Finite Group Databasebeta

← #16854540

Subgroups — #16854540

Order 3968.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758488C211yesmaximal, characteristic
2#14521528C211yesmaximal, characteristic, complemented
3#14521521C211yesmaximal, characteristic, complemented
4#14521158C211yesmaximal, characteristic, complemented
5#14521151C211yesmaximal, characteristic, complemented
6#14521146C211yesmaximal, characteristic, complemented
7#14521136C211yesmaximal, characteristic, complemented
8C4×#92480C211yesmaximal, characteristic, complemented
9D8×Dic31C211yesmaximal, characteristic, complemented
10C4×#92479C211yesmaximal, characteristic, complemented
11#14521088C211yesmaximal, characteristic, complemented
12#14521042C211yesmaximal, characteristic, complemented
13#102×D31C211yesmaximal, characteristic, complemented
14#14521006C211yesmaximal, characteristic, complemented
15C31×#442C211yesmaximal, characteristic, complemented
16C2×#92497C2211yescharacteristic
17C2×D4×D31C2211yescharacteristic
18C2×D8×C31C2211yescharacteristic
19C4×D4×C31C2211yescharacteristic
20C4×D4×C31C2211yescharacteristic, complemented
21D4×Dic31C2211yescharacteristic
22D4×Dic31C2211yescharacteristic, complemented
23C2×#92480C2211yescharacteristic
24C2×#92479C2211yescharacteristic
25C4×C22⋊D31—21no—
26C4×C22⋊D31—21no—
27C2×(C2×D31)⋊C4—21no—
28C2×Dic31⋊C4—21no—
29#11758488C418yescomplemented
30C2×#92470C2211yescharacteristic
31C2×#92469C2211yescharacteristic
32C2×C8×D31C2211yescharacteristic
33#11758470C2211yescharacteristic
34#11758469C2211yescharacteristic
35C4⋊C4×D31C2211yescharacteristic
36#11758464C2211yescharacteristic, complemented
37(C22⋊D31)⋊C4—21no—
38(C22⋊D31)⋊C4—21no—
39C22⋊C4×D31—21no—
40#11758454—21no—
41C4×D124C2211yescharacteristic
42#11758449C2211yescharacteristic
43C4×Dic62C2211yescharacteristic, complemented
44C31×#106C2211yescharacteristic, complemented
45C31×#102C2211yescharacteristic
46C31×#102C2211yescharacteristic, complemented
47C31×#97C2211yescharacteristic, complemented
48#11758420C2211yescharacteristic
49#11758420C2211yescharacteristic, complemented
50#11758409C2211yescharacteristic
51#11758406C2211yescharacteristic, complemented
52#11758404C2211yescharacteristic, complemented
53#11758403C2211yescharacteristic, complemented
54C8×Dic31C2211yescharacteristic, complemented
55#11758398C2211yescharacteristic, complemented
56#11758397C2211yescharacteristic
57#11758396C2211yescharacteristic, complemented
58#11758395C2211yescharacteristic, complemented
59C4×C31⋊C8C2211yescharacteristic, complemented
60C23×D31—21no—
61C2×D4×C31C2311yescharacteristic
62C2×D4×C31C2311yescharacteristic
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C22×Dic31—21no—
68C22×Dic31—21no—
69C22×Dic31—21no—
70#92497—22no—
71#92497C2×C414yescomplemented
72D4×D31—22no—
73D4×D31C2×C414yescomplemented
74C2×D124C2311yescharacteristic
75C2×C4×D31—21no—
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×C4×D31C2311yescharacteristic
79C2×Dic62C2311yescharacteristic
80D8×C31C2×C414yescomplemented
81C2×C8×C31C2311yescharacteristic
82C2×C8×C31C2311yescharacteristic
83C4⋊C4×C31C2311yescharacteristic
84C4⋊C4×C31C2311yescharacteristic
85C22⋊C4×C31—21no—
86C22⋊C4×C31—21no—
87C42×C31C2311yescharacteristic
88C22⋊Dic31—21no—
89C22⋊Dic31—21no—
90#92480C2×C414yescomplemented
91#92479C2×C414yescomplemented
92(C2×D31)⋊C4—21no—
93(C2×D31)⋊C4—21no—
94(C2×D31)⋊C4—21no—
95(C2×D31)⋊C4D411yescharacteristic
96(C2×D31)⋊C4D411yescharacteristic
97C4⋊Dic31C2311yescharacteristic
