Finite Group Databasebeta

← #27182726

Subgroups — #27182726

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×#15692279C211yesmaximal, characteristic, complemented
2#20265826C211yesmaximal, characteristic, complemented
3D4×C22⋊D101C211yesmaximal, characteristic, complemented
4#20265548C211yesmaximal, characteristic, complemented
5#20265423C211yesmaximal, characteristic, complemented
6#20265420C211yesmaximal, characteristic, complemented
7#20265380C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265328C211yesmaximal, characteristic, complemented
10#20265326C211yesmaximal, characteristic, complemented
11#20265322C211yesmaximal, characteristic, complemented
12#20265319C211yesmaximal, characteristic, complemented
13D8×Dic101C211yesmaximal, characteristic, complemented
14#20265253C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C22×C22⋊D101—21no—
18C2×D4×D101C2211yescharacteristic
19#15692279—24no—
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108061C2211yescharacteristic, complemented
24C2×#13108059C2211yescharacteristic, complemented
25C2×#13108057C2211yescharacteristic, complemented
26#15692253C2211yescharacteristic, complemented
27#15692251C2211yescharacteristic
28#15692238—21no—
29#15692236—21no—
30#15692234C2211yescharacteristic, complemented
31#15692231C2211yescharacteristic, complemented
32#15692229C2211yescharacteristic
33#15692227—21no—
34#15692227—21no—
35#15692226C2211yescharacteristic
36#15692224—21no—
37#15692223—21no—
38D4×Dic101C2211yescharacteristic
39D4×Dic101C2211yescharacteristic, complemented
40C4×C22⋊D101C2211yescharacteristic
41C2×#13108053—21no—
42#15692189C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic
55AGL(1,ℤ/8)×C101—24no—
56C2×SD16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic, complemented
58C2×D8×C101C2211yescharacteristic
59C2×D8×C101C2211yescharacteristic, complemented
60C2×M16×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108073—22no—
63#13108073—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—44no—
73C2×C22⋊D101D411yescharacteristic
74D4×D101—22no—
75D4×D101—41no—
76C22×Dic101—21no—
77C22×Dic101—21no—
78C22×Dic101—21no—
79C2×C4×D101C2311yescharacteristic
80#13108063—22no—
81#13108062—22no—
82#13108062—41no—
83#13108061—41no—
84#13108059—22no—
85#13108057—22no—
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C4⋊Dic101C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91#13108053—21no—
92#13108053—21no—
93#13108053—21no—
94#13108053D411yescharacteristic
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101—21no—
99C22⋊Dic101D411yescharacteristic
100#13108051—21no—
101#13108051—21no—
102#13108051D411yescharacteristic
103C4×Dic101C2311yescharacteristic
104C24×C101—21no—
105#41×C101—22no—
106#41×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101—22no—
109C2×D4×C101—21no—
110C2×D4×C101—21no—
111C2×D4×C101—41no—
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101—21no—
116C22×C4×C101C2311yescharacteristic
117SD16×C101—22no—
118SD16×C101—22no—
119D8×C101—22no—
120D8×C101D414yescomplemented
121M16×C101—22no—
122C2×C8×C101C2311yescharacteristic
123C2×C8×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—42no—
128C23×C101—42no—
129C23×C101—81no—
130C23×C101C2×D411yescharacteristic
131C22×D101—21no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—81no—
135C22×D101C2×D411yescharacteristic
136Q8×C101—21no—
137Q8×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—42no—
143D4×C101—42no—
144D4×C101C2×D412yes—
145D4×C101C2×D412yescomplemented
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—42no—
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2411yescharacteristic
152C22⋊D101—22no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—42no—
157C22⋊D101—42no—
158C22⋊D101—81no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—42no—
163C2×Dic101—42no—
164C2×Dic101—42no—
165C2×Dic101C2×D411yescharacteristic
166C2×Dic101C2×D411yescharacteristic
167C2×Dic101C2×D411yescharacteristic
168D404—21no—
169D404—21no—
170C4×D101—21no—
171C4×D101—21no—
172Dic202—21no—
173Dic202—21no—
174C8×C101—21no—
175C8×C101C2×D412yescomplemented
176#11717579—21no—
177#11717579—21no—
178C22×C101—22no—
179C22×C101—22no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—84no—
191C22×C101C22×D411yescharacteristic
192C2×D101—21no—
193C2×D101—21no—
194C2×D101—21no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—84no—
199C4×C101—21no—
200C4×C101—41no—
201C4×C101C22×D411yescharacteristic
202C4×C101H(5,𝔽2)11yescharacteristic
203Dic101—21no—
204Dic101—21no—
205Dic101—41no—
206Dic101—41no—
207Dic101—41no—
208Dic101AGL(1,ℤ/8)12yescomplemented
209C2×C101—42no—
210C2×C101—41no—
211C2×C101—42no—
212C2×C101—81no—
213C2×C101D4211yescharacteristic
214C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C101#57711yescharacteristic
216D101—41no—
217D101—81no—
218#3272—1011nomaximal, Sylow
219C101#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
222#547—1011no—
223D42—1011no—
224#492—1011no—
