Finite Group Databasebeta

← #27182736

Subgroups — #27182736

Order 12928.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265818C211yesmaximal, characteristic, complemented
4#20265815C211yesmaximal, characteristic, complemented
5#20265562C211yesmaximal, characteristic, complemented
6#20265426C211yesmaximal, characteristic, complemented
7#20265425C211yesmaximal, characteristic, complemented
8#20265395C211yesmaximal, characteristic, complemented
9#20265394C211yesmaximal, characteristic, complemented
10#20265338C211yesmaximal, characteristic, complemented
11#20265313C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265257C211yesmaximal, characteristic, complemented
14#20265252C211yesmaximal, characteristic, complemented
15#452×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C22×D404C2211yescharacteristic, complemented
18C2×D4×D101C2211yescharacteristic, complemented
19#15692282—24no—
20#15692280—24no—
21C2×#13108061C2211yescharacteristic, complemented
22C2×#13108061C2211yescharacteristic, complemented
23C2×#13108061C2211yescharacteristic, complemented
24C2×D808C2211yescharacteristic
25C2×D808C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108058C2211yescharacteristic, complemented
29C2×#13108057C2211yescharacteristic, complemented
30#15692244C2211yescharacteristic, complemented
31#15692244C2211yescharacteristic, complemented
32#15692240—21no—
33#15692238—21no—
34#15692231C2211yescharacteristic, complemented
35#15692230—21no—
36#15692230C2211yescharacteristic
37#15692229C2211yescharacteristic
38#15692227—21no—
39#15692225—21no—
40#15692223—21no—
41D4×Dic101C2211yescharacteristic, complemented
42#15692217C2211yescharacteristic, complemented
43C4×C22⋊D101C2211yescharacteristic
44C2×#13108051—21no—
45C22⋊C4×D101—21no—
46#15692188C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692179C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53C8×Dic101C2211yescharacteristic, complemented
54C2×#41×C101C2211yescharacteristic
55C2×Q16×C101C2211yescharacteristic, complemented
56C2×SD16×C101C2211yescharacteristic, complemented
57C22⋊Q8×C101C2211yescharacteristic, complemented
58#103×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108075—22no—
64#13108075—41no—
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—41no—
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101D411yescharacteristic
76D4×D101—22no—
77D4×D101—41no—
78C22×Dic101—21no—
79C2×C4×D101—21no—
80C2×C4×D101—21no—
81C2×C4×D101C2311yescharacteristic
82#13108061—22no—
83#13108061—22no—
84#13108061—22no—
85D808—22no—
86D808—41no—
87#13108059—22no—
88#13108059D414yescomplemented
89#13108058—22no—
90#13108057—22no—
91C2×#11717579C2311yescharacteristic
92C2×#11717579C2311yescharacteristic
93C4⋊Dic101C2311yescharacteristic
94#13108053—21no—
95#13108053D411yescharacteristic
96C22⋊Dic101—21no—
97C22⋊Dic101D411yescharacteristic
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051D411yescharacteristic
107C4×Dic101C2311yescharacteristic
108#41×C101—22no—
109#41×C101—41no—
110C2×Q8×C101C2311yescharacteristic
111C2×Q8×C101C2311yescharacteristic
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116Q16×C101—41no—
117SD16×C101—22no—
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101—21no—
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—81no—
133C22×D101—81no—
134C22×D101C2×D411yescharacteristic
135Q8×C101—21no—
136Q8×C101—21no—
