Finite Group Databasebeta

← #16857568

Subgroups — #16857568

Order 3968.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521749C211yesmaximal, characteristic, complemented
2#14521692C211yesmaximal, characteristic, complemented
3#14521403C211yesmaximal, characteristic, complemented
4#14521383C211yesmaximal, characteristic, complemented
5#14521372C211yesmaximal, characteristic, complemented
6#14521350C211yesmaximal, characteristic, complemented
7#14521282C211yesmaximal, characteristic, complemented
8#14521275C211yesmaximal, characteristic, complemented
9#14521193C211yesmaximal, characteristic, complemented
10#14521181C211yesmaximal, characteristic, complemented
11C4×#92479C211yesmaximal, characteristic, complemented
12#14521114C211yesmaximal, characteristic, complemented
13#14521044C211yesmaximal, characteristic, complemented
14#102×D31C211yesmaximal, characteristic, complemented
15C31×#463C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×D4×D31C2211yescharacteristic
18C2×SD16×C31C2211yescharacteristic, complemented
19C2×D8×C31C2211yescharacteristic
20C31×#124C2211yescharacteristic
21C4×D4×C31C2211yescharacteristic, complemented
22#11758523C2211yescharacteristic
23#11758515C2211yescharacteristic
24#11758515C2211yescharacteristic
25#11758515C2211yescharacteristic, complemented
26C2×#92479C2211yescharacteristic
27#11758506—21no—
28C4×C22⋊D31—21no—
29C2×C8×D31C2211yescharacteristic
30C2×C8×D31C2211yescharacteristic, complemented
31#11758469C2211yescharacteristic, complemented
32C4⋊C4×D31C2211yescharacteristic
33#11758461—21no—
34#11758461—21no—
35#11758460—21no—
36#11758459—21no—
37C22⋊C4×D31—21no—
38#11758456—21no—
39C4×D124C2211yescharacteristic
40C4×D124C2211yescharacteristic
41#11758447C2211yescharacteristic, complemented
42C31×C4⋊C8C2211yescharacteristic, complemented
43C31×#103C2211yescharacteristic, complemented
44C31×#102C2211yescharacteristic
45#11758423C2211yescharacteristic, complemented
46#11758423C2211yescharacteristic, complemented
47#11758420C2211yescharacteristic
48#11758420C2211yescharacteristic
49#11758420C2211yescharacteristic, complemented
50#11758409C2211yescharacteristic
51#11758409C2211yescharacteristic
52(C2×D31)⋊C8C2211yescharacteristic
53(C2×D31)⋊C8C2211yescharacteristic, complemented
54#11758406C2211yescharacteristic, complemented
55#11758405C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic
57#11758395C2211yescharacteristic, complemented
58C4×C31⋊C8C2211yescharacteristic, complemented
59C23×D31—21no—
60C2×Q8×C31C2311yescharacteristic
61C2×D4×C31C2311yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C2×C22⋊D31—21no—
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68#92499—22no—
69#92499—41no—
70D4×D31—22no—
71D4×D31—41no—
72C2×D124—21no—
73C2×D124D411yescharacteristic
74C2×D124C2311yescharacteristic
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×C4×D31C2311yescharacteristic
79C2×C4×D31C2311yescharacteristic
80SD16×C31—41no—
81D8×C31—41no—
82C2×C8×C31C2311yescharacteristic
83C2×C8×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C22⋊C4×C31—21no—
86C22⋊C4×C31—21no—
87C22⋊C4×C31—21no—
88C42×C31C2311yescharacteristic
89C22⋊Dic31—21no—
90C22⋊Dic31—21no—
91C22⋊Dic31—21no—
92#92479—22no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4—21no—
