Finite Group Databasebeta

← #16859368

Subgroups — #16859368

Order 3968.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758564C211yesmaximal, characteristic, complemented
2C2×#41×D31C211yesmaximal, characteristic, complemented
3#14521721C211yesmaximal, characteristic, complemented
4#14521720C213yesmaximal, complemented
5#14521718C211yesmaximal, characteristic, complemented
6#14521717C213yesmaximal, complemented
7#14521710C211yesmaximal, characteristic, complemented
8#14521709C213yesmaximal, complemented
9#14521707C213yesmaximal, complemented
10#14521617C213yesmaximal, complemented
11#14521615C213yesmaximal, complemented
12#14521562C213yesmaximal, complemented
13#14521558C213yesmaximal, complemented
14#14521525C211yesmaximal, characteristic, complemented
15C31×#548C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic, complemented
18#41×D31—28no—
19#11758564—28no—
20C2×#92499C2213yes—
21C2×#92499C2211yescharacteristic
22C2×#92499C2211yescharacteristic, complemented
23C2×Q8×D31C2211yescharacteristic
24C2×Q8×D31C2213yescomplemented
25C2×#92497C2213yes—
26C2×D4×D31C2213yes—
27C2×#92495C2213yes—
28C2×#92495C2213yes—
29C2×#92495C2213yescomplemented
30C22×C4×D31C2213yes—
31C31×#124C2213yescomplemented
32C31×C22⋊Q8C2213yescomplemented
33C31×#121C2213yescomplemented
34C4×Q8×C31C2211yescharacteristic, complemented
35C4×D4×C31C2213yescomplemented
36#11758524C2213yescomplemented
37#11758523C2213yescomplemented
38Q8×Dic31C2211yescharacteristic, complemented
39#11758517C2216yescomplemented
40#11758517C2213yescomplemented
41#11758516C2213yescomplemented
42#11758515C2213yescomplemented
43#11758514C2213yes—
44#11758507C2213yescomplemented
45#11758507C2213yescomplemented
46C4×C22⋊D31C2213yescomplemented
47C4×C22⋊D31C2213yescomplemented
48#11758473C2213yescomplemented
49#11758472C2213yescomplemented
50#11758472C2213yescomplemented
51#11758471C2213yes—
52#11758470C2216yescomplemented
53#11758469C2213yes—
54#11758464C2213yescomplemented
55#11758462C2216yescomplemented
56#11758461C2216yescomplemented
57#11758461C2216yescomplemented
58#11758461C2213yescomplemented
59#11758460C2211yescharacteristic
60#11758459C2216yes—
61#11758459C2213yes—
62(C22⋊D31)⋊C4C2216yescomplemented
63#11758455C2213yescomplemented
64#11758451C2213yescomplemented
65C4×D124C2213yescomplemented
66#11758449C2213yes—
67C4×Dic62C2213yescomplemented
68C23×D31C2313yes—
69#41×C31—24no—
70C2×Q8×C31—24no—
71C2×Q8×C31C2313yes—
72C2×Q8×C31C2311yescharacteristic
73C2×D4×C31C2313yes—
74C2×D4×C31C2313yes—
75C22×C4×C31C2313yes—
76C22×C4×C31C2313yes—
77C2×C22⋊D31C2311yescharacteristic
78C2×C22⋊D31C2316yes—
79C2×C22⋊D31C2313yes—
80C2×C22⋊D31C2316yes—
81C2×C22⋊D31C2313yes—
82C2×C22⋊D31C2313yes—
83C22×Dic31C2313yes—
84#92499—24no—
