Finite Group Databasebeta

← #27182673

Subgroups — #27182673

Order 12928.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692280C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic, complemented
4#20265564C211yesmaximal, characteristic, complemented
5#20265421C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265369C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265320C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265275C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#550×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic
18#15692280—24no—
19#15692275—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108061C2211yescharacteristic
24C2×#13108061C2211yescharacteristic
25C2×#13108059C2211yescharacteristic
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108058C2211yescharacteristic
30C22×#11717579C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic
32#15692240—21no—
33#15692234C2211yescharacteristic
34#15692229C2211yescharacteristic
35#15692228—21no—
36#15692223—21no—
37C4×D404C2211yescharacteristic
38C22⋊C4×D101—21no—
39#15692188C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692178C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic
51C2×#41×C101C2211yescharacteristic, complemented
52#127×C101—21no—
53#127×C101C2211yescharacteristic, complemented
54#121×C101—21no—
55#121×C101—21no—
56#121×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic, complemented
58C4×Q8×C101C2211yescharacteristic, complemented
59C4×D4×C101—21no—
60C4×D4×C101C2211yescharacteristic
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—22no—
75#13108062—22no—
76#13108062—41no—
77#13108061—22no—
78#13108061D414yescomplemented
79#13108059—22no—
80#13108059—22no—
81#13108059—22no—
82#13108059—41no—
83#13108058—22no—
84C2×#11717579—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—
96#41×C101—22no—
97#41×C101—22no—
98#41×C101—41no—
99#41×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—21no—
107C2×D4×C101D411yescharacteristic
108C2×D4×C101C2311yescharacteristic
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101—21no—
111C22×C4×C101—21no—
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114C4⋊C4×C101—21no—
115C4⋊C4×C101D411yescharacteristic
116C4⋊C4×C101C2311yescharacteristic
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C42×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C23×C101C2×D411yescharacteristic
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—42no—
130C22×D101—42no—
131C22×D101—81no—
132Q8×C101—21no—
133Q8×C101C2×D412yes—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—21no—
138D4×C101—41no—
139D4×C101—42no—
140D4×C101—42no—
141D4×C101—42no—
142D4×C101—81no—
143D4×C101—81no—
144D4×C101—81no—
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—42no—
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—21no—
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101C22×D411yescharacteristic
188C2×D101—22no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—84no—
195C4×C101—21no—
196C4×C101—21no—
197C4×C101—22no—
198C4×C101—21no—
199C4×C101—21no—
200C4×C101—41no—
201C4×C101C22×D411yescharacteristic
202C4×C101H(5,𝔽2)11yescharacteristic
203Dic101—41no—
204Dic101—41no—
205C2×C101—41no—
206C2×C101—41no—
207C2×C101—81no—
208C2×C101—81no—
209C2×C101D4211yescharacteristic
210C2×C101#57511yescharacteristic
211C2×C101#57611yescharacteristic
212D101—42no—
213D101—81no—
214#3268—1011nomaximal, Sylow
215C101#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1011no—
217C2×#135—1011no—
218#550D10111yescharacteristic, complemented
219#546—1011no—
220#496—1011no—
221#468—1011no—
222#466—1011no—
223#464—1011no—
224#459—1011no—
225#452—1011no—
226#449—1011no—
227#449—1011no—
228#447—1011no—
229#440—1011no—
230#435—1011no—
231C2×#41—1011no—
232C2×#41C2×D10111yescharacteristic
233C2×#41C2×D10111yescharacteristic, complemented
234C22×D4—1011no—
235AGL(1,ℤ/8)—2024no—
236#135—2024no—
237C2×Q16—1011no—
238C2×SD16—1011no—
239C2×SD16—1011no—
240C2×SD16—1011no—
241C2×SD16—1011no—
242C2×D8—1011no—
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8—1011no—
246C2×M16—1011no—
247C22×C8—1011no—
248#127—21no—
