Finite Group Databasebeta

← #16857235

Subgroups — #16857235

Order 3968.

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×#11758491C211yesmaximal, characteristic, complemented
2C2×#11758496C211yesmaximal, characteristic, complemented
3#14521702C211yesmaximal, characteristic, complemented
4#14521692C211yesmaximal, characteristic, complemented
5#14521304C211yesmaximal, characteristic, complemented
6#14521275C211yesmaximal, characteristic, complemented
7#14521261C211yesmaximal, characteristic, complemented
8#14521251C211yesmaximal, characteristic, complemented
9#14521210C211yesmaximal, characteristic, complemented
10#14521210C211yesmaximal, characteristic, complemented
11#14521198C211yesmaximal, characteristic, complemented
12#14521193C211yesmaximal, characteristic, complemented
13#14521162C211yesmaximal, characteristic, complemented
14#14521122C211yesmaximal, characteristic, complemented
15C31×#497C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92499C2211yescharacteristic
18C2×#92499C2211yescharacteristic, complemented
19C2×D4×D31C2211yescharacteristic
20C2×Q16×C31C2211yescharacteristic, complemented
21C2×Q16×C31C2211yescharacteristic, complemented
22C2×SD16×C31C2211yescharacteristic
23C2×SD16×C31C2211yescharacteristic
24C31×C4⋊Q8C2211yescharacteristic, complemented
25C31×#124C2211yescharacteristic
26#11758523C2211yescharacteristic
27#11758523C2211yescharacteristic
28#11758523C2211yescharacteristic, complemented
29C2×#92481C2211yescharacteristic
30C2×#92481C2211yescharacteristic
31C2×#92481C2211yescharacteristic, complemented
32#11758515C2211yescharacteristic
33C2×#92479C2211yescharacteristic
34#11758496—24no—
35#11758491—24no—
36C2×D248C2211yescharacteristic
37C2×#92469C2211yescharacteristic
38C2×C8×D31C2211yescharacteristic, complemented
39#11758472—21no—
40#11758471—21no—
41#11758471—21no—
42C4⋊C4×D31—21no—
43#11758461—21no—
44#11758460—21no—
45#11758459—21no—
46C22⋊C4×D31—21no—
47C4×D124C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×D124C2211yescharacteristic, complemented
50C31×#97C2211yescharacteristic, complemented
51#11758423C2211yescharacteristic, complemented
52#11758423C2211yescharacteristic, complemented
53#11758423C2211yescharacteristic, complemented
54#11758420C2211yescharacteristic
55#11758409C2211yescharacteristic
56#11758409C2211yescharacteristic, complemented
57(C2×D31)⋊C8C2211yescharacteristic
58(C2×D31)⋊C8C2211yescharacteristic, complemented
59#11758405C2211yescharacteristic, complemented
60C4⋊(C31⋊C8)C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31C2311yescharacteristic
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68#92499—22no—
69#92499—22no—
70#92499—22no—
71#92499—41no—
72#92499—41no—
73#92499—41no—
74D4×D31—22no—
75D4×D31—41no—
76C2×D124—21no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124D411yescharacteristic
80C2×D124C2311yescharacteristic
81C2×D124C2311yescharacteristic
82C2×C4×D31—21no—
83C2×C4×D31—21no—
84C2×C4×D31—21no—
85C2×C4×D31—21no—
86C2×C4×D31—21no—
87C2×C4×D31C2311yescharacteristic
88C2×C4×D31C2311yescharacteristic
89Q16×C31—22no—
90Q16×C31—41no—
91SD16×C31—22no—
92SD16×C31—41no—
93C2×C8×C31C2311yescharacteristic
94C2×C8×C31C2311yescharacteristic
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C22⋊C4×C31—21no—
98C22⋊C4×C31—21no—
99C42×C31C2311yescharacteristic
100C22⋊Dic31—21no—
101#92481—22no—
102#92481—22no—
103#92481—22no—
104#92479—22no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4—21no—