85#92499—24no—
86#92499—212no—
87Q8×D31—212no—
88Q8×D31D418yescomplemented
89#92497—212no—
90D4×D31—212no—
91#92495—212no—
92#92495—212no—
93#92495—212no—
94C2×D124C2313yes—
95C2×D124C2313yes—
96C2×D124C2313yes—
97C2×D124C2313yes—
98C2×D124C2316yes—
99C2×C4×D31—212no—
100C2×C4×D31C2313yes—
101C2×C4×D31C2313yes—
102C2×C4×D31C2316yes—
103C2×C4×D31C2311yescharacteristic
104C2×C4×D31C2313yes—
105C2×C4×D31C2313yes—
106C2×Dic62C2313yes—
107C2×Dic62C2313yes—
108C4⋊C4×C31C2313yes—
109C4⋊C4×C31C2313yes—
110C22⋊C4×C31C2313yes—
111C22⋊C4×C31C2316yes—
112C22⋊C4×C31C2311yescharacteristic
113C42×C31C2313yes—
114C22⋊Dic31C2313yes—
115C22⋊Dic31C2311yescharacteristic
116(C2×D31)⋊C4C2311yescharacteristic
117(C2×D31)⋊C4C2313yes—
118(C2×D31)⋊C4C2313yes—
119(C2×D31)⋊C4C2313yes—
120(C2×D31)⋊C4C2313yes—
121(C2×D31)⋊C4C2316yes—
122(C2×D31)⋊C4C2313yes—
123C4⋊Dic31C2313yes—
124C4⋊Dic31C2311yescharacteristic
125Dic31⋊C4C2313yes—
126Dic31⋊C4C2313yes—
127Dic31⋊C4C2313yes—
128Dic31⋊C4C2311yescharacteristic
129C4×Dic31C2316yes—
130C4×Dic31C2313yes—
131C23×C31C2413yes—
132C23×C31C2411yescharacteristic
133C22×D31—26no—
134C22×D31—43no—
135C22×D31—43no—
136C22×D31C2413yes—
137C22×D31C2413yes—
138C22×D31C2411yescharacteristic
139Q8×C31—26no—
140Q8×C31C2×D414yes—
141D4×C31—26no—
142D4×C31—43no—
143C2×C4×C31—26no—
144C2×C4×C31—26no—
145C2×C4×C31C2413yes—
146C2×C4×C31C2411yescharacteristic
147C2×C4×C31C2413yes—
148C2×C4×C31C2411yescharacteristic
149C2×C4×C31C2413yes—
150C22⋊D31—22no—
151C22⋊D31—26no—
152C22⋊D31—212no—
153C22⋊D31—43no—
154C22⋊D31—46no—
155C22⋊D31—43no—
156C2×Dic31—26no—
157C2×Dic31C2411yescharacteristic
158C2×Dic31C2413yes—
159C2×Dic31C2411yescharacteristic
160C2×Dic31C2411yescharacteristic
161C2×Dic31C2413yes—
162D124—26no—
163D124—26no—
164D124—26no—
165D124—43no—
166D124—46no—
167C4×D31—22no—
168C4×D31—26no—
169C4×D31—26no—
170C4×D31—26no—
171C4×D31—212no—
172C4×D31C2×D4112yes—
173Dic62—26no—
174Dic62C2×D4112yes—
175#3487—311nomaximal, Sylow
176C22×C31—23no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31—23no—
180C22×C31—23no—
181C22×C31—21no—
182C22×C31C2511yescharacteristic
183C2×D31—21no—
184C2×D31—21no—
185C2×D31—23no—
186C2×D31—23no—
187C2×D31—23no—
188C2×D31—23no—
189C2×D31—23no—
190C2×D31—23no—
191C2×D31—412no—
192C2×D31C22×D412yes—
193C4×C31—23no—
194C4×C31—21no—
195C4×C31—21no—
196C4×C31—23no—
197C4×C31C22×D416yes—
198Dic31—21no—
199Dic31—21no—
200Dic31—21no—
201Dic31—23no—
202Dic31C22×D416yes—
203C2×2−1+4—311no—
204C22×#41—311no—
205#550—311no—
206#550—313no—
207#550—311no—
208#550—313no—
209#548—311no—
210#548—313no—
211#548—313no—
212#548D3111yescharacteristic, complemented
213#535—313no—
214#535—313no—
215#525—313no—
216#525—313no—
217#519—311no—
218C2×C31—43no—
219C2×C31—41no—
