Finite Group Databasebeta

← #16857227

Subgroups — #16857227

Order 3968.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758492C211yesmaximal, characteristic, complemented
2C2×SD16×D31C211yesmaximal, characteristic, complemented
3#14521714C211yesmaximal, characteristic, complemented
4#14521694C211yesmaximal, characteristic
5#14521441C211yesmaximal, characteristic, complemented
6#14521298C211yesmaximal, characteristic, complemented
7#14521289C211yesmaximal, characteristic, complemented
8#14521272C211yesmaximal, characteristic, complemented
9#14521208C211yesmaximal, characteristic, complemented
10#14521206C211yesmaximal, characteristic, complemented
11#14521198C211yesmaximal, characteristic, complemented
12#14521189C211yesmaximal, characteristic, complemented
13#14521149C211yesmaximal, characteristic, complemented
14#14521119C211yesmaximal, characteristic, complemented
15C31×#461C211yesmaximal, characteristic, complemented
16C2×Q8×D31C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic, complemented
18C2×#92497C2211yescharacteristic
19C2×D4×D31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic
22C31×C4⋊Q8C2211yescharacteristic, complemented
23C4×D4×C31C2211yescharacteristic
24#11758523C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26C2×#92482C2211yescharacteristic, complemented
27C2×#92482C2211yescharacteristic, complemented
28C2×#92481C2211yescharacteristic
29C2×#92481C2211yescharacteristic
30C2×#92480C2211yescharacteristic
31#11758507—21no—
32C2×(C2×D31)⋊C4—21no—
33#11758492—24no—
34SD16×D31—24no—
35C2×Dic124C2211yescharacteristic, complemented
36C2×#92470C2211yescharacteristic
37C2×#92469C2211yescharacteristic, complemented
38C2×C8×D31C2211yescharacteristic
39#11758474C2211yescharacteristic
40#11758473—21no—
41#11758473—21no—
42#11758472C2211yescharacteristic
43#11758470—21no—
44#11758465—21no—
45#11758463—21no—
46#11758459—21no—
47#11758452C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×Dic62C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#103C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic
54#11758409C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758398C2211yescharacteristic, complemented
59#11758395C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31C2311yescharacteristic
65C22×C4×C31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C22×Dic31—21no—
69Q8×D31—22no—
70Q8×D31—22no—
71Q8×D31—41no—
72Q8×D31—41no—
73#92497—22no—
74#92497—41no—
75D4×D31—22no—
76D4×D31—41no—
77C2×D124—21no—
78C2×D124D411yescharacteristic
79C2×D124C2311yescharacteristic
80C2×C4×D31—21no—
81C2×C4×D31—21no—
82C2×C4×D31C2311yescharacteristic
83C2×C4×D31C2311yescharacteristic
84C2×Dic62—21no—
85C2×Dic62—21no—
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88SD16×C31—22no—
89SD16×C31—22no—
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31C2311yescharacteristic
95C22⋊C4×C31—21no—
96C42×C31C2311yescharacteristic
97#92482—22no—
98#92482—41no—
99#92481—22no—
100#92481—41no—
101#92480D414yescomplemented
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—21no—
104(C2×D31)⋊C4—21no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4D411yescharacteristic
109(C2×D31)⋊C4D411yescharacteristic
110C4⋊Dic31—21no—
111C4⋊Dic31D411yescharacteristic
112C4⋊Dic31C2311yescharacteristic
113Dic31⋊C4—21no—
114Dic31⋊C4—21no—
115Dic31⋊C4—21no—
116C4×Dic31—21no—
