Finite Group Databasebeta

← #27182840

Subgroups — #27182840

Order 12928.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692275C211yesmaximal, characteristic, complemented
2C22×#13108061C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic, complemented
4#20265559C211yesmaximal, characteristic, complemented
5#20265408C211yesmaximal, characteristic, complemented
6#20265404C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265337C211yesmaximal, characteristic, complemented
10#20265333C211yesmaximal, characteristic, complemented
11#20265322C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13C4×#13108061C211yesmaximal, characteristic, complemented
14D4×#11717579C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic
18#15692275—24no—
19C2×#13108064C2211yescharacteristic, complemented
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108061—24no—
23C2×#13108061C2211yescharacteristic
24C2×#13108061C2211yescharacteristic
25C2×#13108061C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C22×#11717579C2211yescharacteristic
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#15692186C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692178C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic, complemented
49C4×#11717579C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic, complemented
51C22×Q8×C101C2211yescharacteristic
52#124×C101—21no—
53#124×C101C2211yescharacteristic, complemented
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101C2211yescharacteristic, complemented
56#121×C101—21no—
57#121×C101C2211yescharacteristic
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#13108064—41no—
76#13108062—22no—
77#13108061—22no—
78#13108061—22no—
79#13108061—22no—
80#13108061D414yescomplemented
81#13108059—22no—
82#13108059—41no—
83C2×#11717579—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—41no—
98C2×Q8×C101—21no—
99C2×Q8×C101—22no—
100C2×Q8×C101—41no—
101C2×Q8×C101C2311yescharacteristic
102C2×Q8×C101C2311yescharacteristic
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101D411yescharacteristic
106C2×D4×C101C2311yescharacteristic
107C22×C4×C101—21no—
108C22×C4×C101—21no—
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C4⋊C4×C101—21no—
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×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C42×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—81no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—42no—
135Q8×C101C2×D412yes—
136D4×C101—22no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—42no—
140D4×C101—42no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—42no—
149C2×C4×C101—42no—
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2×D411yescharacteristic
154C2×C4×C101C2411yescharacteristic
155C22⋊D101—42no—
156C22⋊D101—81no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159D404—21no—
160D404—21no—
161D404—21no—
162D404—42no—
163D404C2×D412yes—
164C4×D101—42no—
165C4×D101—81no—
166Dic202—21no—
167Dic202—21no—
168#11717579—21no—
169#11717579—21no—
170#11717579—21no—
171#11717579C2×D412yes—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101#13512yescomplemented
181C22×C101C22×D411yescharacteristic
182C2×D101—22no—
183C2×D101—41no—
184C2×D101—41no—
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—84no—
189C4×C101—21no—
190C4×C101—21no—
191C4×C101—22no—
192C4×C101—21no—
193C4×C101—21no—
194C4×C101—41no—
195C4×C101—41no—
196C4×C101C22×D411yescharacteristic
197C4×C101H(5,𝔽2)11yescharacteristic
198Dic101—41no—
199Dic101—41no—
200C2×C101—22no—
201C2×C101—41no—
202C2×C101—81no—
203C2×C101D4211yescharacteristic
204C2×C101C2×#13511yescharacteristic
205C2×C101#57711yescharacteristic
206D101—42no—
207D101—81no—
208#3282—1011nomaximal, Sylow
209C101#328211yescharacteristic, complemented, Sylow
210C2×#135—1011no—
211C22×SD16—1011no—
212#548D10111yescharacteristic, complemented
213#546—1011no—
214#495—1011no—
215#465—1011no—
216#465—1011no—
217#464—1011no—
218#463—1011no—
219#452—1011no—
220#451—1011no—
221#449—1011no—
222#447—1011no—
223C4×SD16—1011no—
224C8×D4—1011no—
225C2×#41—1011no—
226C2×#41C2×D10111yescharacteristic, complemented
227C22×Q8C2×D10111yescharacteristic
228C22×D4—1011no—
229#135—2024no—
230C2×Q16—1011no—
231C2×Q16—1011no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×SD16—2024no—
