Finite Group Databasebeta

← #27182653

Subgroups — #27182653

Order 12928.

500 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×#15692281C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265832C211yesmaximal, characteristic, complemented
4#20265572C211yesmaximal, characteristic, complemented
5#20265433C211yesmaximal, characteristic, complemented
6#20265404C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265382C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265337C211yesmaximal, characteristic, complemented
11#20265328C211yesmaximal, characteristic, complemented
12#20265320C211yesmaximal, characteristic, complemented
13#20265287C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#546×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108074C2211yescharacteristic, complemented
18#15692281—24no—
19#15692275—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic
23C2×#13108062C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic
26C2×#13108061C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic
29C2×#13108058C2211yescharacteristic
30C22×#11717579C2211yescharacteristic, complemented
31#15692255C2211yescharacteristic, complemented
32#15692237—21no—
33#15692237—21no—
34#15692234C2211yescharacteristic
35#15692234C2211yescharacteristic, complemented
36#15692226—21no—
37C4×D404C2211yescharacteristic, complemented
38C4⋊C4×D101—21no—
39#15692186C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692178C2211yescharacteristic, complemented
43#15692178C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic
51C22×D4×C101C2211yescharacteristic
52#124×C101C2211yescharacteristic
53#123×C101—21no—
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101—21no—
56#121×C101C2211yescharacteristic
57#120×C101—21no—
58C4×D4×C101C2211yescharacteristic
59C4×D4×C101C2211yescharacteristic
60C2×C22⋊C4×C101—21no—
61#13108074—22no—
62#13108074—22no—
63#13108074—41no—
64#13108074—41no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67C2×D404C2311yescharacteristic
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73#13108064—22no—
74#13108064—41no—
75#13108062—22no—
76#13108062—22no—
77#13108062—22no—
78#13108062—41no—
79#13108061—22no—
80#13108059—22no—
81#13108059D414yescomplemented
82#13108058—22no—
83C2×#11717579—22no—
84C2×#11717579C2311yescharacteristic
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C4⋊Dic101—21no—
89C4⋊Dic101—21no—
90C4⋊Dic101C2311yescharacteristic
91#13108053—21no—
92#13108053—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95#13108051—21no—
96C24×C101—21no—
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×D4×C101—22no—
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—41no—
105C2×D4×C101D411yescharacteristic
106C2×D4×C101C2311yescharacteristic
107C2×D4×C101C2311yescharacteristic
108C22×C4×C101—21no—
109C22×C4×C101—21no—
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
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×C101—21no—
120C22⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C42×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—42no—
126C23×C101—42no—
127C23×C101—81no—
128C23×C101C2×D411yescharacteristic
129C23×C101C2×D411yescharacteristic
130C22×D101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—41no—
137D4×C101—42no—
138D4×C101—42no—
139D4×C101—81no—
140D4×C101C2×D412yes—
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—42no—
148C2×C4×C101—81no—
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2411yescharacteristic
154C22⋊D101—42no—
155C22⋊D101—42no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159D404—21no—
160D404C2×D412yes—
161C4×D101—42no—
162C4×D101—42no—
163C4×D101—42no—
164Dic202—21no—
165Dic202—21no—
166Dic202—21no—
167Dic202—21no—
168#11717579—21no—
169#11717579—21no—
170#11717579—21no—
171#11717579C2×D412yes—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—22no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101—21no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—84no—
185C22×C101C22×D411yescharacteristic
186C2×D101—22no—
187C2×D101—41no—
188C2×D101—41no—
189C4×C101—21no—
190C4×C101—21no—
191C4×C101—21no—
192C4×C101—21no—
193C4×C101—41no—
194C4×C101—41no—
195C4×C101C22×D411yescharacteristic
196C4×C101H(5,𝔽2)11yescharacteristic
197Dic101—41no—
198Dic101—41no—
199Dic101—41no—
200C2×C101—41no—
201C2×C101—41no—
202C2×C101—42no—
203C2×C101—81no—
