Finite Group Databasebeta

← #27182639

Subgroups — #27182639

Order 12928.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692287C211yesmaximal, characteristic, complemented
2C22×#13108063C211yesmaximal, characteristic, complemented
3#20265840C211yesmaximal, characteristic, complemented
4#20265818C211yesmaximal, characteristic
5#20265570C211yesmaximal, characteristic, complemented
6#20265424C211yesmaximal, characteristic, complemented
7#20265402C211yesmaximal, characteristic, complemented
8#20265381C211yesmaximal, characteristic, complemented
9#20265342C211yesmaximal, characteristic, complemented
10#20265332C211yesmaximal, characteristic, complemented
11#20265327C211yesmaximal, characteristic, complemented
12#20265317C211yesmaximal, characteristic, complemented
13#20265276C211yesmaximal, characteristic, complemented
14#20265248C211yesmaximal, characteristic, complemented
15#467×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18C2×Q8×D101C2211yescharacteristic, complemented
19C22×D404C2211yescharacteristic
20#15692287—24no—
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063—24no—
24C2×#13108063C2211yescharacteristic
25C2×#13108063C2211yescharacteristic
26C2×#13108063C2211yescharacteristic
27C2×#13108063C2211yescharacteristic
28C2×#13108061C2211yescharacteristic
29C2×#13108061C2211yescharacteristic
30C2×#13108057C2211yescharacteristic, complemented
31#15692254—21no—
32#15692253C2211yescharacteristic, complemented
33#15692244C2211yescharacteristic
34#15692240—21no—
35#15692235C2211yescharacteristic, complemented
36#15692233—21no—
37#15692232—21no—
38#15692230—21no—
39#15692230C2211yescharacteristic
40#15692225—21no—
41#15692220C2211yescharacteristic, complemented
42#15692217C2211yescharacteristic
43C4×C22⋊D101C2211yescharacteristic
44#15692215—21no—
45C22⋊C4×D101—21no—
46#15692184C2211yescharacteristic, complemented
47#15692178C2211yescharacteristic, complemented
48#15692178C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic
50#15692175C2211yescharacteristic
51C22⋊#11717579C2211yescharacteristic
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C2×Q16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic
56C22⋊Q8×C101C2211yescharacteristic
57C22⋊Q8×C101C2211yescharacteristic
58#107×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#103×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108075—22no—
63#13108075—41no—
64C2×Dic202—22no—
65C2×Dic202—21no—
66C2×Dic202—21no—
67C2×Dic202—41no—
68C2×Dic202C2311yescharacteristic
69C2×Dic202C2311yescharacteristic
70Q8×D101—22no—
71Q8×D101—41no—
72C2×D404—22no—
73C2×D404—21no—
74C2×D404—21no—
75C2×D404—21no—
76C2×D404—41no—
77C2×D404C2311yescharacteristic
78C2×D404C2311yescharacteristic
79C2×C22⋊D101D411yescharacteristic
80C22×Dic101—21no—
81C2×C4×D101—21no—
82C2×C4×D101—21no—
83C2×C4×D101—21no—
84C2×C4×D101C2311yescharacteristic
85#13108064—22no—
86#13108064—41no—
87#13108063—22no—
88#13108063—22no—
89#13108063—22no—
90#13108063D414yescomplemented
91#13108061—22no—
92#13108061—41no—
93#13108057—22no—
94C2×#11717579C2311yescharacteristic
95C2×#11717579C2311yescharacteristic
96C4⋊Dic101—21no—
97#13108053—21no—
98#13108053—21no—
99#13108053—21no—
100#13108053D411yescharacteristic
101C22⋊Dic101D411yescharacteristic
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051D411yescharacteristic
108C4×Dic101—21no—
109C4×Dic101C2311yescharacteristic
110C2×Q8×C101C2311yescharacteristic