220C2×C31C23×D411yescharacteristic
221C2×C31#58511yescharacteristic
222C2×C31#58511yescharacteristic
223D31—22no—
224D31—43no—
225D31—43no—
2262−1+4—628no—
227C2×#41—313no—
228C2×#41—311no—
229C2×#41—311no—
230C2×#41—313no—
231C2×#41—313no—
232C2×#41—313no—
233C2×#41—313no—
234C2×#41—628no—
235C2×#41C2×D3111yescharacteristic
236C22×Q8—311no—
237C22×Q8—313no—
238C22×Q8C2×D3111yescharacteristic, complemented
239C22×D4—313no—
240C23×C4—313no—
241#127—316no—
242#127—313no—
243#127—313no—
244#124—313no—
245#124—316no—
246#124C2×D3113yescomplemented
247#123—313no—
248#123—311no—
249C22⋊Q8—313no—
250C22⋊Q8—313no—
251C22⋊Q8—313no—
252C22⋊Q8C2×D3113yescomplemented
253#121—313no—
254#121—313no—
255#121—313no—
256#121—313no—
257#121—313no—
258#121—313no—
259#121—313no—
260#121—316no—
261#121—316no—
262#121C2×D3113yescomplemented
263#120—313no—
264#120—316no—
265#120—313no—
266C4×Q8—311no—
267C4×Q8—313no—
268C4×Q8—313no—
269C4×Q8C2×D3111yescharacteristic, complemented
270C4×D4—313no—
271C4×D4—313no—
272C4×D4—316no—
273C4×D4—316no—
274C4×D4—313no—
275C4×D4C2×D3113yescomplemented
276#117—313no—
277#117—313no—
278C31#348711yescharacteristic, complemented, Sylow
279C24—313no—
280#41—24no—
281#41—6212no—
282#41—6212no—
283#41—6212no—
284#41—6212no—
285#41—6212no—
286#41—624no—
287#41—624no—
288C2×Q8—24no—
289C2×Q8—318no—
290C2×Q8—313no—
291C2×Q8—313no—
292C2×Q8—6212no—
293C2×Q8C22×D3113yes—
294C2×Q8C22×D3111yescharacteristic
295C2×D4—311no—
296C2×D4—316no—
297C2×D4—313no—
298C2×D4—313no—
299C2×D4—316no—
300C2×D4—313no—
301C2×D4—313no—
302C2×D4—313no—
303C2×D4—313no—
304C2×D4—316no—
305C2×D4—313no—
306C2×D4—6212no—
307C2×D4C22×D3113yes—
308C2×D4C22×D3113yes—
309C22×C4—313no—
310C22×C4—313no—
311C22×C4—316no—
312C22×C4—313no—
313C22×C4—313no—
314C22×C4—313no—
315C22×C4—311no—
316C22×C4—6212no—
317C22×C4C22×D3113yes—
318C22×C4C22×D3113yes—
319C4⋊C4—313no—
320C4⋊C4—313no—
321C4⋊C4—311no—
322C4⋊C4—313no—
323C4⋊C4—313no—
324C4⋊C4—311no—
325C4⋊C4C22×D3113yes—
326C4⋊C4C22×D3113yes—
327C22⋊C4—311no—
328C22⋊C4—313no—
329C22⋊C4—313no—
330C22⋊C4—311no—
331C22⋊C4—313no—
332C22⋊C4—313no—
333C22⋊C4—313no—
334C22⋊C4—316no—
335C22⋊C4—313no—
336C22⋊C4C22×D3116yes—
337C22⋊C4C22×D3111yescharacteristic
338C22⋊C4C22×D3113yes—
339C42—316no—
340C42—313no—
341C42C22×D3113yes—
342C23—311no—
343C23—313no—
344C23—313no—
345C23—626no—
346C23—1243no—
347C23—1243no—
348C23C23×D3113yes—
349C23C23×D3111yescharacteristic
350Q8—26no—
351Q8—3112no—
352Q8—626no—
353Q8D4×D3114yes—
354D4—26no—
355D4—43no—
356D4—626no—
357D4—6212no—
358D4—626no—
359D4—622no—
360D4—626no—
361D4—626no—
362D4—1246no—
363D4—1243no—
364D4—1243no—
365D4—1243no—
366D4—1246no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—313no—
370C2×C4—3112no—