117C2×C31⋊C8C2311yescharacteristic
118C2×C31⋊C8C2311yescharacteristic
119Dic124—22no—
120#92470—22no—
121#92469—22no—
122C8×D31—22no—
123C23×C31—21no—
124C22×D31—21no—
125C22×D31—42no—
126C22×D31—42no—
127C22×D31—81no—
128C22×D31C2×D411yescharacteristic
129C22×D31C2×D411yescharacteristic
130Q8×C31—21no—
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134D4×C31—21no—
135D4×C31C2×D412yes—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—81no—
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2×D411yescharacteristic
142C2×C4×C31C2411yescharacteristic
143C22⋊D31—42no—
144C22⋊D31—42no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151D124—22no—
152D124—21no—
153D124—21no—
154D124—81no—
155C4×D31—21no—
156C4×D31—21no—
157C4×D31—21no—
158C4×D31—21no—
159C4×D31—42no—
160C4×D31—42no—
161Dic62—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62—42no—
165Dic62—42no—
166Dic62C2×D412yes—
167C8×C31—21no—
168C8×C31—21no—
169C31⋊C8—21no—
170C31⋊C8C2×D412yes—
171#3266—311nomaximal, Sylow
172C22×C31—22no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31C22×D411yescharacteristic
176C2×D31—21no—
177C2×D31—21no—
178C2×D31—21no—
179C2×D31—21no—
180C2×D31—21no—
181C2×D31—41no—
182C2×D31—41no—
183C2×D31—41no—
184C2×D31—84no—
185C2×D31#13712yescomplemented
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31—41no—
191C4×C31C22×D411yescharacteristic
192C4×C31H(5,𝔽2)11yescharacteristic
193Dic31—22no—
194Dic31—21no—
195Dic31—21no—
196Dic31—41no—
197Dic31—41no—
198C2×#137—311no—
199C22×SD16—311no—
200D4×Q8—311no—
201#546—311no—
202#497—311no—
203#468—311no—
204#467—311no—
205#464—311no—
206#461D3111yescharacteristic, complemented
207#452—311no—
208#451—311no—
209#450—311no—
210#448—311no—
211#440—311no—
212#435—311no—
213C2×C31—42no—
214C2×C31D4211yescharacteristic
215C2×C31C2×#13711yescharacteristic
216C2×C31#57711yescharacteristic
217D31—22no—
218D31—41no—
219D31—81no—
220C2×#41—311no—
221C22×Q8—311no—
222C22×Q8—311no—
223C22×D4—311no—
224#137—624no—
225C2×Q16—311no—
226C2×Q16—311no—
227C2×Q16—311no—
228C2×SD16—311no—
229C2×SD16—311no—
230C2×SD16—311no—
231C2×SD16—311no—
232C2×SD16—624no—
233C2×SD16C2×D3111yescharacteristic
234C2×SD16C2×D3111yescharacteristic
235C2×M16—311no—
236C22×C8—311no—
237C4⋊Q8—621no—
238C4⋊Q8C2×D3111yescharacteristic, complemented
239#124—311no—
240#123—621no—
241C22⋊Q8—311no—
242C22⋊Q8—311no—
243C22⋊Q8—311no—
244C22⋊Q8—621no—
245C22⋊Q8—621no—
246#121—311no—
247#121—621no—
248#120—621no—
249C4×Q8—311no—
250C4×D4—311no—
251C4×D4—621no—
252C4×D4C2×D3111yescharacteristic
253C2×C22⋊C4—621no—
254#107—311no—
255C4⋊C8C2×D3111yescharacteristic, complemented
256#103—311no—
257#103—311no—
258#103C2×D3111yescharacteristic, complemented
259#103C2×D3111yescharacteristic, complemented
260#102—311no—
261#102—311no—
262C22⋊C8—311no—
263C22⋊C8—311no—
264#97—311no—
265C31#326611yescharacteristic, complemented, Sylow
266C24—621no—
267#41—622no—
268#41—1241no—
269C2×Q8—311no—
270C2×Q8—311no—
271C2×Q8—621no—
272C2×Q8—621no—
273C2×Q8—622no—
274C2×Q8—622no—
275C2×Q8—1241no—
276C2×Q8—1241no—
277C2×Q8C22×D3111yescharacteristic
278C2×Q8C22×D3111yescharacteristic
279C2×D4—311no—
280C2×D4—311no—
281C2×D4—621no—
282C2×D4—621no—
283C2×D4—622no—
284C2×D4—621no—
285C2×D4—1241no—
286C2×D4C22×D3111yescharacteristic
287C22×C4—21no—
288C22×C4—311no—
289C22×C4—311no—
290C22×C4—621no—
291C22×C4—621no—