110(C2×D31)⋊C4D411yescharacteristic
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31D411yescharacteristic
113C4⋊Dic31C2311yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117Dic31⋊C4—21no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120D248D414yescomplemented
121#92469—22no—
122C8×D31—22no—
123C23×C31—21no—
124C22×D31—21no—
125C22×D31—21no—
126C22×D31—42no—
127C22×D31—42no—
128C22×D31—81no—
129C22×D31C2×D411yescharacteristic
130C22×D31C2×D411yescharacteristic
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135Q8×C31—21no—
136Q8×C31—21no—
137D4×C31—21no—
138D4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31C2×D411yescharacteristic
143C2×C4×C31C2×D411yescharacteristic
144C2×C4×C31C2411yescharacteristic
145C22⋊D31—42no—
146C22⋊D31—81no—
147C2×Dic31—21no—
148C2×Dic31—21no—
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151D124—21no—
152D124—21no—
153D124—41no—
154D124—42no—
155D124—42no—
156D124—42no—
157D124C2×D412yes—
158D124C2×D412yes—
159C4×D31—21no—
160C4×D31—21no—
161C4×D31—21no—
162C4×D31—21no—
163C4×D31—42no—
164C4×D31—42no—
165C4×D31—42no—
166C4×D31—42no—
167C4×D31—81no—
168C8×C31—21no—
169C8×C31C2×D412yes—
170C31⋊C8—21no—
171C31⋊C8—21no—
172#3267—311nomaximal, Sylow
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31C22×D411yescharacteristic
177C2×D31—21no—
178C2×D31—21no—
179C2×D31—22no—
180C2×D31—22no—
181C2×D31—21no—
182C2×D31—21no—
183C2×D31—21no—
184C2×D31—21no—
185C2×D31—41no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C2×D31—84no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—41no—
193C4×C31—41no—
194C4×C31—41no—
195C4×C31C22×D411yescharacteristic
196C4×C31H(5,𝔽2)11yescharacteristic
197Dic31—21no—
198Dic31—21no—
199Dic31—41no—
200Dic31—41no—
201C2×AGL(1,ℤ/8)—311no—
202C2×#135—311no—
203#547—311no—
204#546—311no—
205#497D3111yescharacteristic, complemented
206#470—311no—
207#465—311no—
208#463—311no—
209#461—311no—
210#452—311no—
211#452—311no—
212#450—311no—
213#449—311no—
214#442—311no—
215#435—311no—
216C2×C31—81no—
217C2×C31D4211yescharacteristic
218C2×C31#57511yescharacteristic
219C2×C31#57711yescharacteristic
220D31—41no—
221D31—41no—
222D31—42no—
223D31—42no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41—311no—
227C22×D4—311no—
228AGL(1,ℤ/8)—624no—
229#135—624no—
230C2×Q16C2×D3111yescharacteristic, complemented
231C2×Q16C2×D3111yescharacteristic, complemented
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16C2×D3111yescharacteristic
236C2×SD16C2×D3111yescharacteristic
237C2×D8—311no—
238C2×D8—311no—
239C2×M16—311no—
240C22×C8—311no—
241C4⋊Q8C2×D3111yescharacteristic, complemented
242#124C2×D3111yescharacteristic
243#123—621no—
244#123—621no—
245#123—621no—
246C22⋊Q8—311no—
247C22⋊Q8—311no—
248C22⋊Q8—311no—
249#121—311no—
250#121—621no—
251#121—621no—
252#120—621no—
253C4×D4—311no—
254C4×D4—311no—
255C4×D4—311no—
256C2×C4⋊C4—621no—
257C2×C22⋊C4—621no—
258#106—311no—
259C4⋊C8—311no—
260#103—311no—
261#103—311no—
262#103—311no—
263#102—311no—
264#102—311no—
265#102—311no—
266C22⋊C8—311no—
267C22⋊C8—311no—
268#97C2×D3111yescharacteristic, complemented
269C31#326711yescharacteristic, complemented, Sylow
270C24—621no—
271#41—622no—
272#41—622no—
273#41—622no—
274#41—1241no—
275#41—1241no—
276#41—1241no—
277C2×Q8C22×D3111yescharacteristic
278C2×Q8C22×D3111yescharacteristic
279C2×Q8C22×D3111yescharacteristic
280C2×D4—311no—
281C2×D4—311no—
