Finite Group Databasebeta

← #27182623

Subgroups — #27182623

Order 12928.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692281C211yesmaximal, characteristic, complemented
2C22×#13108062C211yesmaximal, characteristic, complemented
3#20265836C211yesmaximal, characteristic, complemented
4#20265564C211yesmaximal, characteristic, complemented
5#20265421C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265374C211yesmaximal, characteristic, complemented
9#20265328C211yesmaximal, characteristic, complemented
10#20265320C211yesmaximal, characteristic, complemented
11#20265315C211yesmaximal, characteristic, complemented
12#20265311C211yesmaximal, characteristic, complemented
13#20265277C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15D42×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×Dic202C2211yescharacteristic, complemented
18#15692281—24no—
19C2×#13108062—24no—
20C2×#13108062C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108062C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108062C2211yescharacteristic, complemented
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108058C2211yescharacteristic, complemented
30C22×#11717579C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic, complemented
32#15692241—21no—
33#15692234C2211yescharacteristic, complemented
34#15692233—21no—
35#15692229C2211yescharacteristic, complemented
36#15692227—21no—
37C4×Dic202C2211yescharacteristic, complemented
38#15692215—21no—
39#15692188C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692178C2211yescharacteristic, complemented
42#15692178C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic
44#15692176C2211yescharacteristic, complemented
45#15692176C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic, complemented
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C22×D4×C101—21no—
51C22×D4×C101C2211yescharacteristic
52C22×D4×C101C2211yescharacteristic
53#127×C101C2211yescharacteristic
54#121×C101—21no—
55#121×C101C2211yescharacteristic
56#121×C101C2211yescharacteristic
57#120×C101—21no—
58#120×C101—21no—
59C4×D4×C101C2211yescharacteristic
60C4×D4×C101C2211yescharacteristic
61#13108074—22no—
62#13108074—41no—
63C2×Dic202—22no—
64C2×Dic202—21no—
65C2×Dic202—41no—
66C2×Dic202C2311yescharacteristic
67C2×Dic202C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C22×Dic101—21no—
72C2×C4×D101—21no—
73#13108062—22no—
74#13108062—22no—
75#13108062—22no—
76#13108062—22no—
77#13108062—41no—
78#13108062D414yescomplemented
79#13108059—22no—
80#13108059—22no—
81#13108059—41no—
82#13108058—22no—
83C2×#11717579—22no—
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C4⋊Dic101C2311yescharacteristic
89#13108053—21no—
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95C4×Dic101—21no—
96C24×C101—21no—
97C24×C101—21no—
98C2×D4×C101—22no—
99C2×D4×C101—22no—
100C2×D4×C101—21no—
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—41no—
107C2×D4×C101—41no—
108C2×D4×C101—44no—
109C2×D4×C101D411yescharacteristic
110C2×D4×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C4⋊C4×C101D411yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C22⋊C4×C101—21no—
121C22⋊C4×C101D411yescharacteristic
122C22⋊C4×C101D411yescharacteristic
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—42no—
128C23×C101—42no—
129C23×C101—42no—
130C23×C101—42no—
131C23×C101—81no—
132C23×C101—81no—
133C23×C101C2×D411yescharacteristic
134C23×C101C2×D411yescharacteristic
135C22×D101—21no—
136D4×C101—22no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—42no—
143D4×C101—42no—
144D4×C101—42no—
145D4×C101—42no—
146D4×C101—42no—
147D4×C101—81no—
148D4×C101C2×D412yes—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101—21no—
153C2×C4×C101—21no—
154C2×C4×C101—42no—
155C2×C4×C101C2×D411yescharacteristic
156C2×C4×C101C2×D411yescharacteristic
157C2×C4×C101C2×D411yescharacteristic
158C2×C4×C101C2×D411yescharacteristic
159C2×C4×C101C2411yescharacteristic
160C22⋊D101—42no—
161C22⋊D101—42no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
165C2×Dic101—42no—
166D404—21no—
167D404—21no—
168C4×D101—42no—
169Dic202—21no—
170Dic202—21no—
171Dic202—21no—
172Dic202—42no—
173Dic202C2×D412yescomplemented
174#11717579—21no—
175#11717579—21no—
176#11717579—21no—
177#11717579C2×D412yescomplemented
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—22no—
182C22×C101—21no—
183C22×C101—21no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—41no—
191C22×C101—41no—
192C22×C101—84no—
193C22×C101—84no—
194C22×C101AGL(1,ℤ/8)12yescomplemented
195C22×C101C22×D411yescharacteristic
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C4×C101—21no—
200C4×C101—21no—
201C4×C101—21no—
202C4×C101—21no—
203C4×C101—41no—
204C4×C101C22×D411yescharacteristic
205C4×C101H(5,𝔽2)11yescharacteristic
206Dic101—22no—
207Dic101—41no—
208Dic101—41no—
209C2×C101—22no—
210C2×C101—41no—
211C2×C101—42no—
212C2×C101—81no—
213C2×C101—81no—
214C2×C101D4211yescharacteristic