137Q8×C101—21no—
138Q8×C101—21no—
139D4×C101—21no—
140D4×C101—42no—
141D4×C101C2×D412yes—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—42no—
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2411yescharacteristic
150C22⋊D101—41no—
151C22⋊D101—42no—
152C22⋊D101—81no—
153C2×Dic101—21no—
154C2×Dic101—81no—
155C2×Dic101C2×D411yescharacteristic
156C2×Dic101C2×D411yescharacteristic
157C2×Dic101C2×D411yescharacteristic
158D404—21no—
159D404—21no—
160D404—21no—
161D404—21no—
162D404—21no—
163D404—42no—
164D404—42no—
165D404—81no—
166D404C2×D412yes—
167C4×D101—21no—
168C4×D101—21no—
169C4×D101—42no—
170C4×D101—81no—
171C8×C101—21no—
172C8×C101—21no—
173#11717579—21no—
174#11717579C2×D412yes—
175C22×C101—22no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101—21no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—22no—
184C2×D101—21no—
185C2×D101—21no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—84no—
195C2×D101—84no—
196C4×C101—21no—
197C4×C101—41no—
198C4×C101—41no—
199C4×C101—41no—
200C4×C101C22×D411yescharacteristic
201C4×C101H(5,𝔽2)11yescharacteristic
202Dic101—21no—
203Dic101—21no—
204Dic101—21no—
205Dic101—41no—
206C2×C101—41no—
207C2×C101—42no—
208C2×C101D4211yescharacteristic
209C2×C101#57511yescharacteristic
210C2×C101#57611yescharacteristic
211D101—41no—
212D101—42no—
213D101—81no—
214D101—81no—
215#3273—1011nomaximal, Sylow
216C101#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1011no—
218C2×AGL(1,ℤ/8)—1011no—
219#546—1011no—
220#546—1011no—
221#495—1011no—
222#468—1011no—
223#468—1011no—
224#463—1011no—
225#463—1011no—
226#452—1011no—
227#452D10111yescharacteristic, complemented
228#447—1011no—
229#447—1011no—
230C4×D8—1011no—
231#436—1011no—
232C2×#41—1011no—
233C2×#41C2×D10111yescharacteristic
234C22×D4—1011no—
235C22×D4—1011no—
236AGL(1,ℤ/8)—2024no—
237AGL(1,ℤ/8)—2024no—
238C2×Q16C2×D10111yescharacteristic, complemented
239C2×SD16—1011no—
240C2×SD16—1011no—
241C2×SD16—1011no—
242C2×SD16C2×D10111yescharacteristic, complemented
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8—1011no—
246C2×D8—1011no—
247C2×M16—1011no—
248C2×M16—1011no—
249#124—1011no—
250#124—1011no—
251#123—2021no—
252#123—2021no—
253C22⋊Q8—1011no—
254C22⋊Q8C2×D10111yescharacteristic, complemented
255#121—1011no—
256#121—1011no—
257#121—2021no—
258#121—2021no—
259#120—2021no—
260#120—2021no—
261C4×D4—1011no—
262C4×D4—1011no—
263C4×D4—1011no—
264C2×C22⋊C4—2021no—
265C2×C22⋊C4—2021no—
266#107—1011no—
267C4⋊C8—1011no—
268#103—1011no—
269#103C2×D10111yescharacteristic, complemented
270#102—1011no—
271#102—1011no—
272#102—1011no—
273#102C2×D10111yescharacteristic, complemented
274C22⋊C8—1011no—
275C22⋊C8C2×D10111yescharacteristic, complemented
276C4×C8—1011no—
277C24—2021no—
278C24—2021no—
279#41—22no—
280#41—41no—
281#41—4041no—
282#41—2022no—
283C2×Q8C22×D10111yescharacteristic
284C2×Q8C22×D10111yescharacteristic
285C2×D4—21no—
286C2×D4—1011no—
287C2×D4—1011no—
288C2×D4—1011no—
289C2×D4—1011no—
290C2×D4—4041no—
291C2×D4—4041no—
292C2×D4—2021no—
293C2×D4—2022no—
294C2×D4—2021no—
295C2×D4—2021no—
296C2×D4—2021no—
297C2×D4—2022no—
298C2×D4—2021no—
299C2×D4C22×D10111yescharacteristic
300C22×C4—21no—
301C22×C4—1011no—
302C22×C4—2021no—
303C22×C4—2021no—
304C22×C4—2021no—
305C22×C4C22×D10111yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—2022no—
309SD16—2022no—
310SD16—2022no—
311D8—1014no—
312D8—4041no—
