic101C2×D411yescharacteristic
146C2×Dic101C2×D411yescharacteristic
147D404—21no—
148D404—21no—
149D404—21no—
150D404—42no—
151D404C2×D412yes—
152C4×D101—42no—
153C4×D101—42no—
154C4×D101—81no—
155C4×D101—81no—
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—22no—
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—21no—
180C4×C101—21no—
181C4×C101—41no—
182C4×C101—41no—
183C4×C101—41no—
184C4×C101C22×D411yescharacteristic
185C4×C101C2×#4111yescharacteristic
186Dic101—21no—
187Dic101—21no—
188Dic101—41no—
189Dic101—41no—
190Dic101#12112yes—
191C2×C101—42no—
192C2×C101C2×#12111yescharacteristic
193C2×C101#57711yescharacteristic
194C2×C101#57711yescharacteristic
195D101—22no—
196D101—42no—
197D101—81no—
198#3042—1011nomaximal, Sylow
199C101#304211yescharacteristic, complemented, Sylow
200C2×AGL(1,ℤ/8)—1011no—
201C22×SD16—1011no—
202#536—1011no—
203#518—1011no—
204#463—1011no—
205#463—1011no—
206#463—1011no—
207#463—1011no—
208#461—1011no—
209#461—1011no—
210#461—1011no—
211#461D10111yescharacteristic, complemented
212#424—1011no—
213#419—1011no—
214#416—1011no—
215C2×#41—1011no—
216C22×Q8—1011no—
217C22×D4—1011no—
218AGL(1,ℤ/8)—2024no—
219C2×SD16—1011no—
220C2×SD16—1011no—
221C2×SD16—1011no—
222C2×SD16—1011no—
223C2×SD16—2024no—
224C2×SD16C2×D10111yescharacteristic
225C2×SD16C2×D10111yescharacteristic
226C2×D8—1011no—
227C2×D8—1011no—
228C2×M16—1011no—
229C22×C8—1011no—
230C4⋊Q8—1011no—
231C4⋊Q8C2×D10111yescharacteristic, complemented
232#124—1011no—
233#124—1011no—
234#123—2021no—
235#123—2021no—
236C22⋊Q8—2021no—
237C22⋊Q8—2021no—
238C4×D4—1011no—
239C4×D4—1011no—
240C4×D4—1011no—
241C4×D4—2021no—
242C4×D4—2021no—
243C4×D4C2×D10111yescharacteristic
244#117—1011no—
245#117—1011no—
246C2×C22⋊C4—2021no—
247C2×C22⋊C4—2021no—
248C4⋊C8—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251C4⋊C8C2×D10111yescharacteristic, complemented
252#103—1011no—
253#103—1011no—
254#103—1011no—
255#103—1011no—
256#103C2×D10111yescharacteristic, complemented
257#103C2×D10111yescharacteristic, complemented
258#102—1011no—
259#102—1011no—
260C24—2021no—
261#41—2022no—
262#41—2022no—
263C2×Q8—1011no—
264C2×Q8—2021no—
265C2×Q8—2022no—
266C2×Q8—2022no—
267C2×Q8C22×D10111yescharacteristic
268C2×Q8C22×D10111yescharacteristic
269C2×D4—1014no—
270C2×D4—1011no—
271C2×D4—1011no—
272C2×D4—2021no—
273C2×D4—2022no—
274C2×D4—2021no—
275C2×D4C22×D10111yescharacteristic
276C22×C4—21no—
277C22×C4—1011no—
278C22×C4—2021no—
279C22×C4—2021no—
280C22×C4—2021no—
281C22×C4—2021no—
282C22×C4—2021no—
283SD16—22no—
284SD16—22no—
285SD16—2022no—
286SD16—2022no—
287SD16—2022no—
288SD16—2022no—
289D8—2022no—
290D8—2022no—
291M16—2022no—
292C2×C8—1011no—
293C2×C8—1011no—
294C2×C8—2022no—
295C2×C8C22×D10111yescharacteristic
296C2×C8C22×D10111yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—1011no—
300C4⋊C4—1011no—
301C4⋊C4—1011no—
302C4⋊C4—2021no—
303C4⋊C4—2021no—
304C4⋊C4C22×D10111yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—1011no—
307C22⋊C4—1011no—
308C22⋊C4—2021no—
309C22⋊C4—2021no—
310C22⋊C4—2021no—
311C22⋊C4—2021no—
312C22⋊C4—2021no—
313C22⋊C4—2021no—
314C22⋊C4—2021no—
315C22⋊C4—2021no—
316C42—1011no—
317C42—2021no—
318C42C22×D10111yescharacteristic
319C23—21no—
320C23—8081no—
321C23—8081no—
322C23—1011no—
323C23—2024no—
324C23—2021no—
325C23—2021no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—4042no—
331Q8—2021no—
332Q8—2021no—
333D4—21no—
334D4—1012no—
335D4—4042no—
336D4—2024no—
337D4—2021no—
338D4—2021no—
339D4—2021no—
340D4D4×D10112yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—8081no—
346C2×C4—8081no—
347C2×C4—8081no—
348C2×C4—1011no—
349C2×C4—1012no—
350C2×C4—1012no—
351C2×C4—1011no—
352C2×C4—1011no—
353C2×C4—4042no—
354C2×C4—4042no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4D4×D10111yescharacteristic
358C2×C4D4×D10111yescharacteristic
359C2×C4C23×D10111yescharacteristic
360C8—21no—
361C8—21no—
362C8—2021no—
363C8—2021no—
364C22—22no—
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—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—1012no—
384C4—4041no—
385C4—4041no—
386C4—2021no—
387C4—2021no—
388C4C2×D4×D10111yescharacteristic
389C4#41×D10111yescharacteristic
390C2—42no—
391C2—8081no—
392C2—4042no—
393C2—2022no—
394C2#121×D10111yescharacteristic
395C2#2026598311yescharacteristic
396C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#461×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692282C2
hypercenter#461D101
hypocenterC101#3042
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D101C2semidirect
3. #20265752C2semidirect
4. #20265649C2semidirect
5. #20265395C2semidirect
6. #20265394C2semidirect
7. #20265393C2semidirect
8. #20265392C2semidirect
9. #20265381C2semidirect
10. #20265380C2semidirect
11. #20265379C2semidirect
12. #20265202C2semidirect
13. #20265184C2semidirect
14. #20265172C2semidirect
15. #461×C101C2semidirect
29. #15692253C22semidirect
30. #15692244C22semidirect
31. #15692243C22semidirect
45. C4⋊#11717579C22semidirect
46. #15692184C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692179C22semidirect
51. #15692178C22semidirect
52. #15692177C22semidirect
53. #15692175C22semidirect
56. C4⋊Q8×C101C22semidirect
58. C4⋊C8×C101C22semidirect
59. #103×C101C22semidirect
60. #103×C101C22semidirect
199. C101#3042semidirect
211. #461D101semidirect
231. C4⋊Q8C2×D101semidirect
251. C4⋊C8C2×D101semidirect
256. #103C2×D101semidirect
257. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×SD16×D1012
#202657522
#202656492
#202653952
#202653942
#202653932
#202653922
#202653812
#202653802
#202653792
#202652022
#202651842
#202651722
#461×C1012
#3042101