Finite Group Databasebeta

← #27182739

Subgroups — #27182739

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×#15692281C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic, complemented
4#20265559C211yesmaximal, characteristic, complemented
5#20265425C211yesmaximal, characteristic, complemented
6#20265421C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265337C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13C4×#13108059C211yesmaximal, characteristic, complemented
14#20265255C211yesmaximal, characteristic, complemented
15#546×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic, complemented
18#15692281—24no—
19#15692280—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108061C2211yescharacteristic, complemented
24C2×#13108061C2211yescharacteristic, complemented
25C2×#13108059C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108058C2211yescharacteristic, complemented
30C2×#13108058C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic, complemented
32#15692240—21no—
33#15692234C2211yescharacteristic, complemented
34#15692229C2211yescharacteristic
35#15692228—21no—
36#15692223—21no—
37C4×D404C2211yescharacteristic, complemented
38C22⋊C4×D101—21no—
39#15692188C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692178C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic, complemented
48C22⋊#11717579C2211yescharacteristic, complemented
49C4×#11717579C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic
51C22×D4×C101C2211yescharacteristic, complemented
52#124×C101C2211yescharacteristic, complemented
53#123×C101—21no—
54C22⋊Q8×C101C2211yescharacteristic, complemented
55#121×C101—21no—
56#121×C101C2211yescharacteristic
57#120×C101—21no—
58C4×D4×C101C2211yescharacteristic
59C4×D4×C101C2211yescharacteristic, complemented
60C2×C22⋊C4×C101—21no—
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—41no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74#13108064—41no—
75#13108062—22no—
76#13108062—22no—
77#13108061—22no—
78#13108061—22no—
79#13108059—22no—
80#13108059—22no—
81#13108059—41no—
82#13108059D414yescomplemented
83#13108058—22no—
84#13108058—22no—
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96C24×C101—21no—
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—22no—
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—41no—
105C2×D4×C101D411yescharacteristic
106C2×D4×C101C2311yescharacteristic
107C2×D4×C101C2311yescharacteristic
108C22×C4×C101—21no—
109C22×C4×C101—21no—
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C4⋊C4×C101—21no—
113C4⋊C4×C101D411yescharacteristic
114C4⋊C4×C101C2311yescharacteristic
115C22⋊C4×C101—21no—
116C22⋊C4×C101—21no—
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C22⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—42no—
126C23×C101—42no—
127C23×C101—81no—
128C23×C101C2×D411yescharacteristic
129C23×C101C2×D411yescharacteristic
130C22×D101—21no—
131C22×D101—21no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—81no—
135Q8×C101—21no—
136Q8×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—41no—
141D4×C101—42no—
142D4×C101—42no—
143D4×C101—81no—
144D4×C101C2×D412yes—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—42no—
152C2×C4×C101—81no—
153C2×C4×C101C2×D411yescharacteristic
154C2×C4×C101C2×D411yescharacteristic
155C2×C4×C101C2×D411yescharacteristic
156C2×C4×C101C2×D411yescharacteristic
157C2×C4×C101C2411yescharacteristic
158C22⋊D101—42no—
159C22⋊D101—81no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162D404—21no—
163D404—21no—
164D404—21no—
165D404—42no—
166D404C2×D412yes—
167C4×D101—42no—
168C4×D101—81no—
169Dic202—21no—
170Dic202—21no—
171#11717579—21no—
172#11717579—21no—
173#11717579—21no—
174#11717579C2×D412yes—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—22no—
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—84no—
188C22×C101C22×D411yescharacteristic
189C2×D101—22no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—84no—
196C4×C101—21no—
197C4×C101—21no—
198C4×C101—21no—
199C4×C101—21no—
200C4×C101—41no—
201C4×C101—41no—
202C4×C101C22×D411yescharacteristic
203C4×C101H(5,𝔽2)11yescharacteristic
204Dic101—41no—
205Dic101—41no—
206C2×C101—41no—
207C2×C101—41no—
208C2×C101—42no—
209C2×C101—81no—
210C2×C101D4211yescharacteristic
211C2×C101#57511yescharacteristic
212C2×C101#57611yescharacteristic
213D101—42no—
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#546D10111yescharacteristic, complemented
221#495—1011no—
222#468—1011no—
223#468—1011no—
224#463—1011no—
225#463—1011no—
226#452—1011no—
227#452—1011no—
228#447—1011no—
229#447—1011no—
230C4×D8—1011no—
231#436—1011no—
232C2×#41—1011no—
233C2×#41C2×D10111yescharacteristic