111C2×Q8×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113Q16×C101—22no—
114C2×C8×C101—22no—
115C2×C8×C101C2311yescharacteristic
116C2×C8×C101C2311yescharacteristic
117C4⋊C4×C101—21no—
118C4⋊C4×C101—21no—
119C4⋊C4×C101C2311yescharacteristic
120C4⋊C4×C101C2311yescharacteristic
121C22⋊C4×C101—21no—
122C22⋊C4×C101—21no—
123C23×C101C2×D411yescharacteristic
124C22×D101—21no—
125C22×D101—21no—
126C22×D101—42no—
127C22×D101—42no—
128C22×D101—81no—
129C22×D101C2×D411yescharacteristic
130Q8×C101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101C2×D411yescharacteristic
141C2×C4×C101C2411yescharacteristic
142C22⋊D101—22no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—42no—
146C2×Dic101C2×D411yescharacteristic
147C2×Dic101C2×D411yescharacteristic
148C2×Dic101C2×D411yescharacteristic
149D404—21no—
150D404—21no—
151D404—21no—
152D404—42no—
153D404—42no—
154D404—81no—
155D404C2×D412yes—
156C4×D101—21no—
157C4×D101—21no—
158C4×D101—42no—
159C4×D101—42no—
160C4×D101—81no—
161Dic202—21no—
162Dic202—21no—
163Dic202—21no—
164Dic202—42no—
165Dic202—42no—
166Dic202C2×D412yes—
167C8×C101—21no—
168C8×C101C2×D412yes—
169#11717579—21no—
170#11717579—21no—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101#13712yescomplemented
174C22×C101C22×D411yescharacteristic
175C2×D101—21no—
176C2×D101—22no—
177C2×D101—21no—
178C2×D101—21no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C2×D101—84no—
185C4×C101—21no—
186C4×C101—41no—
187C4×C101—41no—
188C4×C101—41no—
189C4×C101—41no—
190C4×C101C22×D411yescharacteristic
191C4×C101H(5,𝔽2)11yescharacteristic
192Dic101—22no—
193Dic101—21no—
194Dic101—21no—
195Dic101—21no—
196Dic101—41no—
197C2×C101—22no—
198C2×C101D4211yescharacteristic
199C2×C101C2×#13711yescharacteristic
200C2×C101#57711yescharacteristic
201D101—41no—
202D101—42no—
203D101—81no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137—1011no—
207C22×SD16—1011no—
208D4×Q8—1011no—
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467D10111yescharacteristic, complemented
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450—1011no—
218#448—1011no—
219#440—1011no—
220#435—1011no—
221C2×#41—1011no—
222C22×Q8—1011no—
223C22×Q8—1011no—
224C22×D4—1011no—
225#137—2024no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×Q16C2×D10111yescharacteristic, complemented
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—2024no—
236C2×M16—1011no—
237C22×C8C2×D10111yescharacteristic
238C4⋊Q8—1011no—
239C4⋊Q8—2021no—
240#124—1011no—
241#123—2021no—
242C22⋊Q8—1011no—
243C22⋊Q8—2021no—
244C22⋊Q8—2021no—
245C22⋊Q8C2×D10111yescharacteristic
246C22⋊Q8C2×D10111yescharacteristic
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8—1011no—
251C4×D4—1011no—
252C4×D4—1011no—
253C4×D4—2021no—
254C2×C22⋊C4—2021no—
255#107C2×D10111yescharacteristic, complemented
256C4⋊C8—1011no—
257#103—1011no—
258#103—1011no—
259#103C2×D10111yescharacteristic, complemented
260#103C2×D10111yescharacteristic, complemented
261#102—1011no—
262#102—1011no—
263C22⋊C8—1011no—
264C22⋊C8—1011no—
265#97—1011no—
266C24—2021no—
267#41—4041no—
268#41—2022no—
269C2×Q8—1011no—
270C2×Q8—1011no—
271C2×Q8—4041no—
272C2×Q8—4041no—
273C2×Q8—2021no—
274C2×Q8—2022no—
275C2×Q8—2022no—
276C2×Q8—2021no—
277C2×Q8C22×D10111yescharacteristic
278C2×Q8C22×D10111yescharacteristic
279C2×D4—1011no—
280C2×D4—1011no—
281C2×D4—1011no—
282C2×D4—4041no—
283C2×D4—2021no—
284C2×D4—2022no—
285C2×D4—2021no—
286C2×D4—2021no—
287C22×C4—1011no—