204C2×C101D4211yescharacteristic
205C2×C101#57511yescharacteristic
206C2×C101#57711yescharacteristic
207D101—42no—
208#3267—1011nomaximal, Sylow
209C101#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1011no—
211C2×#135—1011no—
212#547—1011no—
213#546D10111yescharacteristic, complemented
214#497—1011no—
215#470—1011no—
216#465—1011no—
217#463—1011no—
218#461—1011no—
219#452—1011no—
220#452—1011no—
221#450—1011no—
222#449—1011no—
223#442—1011no—
224#435—1011no—
225C2×#41—1011no—
226C2×#41—1011no—
227C2×#41C2×D10111yescharacteristic
228C22×D4C2×D10111yescharacteristic
229AGL(1,ℤ/8)—2024no—
230#135—2024no—
231C2×Q16—1011no—
232C2×Q16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×SD16—1011no—
238C2×D8—1011no—
239C2×D8—1011no—
240C2×M16—1011no—
241C22×C8—1011no—
242C4⋊Q8—1011no—
243#124C2×D10111yescharacteristic
244#123—21no—
245#123—2021no—
246#123—2021no—
247C22⋊Q8—1011no—
248C22⋊Q8—1011no—
249C22⋊Q8C2×D10111yescharacteristic
250#121—21no—
251#121—2021no—
252#121C2×D10111yescharacteristic
253#120—21no—
254C4×D4—1011no—
255C4×D4C2×D10111yescharacteristic
256C4×D4C2×D10111yescharacteristic
257C2×C4⋊C4—2021no—
258C2×C22⋊C4—21no—
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—
269#97—1011no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—4041no—
274#41—4041no—
275#41—2022no—
276#41—2022no—
277C2×Q8—1011no—
278C2×Q8—1011no—
279C2×Q8C22×D10111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1011no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4C22⋊D10111yescharacteristic
289C2×D4C22×D10111yescharacteristic
290C2×D4C22×D10111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2021no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4C22×D10111yescharacteristic
297C22×C4C22×D10111yescharacteristic
298Q16—4041no—
299Q16—2022no—
300SD16—4041no—
301SD16—2022no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305D8—1014no—
306D8—2022no—
307M16—2022no—
308C2×C8—1011no—
309C2×C8—1011no—
310C2×C8—1011no—
311C2×C8—1011no—
312C2×C8—2022no—
313C4⋊C4—21no—
314C4⋊C4—1011no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4—2021no—
319C4⋊C4C22⋊D10111yescharacteristic
320C4⋊C4C22×D10111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4C22⋊D10111yescharacteristic
330C22⋊C4C22⋊D10111yescharacteristic
331C42C22×D10111yescharacteristic
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—2021no—
338C23C2×C22⋊D10111yescharacteristic
339C23C2×C22⋊D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345Q8—2021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—1012no—
354D4—4042no—
355D4—4042no—
356D4—2021no—
357D4D4×D10112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—4042no—
367C2×C4—4042no—
368C2×C4—4042no—
369C2×C4—2021no—
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—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—4041no—
395C22—4041no—
396C22—2022no—
397C22C22×C22⋊D10111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—4041no—
405C4—4041no—
406C4—4041no—
407C4C2×D4×D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—81no—
413C2—4042no—
414C2D4×C22⋊D10111yescharacteristic
415C2#2026597511yescharacteristic
416C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#546×C101C2
socleC22×C101C22×D4
hypercenter#546D101
hypocenterC101#3267
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×#15692275C2semidirect
3. #20265832C2semidirect
4. #20265572C2semidirect
5. #20265433C2semidirect
6. #20265404C2semidirect
7. #20265389C2semidirect
8. #20265382C2semidirect
9. #20265339C2semidirect
10. #20265337C2semidirect
11. #20265328C2semidirect
12. #20265320C2semidirect
13. #20265287C2semidirect
14. #20265249C2semidirect
15. #546×C101C2semidirect
17. C2×#13108074C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108064C22semidirect
26. C2×#13108061C22semidirect
30. C22×#11717579C22semidirect
31. #15692255C22semidirect
35. #15692234C22semidirect
37. C4×D404C22semidirect
39. #15692186C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692178C22semidirect
43. #15692178C22semidirect
46. #15692175C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
81. #13108059D4semidirect
209. C101#3267semidirect
213. #546D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C2×#156922752
#202658322
#202655722
#202654332
#202654042
#202653892
#202653822
#202653392
#202653372
#202653282
#202653202
#202652872
#202652492
#546×C1012
#3267101