371C2×C4—311no—
372C2×C4—313no—
373C2×C4—311no—
374C2×C4—311no—
375C2×C4—626no—
376C2×C4—6212no—
377C2×C4—622no—
378C2×C4—626no—
379C2×C4—626no—
380C2×C4—626no—
381C2×C4C23×D3113yes—
382C2×C4C23×D3111yescharacteristic
383C2×C4C23×D3113yes—
384C2×C4C23×D3111yescharacteristic
385C2×C4C23×D3113yes—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—312no—
393C22—621no—
394C22—621no—
395C22—623no—
396C22—623no—
397C22—623no—
398C22—623no—
399C22—623no—
400C22—623no—
401C22—12412no—
402C22C24×D3111yescharacteristic
403C4—21no—
404C4—23no—
405C4—23no—
406C4—21no—
407C4—316no—
408C4—621no—
409C4—621no—
410C4—621no—
411C4—623no—
412C4C2×D4×D3116yes—
413C2—43no—
414C2—41no—
415C2—622no—
416C2—1243no—
417C2—1243no—
418C2C22×D4×D3111yescharacteristic
419C2#1452188511yescharacteristic
420C2#1452188611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D31
commutator, socle, cosocleC22×C31C25
FittingC31×#548C2
hypercenter#548D31
hypocenterC31#3487
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. C2×#41×D31C2semidirect
3. #14521721C2semidirect
4. #14521720C2semidirect
5. #14521718C2semidirect
6. #14521717C2semidirect
7. #14521710C2semidirect
8. #14521709C2semidirect
9. #14521707C2semidirect
10. #14521617C2semidirect
11. #14521615C2semidirect
12. #14521562C2semidirect
13. #14521558C2semidirect
14. #14521525C2semidirect
15. C31×#548C2semidirect
17. C22×Q8×C31C22semidirect
22. C2×#92499C22semidirect
24. C2×Q8×D31C22semidirect
29. C2×#92495C22semidirect
31. C31×#124C22semidirect
32. C31×C22⋊Q8C22semidirect
33. C31×#121C22semidirect
34. C4×Q8×C31C22semidirect
35. C4×D4×C31C22semidirect
36. #11758524C22semidirect
37. #11758523C22semidirect
38. Q8×Dic31C22semidirect
39. #11758517C22semidirect
40. #11758517C22semidirect
41. #11758516C22semidirect
42. #11758515C22semidirect
44. #11758507C22semidirect
45. #11758507C22semidirect
46. C4×C22⋊D31C22semidirect
47. C4×C22⋊D31C22semidirect
48. #11758473C22semidirect
49. #11758472C22semidirect
50. #11758472C22semidirect
52. #11758470C22semidirect
54. #11758464C22semidirect
55. #11758462C22semidirect
56. #11758461C22semidirect
57. #11758461C22semidirect
58. #11758461C22semidirect
62. (C22⋊D31)⋊C4C22semidirect
63. #11758455C22semidirect
64. #11758451C22semidirect
65. C4×D124C22semidirect
67. C4×Dic62C22semidirect
88. Q8×D31D4semidirect
212. #548D31semidirect
238. C22×Q8C2×D31semidirect
246. #124C2×D31semidirect
252. C22⋊Q8C2×D31semidirect
262. #121C2×D31semidirect
269. C4×Q8C2×D31semidirect
275. C4×D4C2×D31semidirect
278. C31#3487semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
C2×#41×D312
#145217212
#145217202
#145217182
#145217172
#145217102
#145217092
#145217072
#145216172
#145216152
#145215622
#145215582
#145215252
C31×#5482
#348731