292C22×C4—621no—
293Q16—622no—
294Q16—622no—
295Q16—1241no—
296SD16—22no—
297SD16—22no—
298SD16—314no—
299SD16—622no—
300SD16—622no—
301SD16—1241no—
302M16—622no—
303C2×C8—311no—
304C2×C8—311no—
305C2×C8—622no—
306C2×C8C22×D3111yescharacteristic
307C2×C8C22×D3111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—311no—
311C4⋊C4—311no—
312C4⋊C4—621no—
313C4⋊C4—621no—
314C4⋊C4—621no—
315C4⋊C4—621no—
316C4⋊C4C22×D3111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—311no—
319C22⋊C4—311no—
320C22⋊C4—621no—
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C42—621no—
327C42C22×D3111yescharacteristic
328C23—21no—
329C23—311no—
330C23—311no—
331C23—621no—
332C23—1242no—
333C23—1242no—
334C23—2481no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—312no—
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—1242no—
344Q8—1242no—
345D4—21no—
346D4—622no—
347D4—621no—
348D4—621no—
349D4—1242no—
350D4—1242no—
351D4—2481no—
352D4D4×D3112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—311no—
358C2×C4—311no—
359C2×C4—621no—
360C2×C4—621no—
361C2×C4—621no—
362C2×C4—621no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—621no—
366C2×C4—1242no—
367C2×C4—1242no—
368C2×C4—1242no—
369C2×C4D4×D3111yescharacteristic
370C2×C4D4×D3111yescharacteristic
371C2×C4C23×D3111yescharacteristic
372C8—21no—
373C8—21no—
374C8—312no—
375C8—621no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—312no—
380C22—621no—
381C22—621no—
382C22—621no—
383C22—621no—
384C22—621no—
385C22—1241no—
386C22—1241no—
387C22—1241no—
388C22—2484no—
389C22C2×D4×D3111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—621no—
396C4—621no—
397C4—622no—
398C4—1241no—
399C4—1241no—
400C4C2×D4×D3111yescharacteristic
401C4#1175856611yescharacteristic
402C2—42no—
403C2—622no—
404C2—1241no—
405C2—2481no—
406C2#1452168311yescharacteristic
407C2#137×D3111yescharacteristic
408C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#461C2
socleC22×C31C22×D4
prime radical#14521694C2
hypercenter#461D31
hypocenterC31#3266
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#11758492C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521714C2semidirect
5. #14521441C2semidirect
6. #14521298C2semidirect
7. #14521289C2semidirect
8. #14521272C2semidirect
9. #14521208C2semidirect
10. #14521206C2semidirect
11. #14521198C2semidirect
12. #14521189C2semidirect
13. #14521149C2semidirect
14. #14521119C2semidirect
15. C31×#461C2semidirect
17. C2×Q8×D31C22semidirect
22. C31×C4⋊Q8C22semidirect
25. #11758523C22semidirect
26. C2×#92482C22semidirect
27. C2×#92482C22semidirect
35. C2×Dic124C22semidirect
37. C2×#92469C22semidirect
49. C4×Dic62C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×#103C22semidirect
52. C31×#103C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758404C22semidirect
58. #11758398C22semidirect
59. #11758395C22semidirect
60. #11758391C22semidirect
101. #92480D4semidirect
185. C2×D31#137semidirect
206. #461D31semidirect
238. C4⋊Q8C2×D31semidirect
255. C4⋊C8C2×D31semidirect
258. #103C2×D31semidirect
259. #103C2×D31semidirect
265. C31#3266semidirect

Maximal subgroups

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

subgroupindex
C2×#117584922
C2×SD16×D312
#145217142
#145216942
#145214412
#145212982
#145212892
#145212722
#145212082
#145212062
#145211982
#145211892
#145211492
#145211192
C31×#4612
#326631