282C2×D4—311no—
283C2×D4—621no—
284C2×D4—621no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—622no—
288C2×D4—621no—
289C2×D4—1241no—
290C2×D4C22×D3111yescharacteristic
291C22×C4—311no—
292C22×C4—311no—
293C22×C4—621no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4—621no—
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—41no—
302SD16—622no—
303SD16—622no—
304SD16—622no—
305D8—314no—
306D8—622no—
307M16—622no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—622no—
311C2×C8C22×D3111yescharacteristic
312C2×C8C22×D3111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—311no—
316C4⋊C4—311no—
317C4⋊C4—311no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4—621no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—311no—
324C22⋊C4—311no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C22⋊C4—621no—
330C22⋊C4—621no—
331C42C22×D3111yescharacteristic
332C23—21no—
333C23—311no—
334C23—311no—
335C23—621no—
336C23—621no—
337C23—1242no—
338C23—1242no—
339C23—2481no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—312no—
349D4—312no—
350D4—621no—
351D4—621no—
352D4—1241no—
353D4—1242no—
354D4—1242no—
355D4—1242no—
356D4—1242no—
357D4—2481no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—311no—
362C2×C4—311no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—1242no—
370C2×C4—1242no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—2481no—
374C2×C4D4×D3111yescharacteristic
375C2×C4D4×D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—621no—
379C8—621no—
380C8D4×D3112yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—621no—
385C22—621no—
386C22—621no—
387C22—621no—
388C22—622no—
389C22—622no—
390C22—621no—
391C22—621no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C2×D4×D3111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—621no—
404C4—621no—
405C4—1241no—
406C4—1241no—
407C4C2×D4×D3111yescharacteristic
408C4#1175856111yescharacteristic
409C2—81no—
410C2—1241no—
411C2—1241no—
412C2—1242no—
413C2—1242no—
414C2#1452168411yescharacteristic
415C2#1452184811yescharacteristic
416C2#1452185811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#497C2
socleC22×C31C22×D4
hypercenter#497D31
hypocenterC31#3267
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C2×#11758496C2semidirect
3. #14521702C2semidirect
4. #14521692C2semidirect
5. #14521304C2semidirect
6. #14521275C2semidirect
7. #14521261C2semidirect
8. #14521251C2semidirect
9. #14521210C2semidirect
10. #14521210C2semidirect
11. #14521198C2semidirect
12. #14521193C2semidirect
13. #14521162C2semidirect
14. #14521122C2semidirect
15. C31×#497C2semidirect
18. C2×#92499C22semidirect
20. C2×Q16×C31C22semidirect
21. C2×Q16×C31C22semidirect
24. C31×C4⋊Q8C22semidirect
28. #11758523C22semidirect
31. C2×#92481C22semidirect
38. C2×C8×D31C22semidirect
49. C4×D124C22semidirect
50. C31×#97C22semidirect
51. #11758423C22semidirect
52. #11758423C22semidirect
53. #11758423C22semidirect
56. #11758409C22semidirect
58. (C2×D31)⋊C8C22semidirect
59. #11758405C22semidirect
60. C4⋊(C31⋊C8)C22semidirect
120. D248D4semidirect
205. #497D31semidirect
230. C2×Q16C2×D31semidirect
231. C2×Q16C2×D31semidirect
241. C4⋊Q8C2×D31semidirect
268. #97C2×D31semidirect
269. C31#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#117584912
C2×#117584962
#145217022
#145216922
#145213042
#145212752
#145212612
#145212512
#145212102
#145211982
#145211932
#145211622
#145211222
C31×#4972
#326731