98C4⋊Dic31C2311yescharacteristic
99Dic31⋊C4—21no—
100Dic31⋊C4—21no—
101Dic31⋊C4—21no—
102Dic31⋊C4D411yescharacteristic
103Dic31⋊C4D411yescharacteristic
104C4×Dic31—21no—
105C4×Dic31—21no—
106C4×Dic31—21no—
107C4×Dic31C2311yescharacteristic
108C2×C31⋊C8C2311yescharacteristic
109C2×C31⋊C8C2311yescharacteristic
110#92470C2×C414yescomplemented
111#92469C2×C414yescomplemented
112C8×D31—22no—
113C23×C31—21no—
114C23×C31—21no—
115C22×D31—21no—
116C22×D31—24no—
117C22×D31—81no—
118C22×D31—81no—
119C22×D31C2×D411yescharacteristic
120D4×C31—21no—
121D4×C31—21no—
122D4×C31C22×C412yes—
123D4×C31C22×C412yes—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—81no—
127C2×C4×C31—81no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2×D411yescharacteristic
130C2×C4×C31C2411yescharacteristic
131C22⋊D31—24no—
132C22⋊D31—24no—
133C2×Dic31—21no—
134C2×Dic31—24no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—81no—
138C2×Dic31—81no—
139C2×Dic31C2×D411yescharacteristic
140C2×Dic31C2×D411yescharacteristic
141C2×Dic31C2×D411yescharacteristic
142D124—21no—
143D124C22×C412yes—
144C4×D31—81no—
145C4×D31—81no—
146C4×D31—81no—
147C4×D31C22×C412yes—
148C4×D31#4112yes—
149Dic62—21no—
150Dic62C22×C412yes—
151C8×C31—21no—
152C8×C31C22×C412yes—
153C31⋊C8—21no—
154C31⋊C8C22×C412yes—
155#2946—311nomaximal, Sylow
156C22×C31—22no—
157C22×C31—22no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31C22×D411yescharacteristic
163C2×D31—21no—
164C2×D31—21no—
165C2×D31—22no—
166C2×D31—41no—
167C2×D31—41no—
168C2×D31—84no—
169C2×D31C4×D412yes—
170C4×C31—21no—
171C4×C31—21no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31C23×C411yescharacteristic
175C4×C31C2×#4111yescharacteristic
176Dic31—22no—
177Dic31—21no—
178Dic31—21no—
179Dic31—41no—
180Dic31—41no—
181Dic31C4×D412yes—
182C2×AGL(1,ℤ/8)—311no—
183#520—311no—
184#518—311no—
185#442—311no—
186#442D3111yescharacteristic, complemented
187#440—311no—
188#440—311no—
189C4×SD16—311no—
190C4×SD16—311no—
191C4×D8—311no—
192C4×D8—311no—
193#428—311no—
194#416—311no—
195C2×#102—311no—
196#405—311no—
197C2×C31—42no—
198C2×C31—42no—
199C2×C31C2×C4×D411yescharacteristic
200C2×C31#57611yescharacteristic
201C2×C31#57711yescharacteristic
202D31—22no—
203D31—42no—
204C2×#41—311no—
205C22×D4—311no—
206AGL(1,ℤ/8)—318no—
207C2×SD16—311no—
208C2×SD16—311no—
209C2×D8—311no—
210C2×D8C2×D3111yescharacteristic
211C2×M16—311no—
212C22×C8—311no—
213C4×Q8—311no—
214C4×Q8—311no—
215C4×D4—311no—
216C4×D4—311no—
217C4×D4—311no—
218C4×D4—311no—
219C4×D4—621no—
220C4×D4—621no—
221C4×D4—621no—
222C4×D4—621no—
223C4×D4C2×D3111yescharacteristic
224C4×D4C2×D3111yescharacteristic, complemented
225#117—311no—
226#117—311no—
227#117—621no—
228C2×C4⋊C4—311no—
229C2×C4⋊C4—621no—
230C2×C22⋊C4—621no—
231C2×C22⋊C4—621no—
232#107—311no—
233#107—311no—
234#106—311no—
235#106C2×D3111yescharacteristic, complemented
236#103—311no—
237#103—311no—
238#102—311no—
239#102—311no—
240#102—311no—
241#102—311no—
242#102C2×D3111yescharacteristic
243#102C2×D3111yescharacteristic, complemented
244#97—311no—
245#97C2×D3111yescharacteristic, complemented
246C4×C8—311no—
247C4×C8—311no—
248C31#294611yescharacteristic, complemented, Sylow
249C24—621no—
250#41—314no—
251#41—622no—
252C2×Q8—311no—
253C2×D4—314no—
254C2×D4—311no—
255C2×D4—622no—
256C2×D4—621no—
257C2×D4—621no—
258C2×D4C22×D3111yescharacteristic
259C2×D4C22×D3111yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—311no—
263C22×C4—621no—
264C22×C4—621no—