249#127C2×D10111yescharacteristic, complemented
250#124—1011no—
251#123—2021no—
252C22⋊Q8—1011no—
253#121—21no—
254#121—21no—
255#121—1011no—
256#121—2021no—
257#121C2×D10111yescharacteristic
258#121C2×D10111yescharacteristic, complemented
259#120—2021no—
260C4×Q8C2×D10111yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—1011no—
263C4×D4C2×D10111yescharacteristic
264C2×C22⋊C4—2021no—
265#107—1011no—
266C4⋊C8—1011no—
267#103—1011no—
268#103—1011no—
269#102—1011no—
270#102—1011no—
271#102—1011no—
272#102—1011no—
273C22⋊C8—1011no—
274C22⋊C8—1011no—
275#97—1011no—
276C24—2021no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—4041no—
282#41—2022no—
283C2×Q8—1011no—
284C2×Q8C22×D10111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—4041no—
294C2×D4—2022no—
295C2×D4—2021no—
296C2×D4—2021no—
297C2×D4—2021no—
298C2×D4C22⋊D10111yescharacteristic
299C2×D4C22×D10111yescharacteristic
300C2×D4C22×D10111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2021no—
304C22×C4—2021no—
305C22×C4C22×D10111yescharacteristic
306C22×C4C22×D10111yescharacteristic
307Q16—2022no—
308SD16—1014no—
309SD16—4041no—
310SD16—2022no—
311SD16—2022no—
312D8—4041no—
313D8—2022no—
314D8—2022no—
315D8—2022no—
316M16—2022no—
317C2×C8—1011no—
318C2×C8—1011no—
319C2×C8—1011no—
320C2×C8—1011no—
321C2×C8—2022no—
322C4⋊C4—21no—
323C4⋊C4—1011no—
324C4⋊C4—2021no—
325C4⋊C4C22⋊D10111yescharacteristic
326C4⋊C4C22×D10111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4C22⋊D10111yescharacteristic
335C22⋊C4C22⋊D10111yescharacteristic
336C42—21no—
337C42C22×D10111yescharacteristic
338C23—21no—
339C23—21no—
340C23—8081no—
341C23—4042no—
342C23—4042no—
343C23—2021no—
344C23—2021no—
345C23C2×C22⋊D10111yescharacteristic
346C23C2×C22⋊D10111yescharacteristic
347Q8—21no—
348Q8—2021no—
349Q8—2021no—
350Q8D4×D10112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—8081no—
363D4—1012no—
364D4—4042no—
365D4—4042no—
366D4—2021no—
367D4—2021no—
368D4—2021no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—8081no—
378C2×C4—4042no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4C2×C22⋊D10111yescharacteristic
382C2×C4C2×C22⋊D10111yescharacteristic
383C2×C4C2×C22⋊D10111yescharacteristic
384C2×C4C2×C22⋊D10111yescharacteristic
385C2×C4C23×D10111yescharacteristic
386C8—1012no—
387C8—2021no—
388C8—2021no—
389C8—2021no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—8084no—
403C22—4041no—
404C22—4041no—
405C22—4041no—
406C22—4041no—
407C22—4041no—
408C22—2022no—
409C22C22×C22⋊D10111yescharacteristic
410C4—21no—
411C4—21no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—4041no—
417C4—4041no—
418C4C2×D4×D10111yescharacteristic
419C4#1569230011yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—8081no—
425C2—4042no—
426C2D4×C22⋊D10111yescharacteristic
427C2#2026597511yescharacteristic
428C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#550×C101C2
socleC22×C101C22×D4
hypercenter#550D101
hypocenterC101#3268
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C2×#15692275C2semidirect
3. #20265823C2semidirect
4. #20265564C2semidirect
5. #20265421C2semidirect
6. #20265411C2semidirect
7. #20265399C2semidirect
8. #20265369C2semidirect
9. #20265339C2semidirect
10. #20265322C2semidirect
11. #20265320C2semidirect
12. #20265310C2semidirect
13. #20265275C2semidirect
14. #20265249C2semidirect
15. #550×C101C2semidirect
20. C2×#13108064C22semidirect
21. C2×#13108062C22semidirect
22. C2×#13108062C22semidirect
30. C22×#11717579C22semidirect
39. #15692188C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692178C22semidirect
43. #15692176C22semidirect
44. #15692176C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
51. C2×#41×C101C22semidirect
53. #127×C101C22semidirect
57. #121×C101C22semidirect
58. C4×Q8×C101C22semidirect
78. #13108061D4semidirect
215. C101#3268semidirect
218. #550D101semidirect
233. C2×#41C2×D101semidirect
249. #127C2×D101semidirect
258. #121C2×D101semidirect
260. C4×Q8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×#156922752
#202658232
#202655642
#202654212
#202654112
#202653992
#202653692
#202653392
#202653222
#202653202
#202653102
#202652752
#202652492
#550×C1012
#3268101