313D8—2022no—
314D8—2022no—
315M16—2022no—
316M16—2022no—
317C2×C8—1011no—
318C2×C8—1011no—
319C2×C8C22×D10111yescharacteristic
320C2×C8C22×D10111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1011no—
323C4⋊C4—1011no—
324C4⋊C4—2021no—
325C4⋊C4C22×D10111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—1011no—
328C22⋊C4—1011no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C42—1011no—
339C23—21no—
340C23—8081no—
341C23—8081no—
342C23—1011no—
343C23—4042no—
344C23—4042no—
345C23—4042no—
346C23—4042no—
347C23—2021no—
348C23—2021no—
349C23—2021no—
350C23D4×D10111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—8081no—
358D4—8081no—
359D4—1012no—
360D4—4042no—
361D4—4042no—
362D4—4042no—
363D4—4041no—
364D4—2021no—
365D4—2021no—
366D4—2021no—
367D4—2021no—
368D4—2021no—
369D4D4×D10112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—8081no—
377C2×C4—8081no—
378C2×C4—1011no—
379C2×C4—1011no—
380C2×C4—1011no—
381C2×C4—4042no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4D4×D10111yescharacteristic
386C2×C4C23×D10111yescharacteristic
387C8—21no—
388C8—21no—
389C8—1012no—
390C8—2021no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—8084no—
398C22—8084no—
399C22—4041no—
400C22—4041no—
401C22—4041no—
402C22—4041no—
403C22—4041no—
404C22—4041no—
405C22—4041no—
406C22—4041no—
407C22—2022no—
408C22—2021no—
409C22—2021no—
410C22—2021no—
411C22C2×D4×D10111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4041no—
417C4—2021no—
418C4—2021no—
419C4—2021no—
420C4C2×D4×D10111yescharacteristic
421C4#1569230011yescharacteristic
422C2—41no—
423C2—42no—
424C2—8081no—
425C2—8081no—
426C2—4041no—
427C2—4042no—
428C2#2026580911yescharacteristic
429C2#2026597611yescharacteristic
430C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#452×C101C2
socleC22×C101C22×D4
hypercenter#452D101
hypocenterC101#3273
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. C2×#15692280C2semidirect
3. #20265818C2semidirect
4. #20265815C2semidirect
5. #20265562C2semidirect
6. #20265426C2semidirect
7. #20265425C2semidirect
8. #20265395C2semidirect
9. #20265394C2semidirect
10. #20265338C2semidirect
11. #20265313C2semidirect
12. #20265312C2semidirect
13. #20265257C2semidirect
14. #20265252C2semidirect
15. #452×C101C2semidirect
17. C22×D404C22semidirect
18. C2×D4×D101C22semidirect
21. C2×#13108061C22semidirect
22. C2×#13108061C22semidirect
23. C2×#13108061C22semidirect
28. C2×#13108058C22semidirect
29. C2×#13108057C22semidirect
30. #15692244C22semidirect
31. #15692244C22semidirect
34. #15692231C22semidirect
41. D4×Dic101C22semidirect
42. #15692217C22semidirect
46. #15692188C22semidirect
47. #15692184C22semidirect
48. #15692179C22semidirect
49. #15692177C22semidirect
50. #15692177C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. C8×Dic101C22semidirect
55. C2×Q16×C101C22semidirect
56. C2×SD16×C101C22semidirect
57. C22⋊Q8×C101C22semidirect
58. #103×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
88. #13108059D4semidirect
216. C101#3273semidirect
227. #452D101semidirect
238. C2×Q16C2×D101semidirect
242. C2×SD16C2×D101semidirect
254. C22⋊Q8C2×D101semidirect
269. #103C2×D101semidirect
273. #102C2×D101semidirect
275. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×#156922802
#202658182
#202658152
#202655622
#202654262
#202654252
#202653952
#202653942
#202653382
#202653132
#202653122
#202652572
#202652522
#452×C1012
#3273101