234C22×D4—1011no—
235C22×D4C2×D10111yescharacteristic, complemented
236AGL(1,ℤ/8)—2024no—
237AGL(1,ℤ/8)—2024no—
238C2×Q16—1011no—
239C2×SD16—1011no—
240C2×SD16—1011no—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8—1011no—
246C2×D8—1011no—
247C2×M16—1011no—
248C2×M16—1011no—
249#124—1011no—
250#124C2×D10111yescharacteristic, complemented
251#123—21no—
252#123—2021no—
253C22⋊Q8—1011no—
254C22⋊Q8C2×D10111yescharacteristic, complemented
255#121—21no—
256#121—1011no—
257#121—2021no—
258#121C2×D10111yescharacteristic
259#120—21no—
260#120—2021no—
261C4×D4—1011no—
262C4×D4C2×D10111yescharacteristic
263C4×D4C2×D10111yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—2021no—
266#107—1011no—
267C4⋊C8—1011no—
268#103—1011no—
269#103—1011no—
270#102—1011no—
271#102—1011no—
272#102—1011no—
273#102—1011no—
274C22⋊C8—1011no—
275C22⋊C8—1011no—
276C4×C8—1011no—
277C24—21no—
278C24—2021no—
279#41—22no—
280#41—41no—
281#41—4041no—
282#41—2022no—
283C2×Q8—1011no—
284C2×Q8C22×D10111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—1011no—
291C2×D4—1011no—
292C2×D4—4041no—
293C2×D4—2022no—
294C2×D4—2021no—
295C2×D4—2021no—
296C2×D4—2021no—
297C2×D4C22⋊D10111yescharacteristic
298C2×D4C22×D10111yescharacteristic
299C2×D4C22×D10111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—2021no—
303C22×C4—2021no—
304C22×C4C22×D10111yescharacteristic
305C22×C4C22×D10111yescharacteristic
306Q16—4041no—
307SD16—2022no—
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×C8—1011no—
320C2×C8—1011no—
321C4⋊C4—21no—
322C4⋊C4—1011no—
323C4⋊C4—2021no—
324C4⋊C4C22⋊D10111yescharacteristic
325C4⋊C4C22×D10111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C22⋊C4C22⋊D10111yescharacteristic
337C22⋊C4C22⋊D10111yescharacteristic
338C42C22×D10111yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—8081no—
345C23—4042no—
346C23—4042no—
347C23—2021no—
348C23—2021no—
349C23C2×C22⋊D10111yescharacteristic
350C23C2×C22⋊D10111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2021no—
354Q8—2021no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—8081no—
363D4—1012no—
364D4—4042no—
365D4—4042no—
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—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—8081no—
379C2×C4—4042no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4C2×C22⋊D10111yescharacteristic
383C2×C4C2×C22⋊D10111yescharacteristic
384C2×C4C2×C22⋊D10111yescharacteristic
385C2×C4C2×C22⋊D10111yescharacteristic
386C2×C4C23×D10111yescharacteristic
387C8—1012no—
388C8—2021no—
389C8—2021no—
390C8—2021no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—8084no—
405C22—4041no—
406C22—4041no—
407C22—4041no—
408C22—4041no—
409C22—4041no—
410C22—2022no—
411C22C22×C22⋊D10111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—4041no—
419C4—4041no—
420C4C2×D4×D10111yescharacteristic
421C4#1569230011yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—8081no—
427C2—4042no—
428C2D4×C22⋊D10111yescharacteristic
429C2#2026597511yescharacteristic
430C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#546×C101C2
socleC22×C101C22×D4
hypercenter#546D101
hypocenterC101#3273
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×#15692280C2semidirect
3. #20265823C2semidirect
4. #20265559C2semidirect
5. #20265425C2semidirect
6. #20265421C2semidirect
7. #20265392C2semidirect
8. #20265389C2semidirect
9. #20265339C2semidirect
10. #20265337C2semidirect
11. #20265311C2semidirect
12. #20265310C2semidirect
13. C4×#13108059C2semidirect
14. #20265255C2semidirect
15. #546×C101C2semidirect
17. C22×D404C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108062C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108061C22semidirect
24. C2×#13108061C22semidirect
29. C2×#13108058C22semidirect
30. C2×#13108058C22semidirect
31. #15692243C22semidirect
33. #15692234C22semidirect
37. C4×D404C22semidirect
39. #15692188C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692178C22semidirect
43. #15692176C22semidirect
44. #15692176C22semidirect
45. #15692175C22semidirect
46. #15692175C22semidirect
47. C22⋊#11717579C22semidirect
48. C22⋊#11717579C22semidirect
49. C4×#11717579C22semidirect
51. C22×D4×C101C22semidirect
52. #124×C101C22semidirect
54. C22⋊Q8×C101C22semidirect
59. C4×D4×C101C22semidirect
82. #13108059D4semidirect
216. C101#3273semidirect
220. #546D101semidirect
235. C22×D4C2×D101semidirect
250. #124C2×D101semidirect
254. C22⋊Q8C2×D101semidirect
263. C4×D4C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C2×#156922802
#202658232
#202655592
#202654252
#202654212
#202653922
#202653892
#202653392
#202653372
#202653112
#202653102
C4×#131080592
#202652552
#546×C1012
#3273101