96(C2×D31)⋊C4D411yescharacteristic
97(C2×D31)⋊C4D411yescharacteristic
98C4⋊Dic31D411yescharacteristic
99C4⋊Dic31C2311yescharacteristic
100C4⋊Dic31C2311yescharacteristic
101C4⋊Dic31C2311yescharacteristic
102Dic31⋊C4—21no—
103Dic31⋊C4—21no—
104Dic31⋊C4—21no—
105Dic31⋊C4—21no—
106C4×Dic31—21no—
107C2×C31⋊C8C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C8×D31—41no—
110C8×D31—41no—
111C23×C31—21no—
112C23×C31—21no—
113C22×D31—21no—
114C22×D31—42no—
115C22×D31—42no—
116C22×D31—81no—
117C22×D31C2×D411yescharacteristic
118C22×D31C2×D411yescharacteristic
119Q8×C31—21no—
120Q8×C31—21no—
121D4×C31—21no—
122D4×C31—21no—
123D4×C31—21no—
124D4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—81no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—42no—
132C22⋊D31—42no—
133C22⋊D31—81no—
134C22⋊D31—81no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
138C2×Dic31C2×D411yescharacteristic
139C2×Dic31C2×D411yescharacteristic
140D124—22no—
141D124—21no—
142D124—42no—
143D124#4112yes—
144C4×D31—21no—
145C4×D31—21no—
146C4×D31—21no—
147C4×D31—21no—
148C4×D31—42no—
149C4×D31—81no—
150C4×D31—81no—
151C8×C31—21no—
152C8×C31—21no—
153C31⋊C8—21no—
154C31⋊C8—21no—
155#3297—311nomaximal, Sylow
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31C22×D411yescharacteristic
163C2×D31—21no—
164C2×D31—21no—
165C2×D31—21no—
166C2×D31—22no—
167C2×D31—21no—
168C2×D31—21no—
169C2×D31—41no—
170C2×D31—41no—
171C2×D31—84no—
172C2×D31#13512yes—
173C4×C31—21no—
174C4×C31—21no—
175C4×C31—41no—
176C4×C31—41no—
177C4×C31C2×#4111yescharacteristic
178C4×C31H(5,𝔽2)11yescharacteristic
179Dic31—21no—
180Dic31—21no—
181Dic31—41no—
182Dic31—41no—
183Dic31—41no—
184#553—311no—
185#546—311no—
186#488—311no—
187#484—311no—
188#482—311no—
189#478—311no—
190#466—311no—
191#465—311no—
192#463D3111yescharacteristic, complemented
193#449—311no—
194#447—311no—
195C4×D8—311no—
196C8×D4—311no—
197#416—311no—
198C2×#102—311no—
199C2×C31—81no—
200C2×C31—81no—
201C2×C31#54611yescharacteristic
202C2×C31C2×#13511yescharacteristic
203C2×C31#57611yescharacteristic
204D31—22no—
205D31—41no—
206D31—42no—
207C2×#41—311no—
208C22×D4—311no—
209C2×SD16C2×D3111yescharacteristic, complemented
210C2×D8—311no—
211C2×D8C2×D3111yescharacteristic
212C22×C8—311no—
213C22×C8—311no—
214#126—621no—
215#125—311no—
216#124C2×D3111yescharacteristic
217#123—621no—
218#123—621no—
219C22⋊Q8—311no—
220#121—311no—
221#121—311no—
222#121—311no—
223#121—621no—
224#121—621no—
225#120—621no—
226C4×D4—311no—
227C4×D4—311no—
228C4×D4—621no—
229C4×D4C2×D3111yescharacteristic, complemented
230#117—311no—
231C2×C4⋊C4—311no—
232C2×C22⋊C4—621no—
233#107—311no—
234#107—311no—
235#106—311no—
236C4⋊C8C2×D3111yescharacteristic, complemented
237#103—311no—
238#103—311no—
239#103C2×D3111yescharacteristic, complemented
240#102—311no—
241#102—311no—
242#102—311no—
243#102—311no—
244#102—311no—
245#102—311no—
246#102C2×D3111yescharacteristic
247C22⋊C8—311no—
248C22⋊C8—311no—
249C4×C8—311no—
250C31#329711yescharacteristic, complemented, Sylow
251C24—621no—
252#41—622no—
253#41—1241no—
254C2×Q8C22×D3111yescharacteristic
255C2×D4—311no—
256C2×D4—311no—
257C2×D4—621no—
258C2×D4—621no—
259C2×D4—622no—
260C2×D4—621no—
261C2×D4—621no—
262C2×D4—621no—
263C2×D4—1241no—
264C2×D4C22×D3111yescharacteristic