265C22×C4—621no—
266C22×C4—621no—
267C22×C4—621no—
268C22×C4—621no—
269SD16—314no—
270SD16—314no—
271D8—314no—
272D8C4×D3114yescomplemented
273M16—314no—
274C2×C8—311no—
275C2×C8—311no—
276C2×C8—622no—
277C2×C8C22×D3111yescharacteristic
278C2×C8C22×D3111yescharacteristic
279C4⋊C4—311no—
280C4⋊C4—311no—
281C4⋊C4—311no—
282C4⋊C4—311no—
283C4⋊C4—621no—
284C4⋊C4—621no—
285C4⋊C4—621no—
286C4⋊C4C22×D3111yescharacteristic
287C4⋊C4C22×D3111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—311no—
291C22⋊C4—311no—
292C22⋊C4—621no—
293C22⋊C4—621no—
294C22⋊C4—621no—
295C22⋊C4—621no—
296C22⋊C4—621no—
297C42—311no—
298C42—621no—
299C42—621no—
300C42—621no—
301C42C22×D3111yescharacteristic
302C23—21no—
303C23—21no—
304C23—311no—
305C23—624no—
306C23—621no—
307C23—2481no—
308C23—2481no—
309Q8—312no—
310Q8—621no—
311D4—21no—
312D4—21no—
313D4—312no—
314D4—624no—
315D4—624no—
316D4—621no—
317D4C2×C4×D3112yes—
318D4C2×C4×D3112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—311no—
324C2×C4—312no—
325C2×C4—312no—
326C2×C4—311no—
327C2×C4—311no—
328C2×C4—624no—
329C2×C4—621no—
330C2×C4—621no—
331C2×C4—621no—
332C2×C4—2481no—
333C2×C4—2481no—
334C2×C4—2481no—
335C2×C4—2481no—
336C2×C4—2481no—
337C2×C4D4×D3111yescharacteristic
338C2×C4D4×D3111yescharacteristic
339C2×C4C23×D3111yescharacteristic
340C8—21no—
341C8—312no—
342C8—621no—
343C8C2×C4×D3112yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—312no—
351C22—621no—
352C22—621no—
353C22—622no—
354C22—1241no—
355C22—1241no—
356C22—2484no—
357C22C2×D4×D3111yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—312no—
363C4—621no—
364C4—622no—
365C4—621no—
366C4—1241no—
367C4—1241no—
368C4C22×C4×D3111yescharacteristic
369C4#41×D3111yescharacteristic
370C2—42no—
371C2—42no—
372C2—622no—
373C2—1242no—
374C2C4×D4×D3111yescharacteristic
375C2#1452185311yescharacteristic
376C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutatorC4×C31C23×C4
FrattiniC2×C4C23×D31
FittingC31×#442C2
socleC22×C31C22×D4
prime radicalC2×#11758488C2
cosocleC2×C4×C31C24
hypercenter#442D31
hypocenterC31#2946
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #14521528C2semidirect
3. #14521521C2semidirect
4. #14521158C2semidirect
5. #14521151C2semidirect
6. #14521146C2semidirect
7. #14521136C2semidirect
8. C4×#92480C2semidirect
9. D8×Dic31C2semidirect
10. C4×#92479C2semidirect
11. #14521088C2semidirect
12. #14521042C2semidirect
13. #102×D31C2semidirect
14. #14521006C2semidirect
15. C31×#442C2semidirect
20. C4×D4×C31C22semidirect
22. D4×Dic31C22semidirect
29. #11758488C4semidirect
36. #11758464C22semidirect
43. C4×Dic62C22semidirect
44. C31×#106C22semidirect
46. C31×#102C22semidirect
47. C31×#97C22semidirect
49. #11758420C22semidirect
51. #11758406C22semidirect
52. #11758404C22semidirect
53. #11758403C22semidirect
54. C8×Dic31C22semidirect
55. #11758398C22semidirect
57. #11758396C22semidirect
58. #11758395C22semidirect
59. C4×C31⋊C8C22semidirect
71. #92497C2×C4semidirect
73. D4×D31C2×C4semidirect
80. D8×C31C2×C4semidirect
90. #92480C2×C4semidirect
91. #92479C2×C4semidirect
110. #92470C2×C4semidirect
111. #92469C2×C4semidirect
186. #442D31semidirect
224. C4×D4C2×D31semidirect
235. #106C2×D31semidirect
243. #102C2×D31semidirect
245. #97C2×D31semidirect
248. C31#2946semidirect
272. D8C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584882
#145215282
#145215212
#145211582
#145211512
#145211462
#145211362
C4×#924802
D8×Dic312
C4×#924792
#145210882
#145210422
#102×D312
#145210062
C31×#4422
#294631