225#468—1011no—
226#468—1011no—
227#461—1011no—
228#459—1011no—
229#450—1011no—
230#450—1011no—
231#449—1011no—
232#449—1011no—
233C4×D8—1011no—
234#436—1011no—
235C2×#41—1011no—
236C2×#41C2×D10111yescharacteristic, complemented
237C22×D4—1011no—
238C22×D4—2021no—
239C22×D4C2×D10111yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—2024no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—1011no—
246C2×SD16C2×D10111yescharacteristic
247C2×SD16C2×D10111yescharacteristic, complemented
248C2×D8—1011no—
249C2×D8C2×D10111yescharacteristic
250C2×D8C2×D10111yescharacteristic, complemented
251C2×M16—1011no—
252C2×M16C2×D10111yescharacteristic, complemented
253C4⋊Q8—1011no—
254#127—1011no—
255#123—2021no—
256#123—2021no—
257C22⋊Q8—1011no—
258C22⋊Q8—1011no—
259#121—1011no—
260#121—1011no—
261#121—2021no—
262#121—2021no—
263#120—2021no—
264#120—2021no—
265C4×D4—1011no—
266C4×D4—1011no—
267C4×D4—1011no—
268C2×C4⋊C4—2021no—
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—
279C4×C8—1011no—
280C24—21no—
281C24—2021no—
282#41—22no—
283#41—41no—
284#41—4041no—
285#41—2022no—
286C2×Q8—1011no—
287C2×Q8C22×D10111yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—1011no—
293C2×D4—1011no—
294C2×D4—4044no—
295C2×D4—4041no—
296C2×D4—2021no—
297C2×D4—2021no—
298C2×D4—2021no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2022no—
303C2×D4C22×D10111yescharacteristic
304C2×D4C22×D10111yescharacteristic
305C2×D4C22×D10111yescharacteristic
306C22×C4—21no—
307C22×C4—1011no—
308C22×C4—2021no—
309C22×C4—2021no—
310C22×C4—2021no—
311C22×C4C22×D10111yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—4041no—
315SD16—4041no—
316SD16—2022no—
317SD16—2022no—
318D8—22no—
319D8—2022no—
320D8C22⋊D10114yescomplemented
321M16—22no—
322M16—2022no—
323C2×C8—1011no—
324C2×C8—1011no—
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—1011no—
328C4⋊C4—1011no—
329C4⋊C4—1011no—
330C4⋊C4—2021no—
331C4⋊C4—2021no—
332C4⋊C4—2021no—
333C4⋊C4—2021no—
334C22⋊C4—1011no—
335C22⋊C4—1011no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4—2021no—
340C22⋊C4—2021no—
341C22⋊C4—2021no—
342C42—1011no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—8081no—
350C23—1011no—
351C23—4042no—
352C23—4042no—
353C23—2021no—
354C23D4×D10111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2021no—
358Q8—2021no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—8081no—
366D4—4042no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—4042no—
371D4—2022no—
372D4—2021no—
373D4—2021no—
374D4C2×C22⋊D10112yes—
375D4C2×C22⋊D10112yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—1011no—
381C2×C4—1011no—
382C2×C4—1011no—
383C2×C4—4042no—
384C2×C4—4042no—
385C2×C4—4042no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4D4×D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—2021no—
395C8—2021no—
396C8C2×C22⋊D10112yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—8084no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—2021no—
415C22—2021no—
416C22—2021no—
417C22C2×D4×D10111yescharacteristic
418C4—21no—
419C4—41no—
420C4—1012no—
421C4—4041no—
422C4—4041no—
423C4—4041no—
424C4—2021no—
425C4—2021no—
426C4C22×C22⋊D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—8081no—
433C2—4041no—
434C2D4×C22⋊D10111yescharacteristic
435C2AGL(1,ℤ/8)×D10111yescharacteristic
436C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3272
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265826C2semidirect
3. D4×C22⋊D101C2semidirect
4. #20265548C2semidirect
5. #20265423C2semidirect
6. #20265420C2semidirect
7. #20265380C2semidirect
8. #20265370C2semidirect
9. #20265328C2semidirect
10. #20265326C2semidirect
11. #20265322C2semidirect
12. #20265319C2semidirect
13. D8×Dic101C2semidirect
14. #20265253C2semidirect
15. C2×AGL(1,ℤ/8)×C101C2semidirect
16. C2×#13108073C22semidirect
20. C2×#13108063C22semidirect
22. C2×#13108062C22semidirect
23. C2×#13108061C22semidirect
24. C2×#13108059C22semidirect
25. C2×#13108057C22semidirect
26. #15692253C22semidirect
30. #15692234C22semidirect
31. #15692231C22semidirect
39. D4×Dic101C22semidirect
42. #15692189C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692177C22semidirect
48. #15692176C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
51. #15692167C22semidirect
52. C8×Dic101C22semidirect
53. C2×#41×C101C22semidirect
57. C2×SD16×C101C22semidirect
59. C2×D8×C101C22semidirect
60. C2×M16×C101C22semidirect
120. D8×C101D4semidirect
145. D4×C101C2×D4semidirect
175. C8×C101C2×D4semidirect
208. Dic101AGL(1,ℤ/8)semidirect
219. C101#3272semidirect
221. C2×AGL(1,ℤ/8)D101semidirect
236. C2×#41C2×D101semidirect
247. C2×SD16C2×D101semidirect
250. C2×D8C2×D101semidirect
252. C2×M16C2×D101semidirect
320. D8C22⋊D101semidirect
375. D4C2×C22⋊D101semidirect
396. C8C2×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658262
D4×C22⋊D1012
#202655482
#202654232
#202654202
#202653802
#202653702
#202653282
#202653262
#202653222
#202653192
D8×Dic1012
#202652532
C2×AGL(1,ℤ/8)×C1012
#3272101