Finite Group Databasebeta

← #27182737

Subgroups — #27182737

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×#15692279C211yesmaximal, characteristic, complemented
2#20265821C211yesmaximal, characteristic, complemented
3#20265815C211yesmaximal, characteristic, complemented
4#20265558C211yesmaximal, characteristic, complemented
5#20265423C211yesmaximal, characteristic, complemented
6#20265420C211yesmaximal, characteristic, complemented
7#20265394C211yesmaximal, characteristic, complemented
8#20265390C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265336C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13D8×Dic101C211yesmaximal, characteristic, complemented
14#20265253C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic, complemented
18#15692279—24no—
19C2×#13108064C2211yescharacteristic, complemented
20C2×#13108063C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108059C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×#13108057C2211yescharacteristic, complemented
25#15692244C2211yescharacteristic, complemented
26#15692242C2211yescharacteristic, complemented
27#15692238—21no—
28#15692236—21no—
29#15692234C2211yescharacteristic, complemented
30#15692231C2211yescharacteristic, complemented
31#15692229C2211yescharacteristic
32#15692227—21no—
33#15692227—21no—
34#15692226C2211yescharacteristic
35#15692224—21no—
36#15692223—21no—
37D4×Dic101C2211yescharacteristic, complemented
38D4×Dic101C2211yescharacteristic, complemented
39C4×C22⋊D101C2211yescharacteristic
40C2×C22⋊Dic101—21no—
41C2×#13108051—21no—
42#15692189C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic, complemented
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic, complemented
52C8×Dic101C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic
54C22×D4×C101C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C101—24no—
56C2×SD16×C101C2211yescharacteristic, complemented
57C2×SD16×C101C2211yescharacteristic, complemented
58C2×D8×C101C2211yescharacteristic
59C2×D8×C101C2211yescharacteristic
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⋊D101D411yescharacteristic
71D4×D101—22no—
72D4×D101—41no—
73C22×Dic101—21no—
74C22×Dic101—21no—
75C22×Dic101—21no—
76C2×C4×D101C2311yescharacteristic
77#13108064—41no—
78#13108063—22no—
79#13108062—22no—
80#13108059—22no—
81#13108059—41no—
82#13108057—22no—
83C2×#11717579C2311yescharacteristic
84C2×#11717579C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053—21no—
89#13108053D411yescharacteristic
90C22⋊Dic101—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96C22⋊Dic101D411yescharacteristic
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051D411yescharacteristic
102C4×Dic101C2311yescharacteristic
103C24×C101—21no—
104#41×C101—22no—
105#41×C101—41no—
106C2×Q8×C101C2311yescharacteristic
107C2×D4×C101—22no—
108C2×D4×C101—21no—
109C2×D4×C101—21no—
110C2×D4×C101—41no—
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116SD16×C101—22no—
117SD16×C101—22no—
118D8×C101—22no—
119D8×C101D414yescomplemented
120M16×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—42no—
127C23×C101—42no—
128C23×C101—81no—
129C23×C101C2×D411yescharacteristic
130C22×D101—21no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—81no—
134C22×D101C2×D411yescharacteristic
135Q8×C101—21no—
136Q8×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—42no—
142D4×C101—42no—
143D4×C101C2×D412yes—
144D4×C101C2×D412yes—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—42no—
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2411yescharacteristic
151C22⋊D101—41no—
152C22⋊D101—42no—
153C22⋊D101—42no—
154C22⋊D101—81no—
155C22⋊D101—81no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—42no—
160C2×Dic101—81no—
161C2×Dic101—81no—
162C2×Dic101C2×D411yescharacteristic
163C2×Dic101C2×D411yescharacteristic
164C2×Dic101C2×D411yescharacteristic
165D404—21no—
166D404—21no—
167C4×D101—21no—
168C4×D101—21no—
169Dic202—21no—
170Dic202—21no—
171C8×C101—21no—
172C8×C101C2×D412yes—
173#11717579—21no—
174#11717579—21no—
175C22×C101—22no—
176C22×C101—22no—
177C22×C101—21no—
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—84no—
188C22×C101C22×D411yescharacteristic
189C2×D101—21no—
190C2×D101—21no—
191C2×D101—21no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—84no—
196C4×C101—21no—
197C4×C101—41no—
198C4×C101C22×D411yescharacteristic
199C4×C101H(5,𝔽2)11yescharacteristic
200Dic101—21no—
201Dic101—21no—
202Dic101—21no—
203Dic101—41no—
204Dic101—41no—
205Dic101—41no—
206C2×C101—42no—
207C2×C101—41no—
208C2×C101—42no—
209C2×C101—81no—
210C2×C101D4211yescharacteristic
211C2×C101#57511yescharacteristic