288C22×C4—2021no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4—2021no—
292C22×C4C22×D10111yescharacteristic
293Q16—22no—
294Q16—4041no—
295Q16—2022no—
296SD16—1014no—
297SD16—4041no—
298SD16—2022no—
299SD16—2022no—
300SD16—2022no—
301SD16—2022no—
302M16—2022no—
303C2×C8—22no—
304C2×C8—1011no—
305C2×C8—1011no—
306C2×C8C22×D10111yescharacteristic
307C2×C8C22×D10111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—1011no—
311C4⋊C4—2021no—
312C4⋊C4—2021no—
313C4⋊C4—2021no—
314C4⋊C4—2021no—
315C4⋊C4C22×D10111yescharacteristic
316C4⋊C4C22×D10111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1011no—
320C22⋊C4—1011no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C42—1011no—
327C42—2021no—
328C23—8081no—
329C23—1011no—
330C23—4042no—
331C23—4042no—
332C23—2021no—
333C23—2021no—
334C23D4×D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—1012no—
340Q8—4042no—
341Q8—4042no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345D4—8081no—
346D4—1012no—
347D4—4042no—
348D4—4042no—
349D4—2022no—
350D4—2021no—
351D4—2021no—
352D4—2021no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—8081no—
360C2×C4—1011no—
361C2×C4—1011no—
362C2×C4—1011no—
363C2×C4—4042no—
364C2×C4—4042no—
365C2×C4—4042no—
366C2×C4—2021no—
367C2×C4—2021no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4D4×D10111yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—21no—
373C8—2021no—
374C8—2021no—
375C8D4×D10112yes—
376C22—21no—
377C22—21no—
378C22—8084no—
379C22—4041no—
380C22—4041no—
381C22—4041no—
382C22—4041no—
383C22—4041no—
384C22—2021no—
385C22—2021no—
386C22—2022no—
387C22—2021no—
388C22#1569228712yescomplemented
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—4041no—
396C4—2021no—
397C4—2021no—
398C4—2022no—
399C4—2021no—
400C4C2×D4×D10111yescharacteristic
401C4#1569230111yescharacteristic
402C2—22no—
403C2—8081no—
404C2—4042no—
405C2—4041no—
406C2#2026581111yescharacteristic
407C2C2×#1569228711yescharacteristic
408C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#467×C101C2
socleC22×C101C22×D4
prime radical#20265818C2
hypercenter#467D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#15692287C2semidirect
2. C22×#13108063C2semidirect
3. #20265840C2semidirect
5. #20265570C2semidirect
6. #20265424C2semidirect
7. #20265402C2semidirect
8. #20265381C2semidirect
9. #20265342C2semidirect
10. #20265332C2semidirect
11. #20265327C2semidirect
12. #20265317C2semidirect
13. #20265276C2semidirect
14. #20265248C2semidirect
15. #467×C101C2semidirect
18. C2×Q8×D101C22semidirect
21. C2×#13108064C22semidirect
22. C2×#13108064C22semidirect
30. C2×#13108057C22semidirect
32. #15692253C22semidirect
35. #15692235C22semidirect
41. #15692220C22semidirect
46. #15692184C22semidirect
47. #15692178C22semidirect
48. #15692178C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
54. C2×Q16×C101C22semidirect
58. #107×C101C22semidirect
59. #103×C101C22semidirect
60. #103×C101C22semidirect
90. #13108063D4semidirect
173. C22×C101#137semidirect
205. C101#3266semidirect
212. #467D101semidirect
228. C2×Q16C2×D101semidirect
255. #107C2×D101semidirect
259. #103C2×D101semidirect
260. #103C2×D101semidirect
388. C22#15692287semidirect

Maximal subgroups

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

subgroupindex
C2×#156922872
C22×#131080632
#202658402
#202658182
#202655702
#202654242
#202654022
#202653812
#202653422
#202653322
#202653272
#202653172
#202652762
#202652482
#467×C1012
#3266101