265C2×D4C22×D3111yescharacteristic
266C22×C4—21no—
267C22×C4—311no—
268C22×C4—311no—
269C22×C4—621no—
270C22×C4—621no—
271C22×C4—621no—
272SD16—41no—
273D8—41no—
274D8—622no—
275C2×C8—311no—
276C2×C8—311no—
277C2×C8—1241no—
278C2×C8—1241no—
279C2×C8C22×D3111yescharacteristic
280C2×C8C22×D3111yescharacteristic
281C4⋊C4—311no—
282C4⋊C4—311no—
283C4⋊C4—311no—
284C4⋊C4—311no—
285C4⋊C4—621no—
286C4⋊C4—621no—
287C4⋊C4—621no—
288C4⋊C4—621no—
289C4⋊C4C22×D3111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—311no—
294C22⋊C4—311no—
295C22⋊C4—621no—
296C22⋊C4—621no—
297C22⋊C4—621no—
298C22⋊C4—621no—
299C22⋊C4—621no—
300C22⋊C4—621no—
301C42—621no—
302C42C22×D3111yescharacteristic
303C23—21no—
304C23—21no—
305C23—311no—
306C23—311no—
307C23—621no—
308C23—1242no—
309C23—1242no—
310C23—2481no—
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—312no—
318D4—622no—
319D4—621no—
320D4—1242no—
321D4—1242no—
322D4—1242no—
323D4—2481no—
324D4—2481no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—311no—
329C2×C4—311no—
330C2×C4—621no—
331C2×C4—621no—
332C2×C4—621no—
333C2×C4—621no—
334C2×C4—621no—
335C2×C4—621no—
336C2×C4—621no—
337C2×C4—1242no—
338C2×C4—2481no—
339C2×C4—2481no—
340C2×C4D4×D3111yescharacteristic
341C2×C4D4×D3111yescharacteristic
342C2×C4C23×D3111yescharacteristic
343C8—21no—
344C8—21no—
345C8—621no—
346C8—621no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—312no—
354C22—621no—
355C22—621no—
356C22—621no—
357C22—622no—
358C22—621no—
359C22—621no—
360C22—1241no—
361C22—1241no—
362C22—2484no—
363C22C2×D4×D3111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—621no—
369C4—621no—
370C4—1241no—
371C4—1241no—
372C4—1241no—
373C4#1175856111yescharacteristic
374C4#41×D3111yescharacteristic
375C2—81no—
376C2—81no—
377C2—622no—
378C2—1241no—
379C2—1242no—
380C2#1452169611yescharacteristic
381C2#135×D3111yescharacteristic
382C2#1452185311yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#463C2
socleC22×C31C22×D4
hypercenter#463D31
hypocenterC31#3297
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #14521749C2semidirect
2. #14521692C2semidirect
3. #14521403C2semidirect
4. #14521383C2semidirect
5. #14521372C2semidirect
6. #14521350C2semidirect
7. #14521282C2semidirect
8. #14521275C2semidirect
9. #14521193C2semidirect
10. #14521181C2semidirect
11. C4×#92479C2semidirect
12. #14521114C2semidirect
13. #14521044C2semidirect
14. #102×D31C2semidirect
15. C31×#463C2semidirect
18. C2×SD16×C31C22semidirect
21. C4×D4×C31C22semidirect
25. #11758515C22semidirect
30. C2×C8×D31C22semidirect
31. #11758469C22semidirect
41. #11758447C22semidirect
42. C31×C4⋊C8C22semidirect
43. C31×#103C22semidirect
45. #11758423C22semidirect
46. #11758423C22semidirect
49. #11758420C22semidirect
53. (C2×D31)⋊C8C22semidirect
54. #11758406C22semidirect
55. #11758405C22semidirect
57. #11758395C22semidirect
58. C4×C31⋊C8C22semidirect
192. #463D31semidirect
209. C2×SD16C2×D31semidirect
229. C4×D4C2×D31semidirect
236. C4⋊C8C2×D31semidirect
239. #103C2×D31semidirect
250. C31#3297semidirect

Maximal subgroups

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

subgroupindex
#145217492
#145216922
#145214032
#145213832
#145213722
#145213502
#145212822
#145212752
#145211932
#145211812
C4×#924792
#145211142
#145210442
#102×D312
C31×#4632
#329731