212C2×C101#57611yescharacteristic
213D101—41no—
214D101—81no—
215#3273—1011nomaximal, Sylow
216C101#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1011no—
218C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
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#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)—24no—
237AGL(1,ℤ/8)—2024no—
238C2×Q16—1011no—
239C2×SD16—1011no—
240C2×SD16—1011no—
241C2×SD16C2×D10111yescharacteristic, complemented
242C2×SD16C2×D10111yescharacteristic, complemented
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8C2×D10111yescharacteristic
246C2×D8C2×D10111yescharacteristic
247C2×M16—1011no—
248C2×M16C2×D10111yescharacteristic, complemented
249#124—1011no—
250#124—1011no—
251#123—2021no—
252#123—2021no—
253C22⋊Q8—1011no—
254C22⋊Q8—1011no—
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#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—41no—
289C2×D4—1011no—
290C2×D4—1011no—
291C2×D4—4041no—
292C2×D4—2021no—
293C2×D4—2021no—
294C2×D4—2021no—
295C2×D4—2021no—
296C2×D4—2022no—
297C2×D4C22×D10111yescharacteristic
298C2×D4C22×D10111yescharacteristic
299C2×D4C22×D10111yescharacteristic
300C22×C4—21no—
301C22×C4—1011no—
302C22×C4—2021no—
303C22×C4—2021no—
304C22×C4—2021no—
305C22×C4C22×D10111yescharacteristic
306Q16—4041no—
307SD16—22no—
308SD16—22no—
309SD16—2022no—
310SD16—2022no—
311D8—22no—
312D8—4041no—
313D8—2022no—
314D8C22⋊D10114yescomplemented
315M16—22no—
316M16—2022no—
317C2×C8—1011no—
318C2×C8—1011no—
319C2×C8C22×D10111yescharacteristic
320C2×C8C22×D10111yescharacteristic
321C4⋊C4—1011no—
322C4⋊C4—1011no—
323C4⋊C4—1011no—
324C4⋊C4—2021no—
325C4⋊C4—2021no—
326C22⋊C4—1011no—
327C22⋊C4—1011no—
328C22⋊C4—2021no—
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—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—8081no—
346C23—1011no—
347C23—4042no—
348C23—4042no—
349C23—2021no—
350C23D4×D10111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2021no—
354Q8—2021no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—8081no—
362D4—8081no—
363D4—4042no—
364D4—4042no—
365D4—4041no—
366D4—2021no—
367D4—2021no—
368D4C2×C22⋊D10112yes—
369D4C2×C22⋊D10112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—8081no—
375C2×C4—8081no—
376C2×C4—1011no—
377C2×C4—1011no—
378C2×C4—1011no—
379C2×C4—4042no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4D4×D10111yescharacteristic
386C2×C4C23×D10111yescharacteristic
387C8—21no—
388C8—2021no—
389C8—2021no—
390C8C2×C22⋊D10112yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—8084no—
405C22—4041no—
406C22—4041no—
407C22—4041no—
408C22—2021no—
409C22—2021no—
410C22—2021no—
411C22C2×D4×D10111yescharacteristic
412C4—21no—
413C4—41no—
414C4—4041no—
415C4—4041no—
416C4—4041no—
417C4—2021no—
418C4—2021no—
419C4—2021no—
420C4C22×C22⋊D10111yescharacteristic
421C4#1569230011yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—8081no—
427C2—4041no—
428C2D4×C22⋊D10111yescharacteristic
429C2#2026597311yescharacteristic
430C2#2026598111yescharacteristic
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#3273
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. #20265821C2semidirect
3. #20265815C2semidirect
4. #20265558C2semidirect
5. #20265423C2semidirect
6. #20265420C2semidirect
7. #20265394C2semidirect
8. #20265390C2semidirect
9. #20265339C2semidirect
10. #20265336C2semidirect
11. #20265311C2semidirect
12. #20265309C2semidirect
13. D8×Dic101C2semidirect
14. #20265253C2semidirect
15. C2×AGL(1,ℤ/8)×C101C2semidirect
17. C2×D4×D101C22semidirect
19. C2×#13108064C22semidirect
20. C2×#13108063C22semidirect
21. C2×#13108062C22semidirect
24. C2×#13108057C22semidirect
25. #15692244C22semidirect
26. #15692242C22semidirect
29. #15692234C22semidirect
30. #15692231C22semidirect
37. D4×Dic101C22semidirect
38. D4×Dic101C22semidirect
42. #15692189C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692177C22semidirect
47. #15692176C22semidirect
48. #15692176C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
51. #15692167C22semidirect
52. C8×Dic101C22semidirect
54. C22×D4×C101C22semidirect
56. C2×SD16×C101C22semidirect
57. C2×SD16×C101C22semidirect
60. C2×M16×C101C22semidirect
119. D8×C101D4semidirect
216. C101#3273semidirect
218. C2×AGL(1,ℤ/8)D101semidirect
235. C22×D4C2×D101semidirect
241. C2×SD16C2×D101semidirect
242. C2×SD16C2×D101semidirect
248. C2×M16C2×D101semidirect
314. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
#202658212
#202658152
#202655582
#202654232
#202654202
#202653942
#202653902
#202653392
#202653362
#202653112
#202653092
D8×Dic1012
#202652532
C2×AGL(1,ℤ/8)×C1012
#3273101