215C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C101#57711yescharacteristic
217D101—81no—
218#3265—1011nomaximal, Sylow
219C101#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C22×SD16—1011no—
222#548—1011no—
223D42D10111yescharacteristic, complemented
224#496—1011no—
225#468—1011no—
226#466—1011no—
227#463—1011no—
228#460—1011no—
229#450—1011no—
230#449—1011no—
231#448—1011no—
232#447—1011no—
233#440—1011no—
234#435—1011no—
235C2×#41—1011no—
236C22×Q8—1011no—
237C22×D4—21no—
238C22×D4C2×D10111yescharacteristic
239C22×D4C2×D10111yescharacteristic
240AGL(1,ℤ/8)—2024no—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—1011no—
246C2×SD16—1011no—
247C2×SD16—2024no—
248C2×D8—1011no—
249C2×D8—1011no—
250C2×D8—1011no—
251C2×M16—1011no—
252C22×C8—1011no—
253#127C2×D10111yescharacteristic
254#124—1011no—
255#124—2021no—
256C22⋊Q8—1011no—
257C22⋊Q8—2021no—
258#121—21no—
259#121—1011no—
260#121—2021no—
261#121C2×D10111yescharacteristic
262#121C2×D10111yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1011no—
266C4×D4—2021no—
267C4×D4C2×D10111yescharacteristic
268C4×D4C2×D10111yescharacteristic
269#107—1011no—
270C4⋊C8—1011no—
271#103—1011no—
272#103—1011no—
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102—1011no—
277C22⋊C8—1011no—
278C22⋊C8—1011no—
279#97—1011no—
280C24—21no—
281C24—21no—
282#41—4041no—
283#41—2022no—
284C2×Q8—1011no—
285C2×Q8—1011no—
286C2×Q8—4041no—
287C2×Q8—2022no—
288C2×Q8—2021no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—1011no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4C22⋊D10111yescharacteristic
304C2×D4C22×D10111yescharacteristic
305C2×D4C22×D10111yescharacteristic
306C2×D4C22×D10111yescharacteristic
307C22×C4—21no—
308C22×C4—2021no—
309C22×C4—2021no—
310C22×C4C22×D10111yescharacteristic
311C22×C4C22×D10111yescharacteristic
312SD16—1014no—
313SD16—4041no—
314SD16—2022no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—4041no—
319D8—2022no—
320D8—2022no—
321M16—2022no—
322C2×C8—1011no—
323C2×C8—1011no—
324C2×C8—1011no—
325C2×C8—1011no—
326C2×C8—2022no—
327C4⋊C4—1011no—
328C4⋊C4—2021no—
329C4⋊C4—2021no—
330C4⋊C4C22⋊D10111yescharacteristic
331C4⋊C4C22×D10111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4C22⋊D10111yescharacteristic
340C22⋊C4C22⋊D10111yescharacteristic
341C42—2021no—
342C42C22×D10111yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—2021no—
353C23C2×C22⋊D10111yescharacteristic
354C23C2×C22⋊D10111yescharacteristic
355Q8—1012no—
356Q8—4042no—
357Q8—2021no—
358Q8—2021no—
359Q8—2021no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—4042no—
373D4—4042no—
374D4—2021no—
375D4—2021no—
376D4D4×D10112yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—4042no—
384C2×C4—4042no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4C2×C22⋊D10111yescharacteristic
389C2×C4C2×C22⋊D10111yescharacteristic
390C2×C4C2×C22⋊D10111yescharacteristic
391C2×C4C2×C22⋊D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—1012no—
394C8—2021no—
395C8—2021no—
396C8—2021no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—21no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—4041no—
414C22—4041no—
415C22—4041no—
416C22#1569228112yescomplemented
417C22C22×C22⋊D10111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—4041no—
424C4—4041no—
425C4—2022no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—8081no—
434C2D4×C22⋊D10111yescharacteristic
435C2C2×#1569228111yescharacteristic
436C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingD42×C101C2
socleC22×C101C22×D4
hypercenterD42D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C22×#13108062C2semidirect
3. #20265836C2semidirect
4. #20265564C2semidirect
5. #20265421C2semidirect
6. #20265411C2semidirect
7. #20265392C2semidirect
8. #20265374C2semidirect
9. #20265328C2semidirect
10. #20265320C2semidirect
11. #20265315C2semidirect
12. #20265311C2semidirect
13. #20265277C2semidirect
14. #20265249C2semidirect
15. D42×C101C2semidirect
16. C2×#13108074C22semidirect
17. C22×Dic202C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108062C22semidirect
25. C2×#13108062C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C22×#11717579C22semidirect
31. #15692243C22semidirect
33. #15692234C22semidirect
35. #15692229C22semidirect
37. C4×Dic202C22semidirect
39. #15692188C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692178C22semidirect
42. #15692178C22semidirect
44. #15692176C22semidirect
45. #15692176C22semidirect
46. #15692175C22semidirect
47. C22⋊#11717579C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
78. #13108062D4semidirect
173. Dic202C2×D4semidirect
177. #11717579C2×D4semidirect
194. C22×C101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
223. D42D101semidirect
416. C22#15692281semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C22×#131080622
#202658362
#202655642
#202654212
#202654112
#202653922
#202653742
#202653282
#202653202
#202653152
#202653112
#202652772
#202652492
D42×C1012
#3265101