238C2×D8—1011no—
239C2×D8—1011no—
240C22×C8—1011no—
241C22×C8—1011no—
242#124—21no—
243#124—1011no—
244#124C2×D10111yescharacteristic, complemented
245#123—2021no—
246C22⋊Q8—21no—
247C22⋊Q8—1011no—
248C22⋊Q8C2×D10111yescharacteristic, complemented
249#121—21no—
250#121—1011no—
251#121—2021no—
252#121C2×D10111yescharacteristic
253#120—2021no—
254C4×Q8C2×D10111yescharacteristic, complemented
255C4×D4—21no—
256C4×D4—1011no—
257C4×D4C2×D10111yescharacteristic
258C2×C22⋊C4—2021no—
259#106—1011no—
260C4⋊C8—1011no—
261#103—1011no—
262#103—1011no—
263#103—1011no—
264#102—1011no—
265#102—1011no—
266#102—1011no—
267C22⋊C8—1011no—
268C22⋊C8—1011no—
269C4×C8—1011no—
270C24—2021no—
271#41—22no—
272#41—41no—
273#41—4041no—
274#41—2022no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1011no—
279C2×Q8C22×D10111yescharacteristic
280C2×Q8C22×D10111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1011no—
284C2×D4—1011no—
285C2×D4—4041no—
286C2×D4—2022no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4C22⋊D10111yescharacteristic
291C2×D4C22×D10111yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4C22×D10111yescharacteristic
297C22×C4C22×D10111yescharacteristic
298Q16—4041no—
299Q16—2022no—
300SD16—1014no—
301SD16—2022no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305D8—4041no—
306D8—2022no—
307C2×C8—1011no—
308C2×C8—1011no—
309C2×C8—1011no—
310C2×C8—1011no—
311C2×C8—2022no—
312C2×C8—2022no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—1011no—
316C4⋊C4—2021no—
317C4⋊C4C22⋊D10111yescharacteristic
318C4⋊C4C22×D10111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4C22⋊D10111yescharacteristic
329C22⋊C4C22⋊D10111yescharacteristic
330C42—21no—
331C42C22×D10111yescharacteristic
332C23—21no—
333C23—8081no—
334C23—4042no—
335C23—4042no—
336C23—2021no—
337C23—2021no—
338C23C2×C22⋊D10111yescharacteristic
339C23C2×C22⋊D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—2021no—
345Q8—2021no—
346Q8D4×D10112yes—
347D4—22no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—8081no—
353D4—1012no—
354D4—4042no—
355D4—4042no—
356D4—2021no—
357D4—2021no—
358D4—2021no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—8081no—
369C2×C4—4042no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4C2×C22⋊D10111yescharacteristic
373C2×C4C2×C22⋊D10111yescharacteristic
374C2×C4C2×C22⋊D10111yescharacteristic
375C2×C4C2×C22⋊D10111yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—1012no—
378C8—2021no—
379C8—2021no—
380C8—2021no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—8084no—
390C22—4041no—
391C22—4041no—
392C22—4041no—
393C22—4041no—
394C22—4041no—
395C22—2022no—
396C22#1569227512yescomplemented
397C22C22×C22⋊D10111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—4041no—
406C4—4041no—
407C4C2×D4×D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—22no—
410C2—41no—
411C2—81no—
412C2—8081no—
413C2—4042no—
414C2D4×C22⋊D10111yescharacteristic
415C2C2×#1569227511yescharacteristic
416C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3282
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. C22×#13108061C2semidirect
3. #20265823C2semidirect
4. #20265559C2semidirect
5. #20265408C2semidirect
6. #20265404C2semidirect
7. #20265399C2semidirect
8. #20265389C2semidirect
9. #20265337C2semidirect
10. #20265333C2semidirect
11. #20265322C2semidirect
12. #20265310C2semidirect
13. C4×#13108061C2semidirect
14. D4×#11717579C2semidirect
15. #548×C101C2semidirect
19. C2×#13108064C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108062C22semidirect
30. C22×#11717579C22semidirect
39. #15692186C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
43. #15692178C22semidirect
44. #15692176C22semidirect
48. C22⋊#11717579C22semidirect
49. C4×#11717579C22semidirect
50. C2×#41×C101C22semidirect
53. #124×C101C22semidirect
55. C22⋊Q8×C101C22semidirect
58. C4×Q8×C101C22semidirect
80. #13108061D4semidirect
180. C22×C101#135semidirect
209. C101#3282semidirect
212. #548D101semidirect
226. C2×#41C2×D101semidirect
244. #124C2×D101semidirect
248. C22⋊Q8C2×D101semidirect
254. C4×Q8C2×D101semidirect
396. C22#15692275semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
C22×#131080612
#202658232
#202655592
#202654082
#202654042
#202653992
#202653892
#202653372
#202653332
#202653222
#202653102
C4×#131080612
D4×#117175792
#548×C1012
#3282101