Finite Group Databasebeta

← #16857228

Subgroups — #16857228

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×#11758486C211yesmaximal, characteristic, complemented
2C2×SD16×D31C211yesmaximal, characteristic, complemented
3#14521712C211yesmaximal, characteristic, complemented
4#14521696C211yesmaximal, characteristic
5#14521440C211yesmaximal, characteristic, complemented
6#14521291C211yesmaximal, characteristic, complemented
7#14521270C211yesmaximal, characteristic, complemented
8#14521253C211yesmaximal, characteristic, complemented
9#14521209C211yesmaximal, characteristic, complemented
10#14521206C211yesmaximal, characteristic, complemented
11#14521201C211yesmaximal, characteristic, complemented
12#14521187C211yesmaximal, characteristic, complemented
13#14521148C211yesmaximal, characteristic, complemented
14#14521118C211yesmaximal, characteristic, complemented
15C31×#468C211yesmaximal, characteristic, complemented
16C2×Q8×D31C2211yescharacteristic
17C2×D4×D31C2211yescharacteristic
18C2×#92495C2211yescharacteristic
19C22×Dic62C2211yescharacteristic, complemented
20C2×SD16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic, complemented
22C31×C22⋊Q8C2211yescharacteristic, complemented
23C31×#121C2211yescharacteristic
24#11758521C2211yescharacteristic, complemented
25C2×#92482C2211yescharacteristic, complemented
26C2×#92481C2211yescharacteristic
27#11758515—21no—
28#11758514—21no—
29#11758513C2211yescharacteristic
30C2×#92480C2211yescharacteristic
31C2×#92480C2211yescharacteristic
32C4×C22⋊D31C2211yescharacteristic
33SD16×D31—24no—
34#11758486—24no—
35C2×Dic124C2211yescharacteristic, complemented
36C2×Dic124C2211yescharacteristic, complemented
37C2×#92470C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×C8×D31C2211yescharacteristic
40#11758474C2211yescharacteristic
41#11758474C2211yescharacteristic
42#11758473—21no—
43#11758470C2211yescharacteristic
44#11758465—21no—
45#11758464C2211yescharacteristic, complemented
46#11758460—21no—
47(C22⋊D31)⋊C4—21no—
48C22⋊C4×D31—21no—
49#11758455—21no—
50C31×#107C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#102C2211yescharacteristic
53C22⋊(C31⋊C8)C2211yescharacteristic, complemented
54(C2×D31)⋊C8C2211yescharacteristic
55#11758403C2211yescharacteristic, complemented
56Dic31⋊C8C2211yescharacteristic, complemented
57#11758398C2211yescharacteristic, complemented
58#11758398C2211yescharacteristic, complemented
59#11758398C2211yescharacteristic, complemented
60#11758397C2211yescharacteristic
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×D4×C31—21no—
64C2×D4×C31C2311yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31D411yescharacteristic
69C22×Dic31—21no—
70Q8×D31—22no—
71Q8×D31—41no—
72D4×D31—22no—
73D4×D31—41no—
74#92495—22no—
75#92495—41no—
76C2×D124C2311yescharacteristic
77C2×C4×D31—21no—
78C2×C4×D31—21no—
79C2×C4×D31—21no—
80C2×C4×D31C2311yescharacteristic
81C2×Dic62—22no—
82C2×Dic62—21no—
83C2×Dic62—21no—
84C2×Dic62—41no—
85C2×Dic62C2311yescharacteristic
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88SD16×C31—22no—
89M16×C31—22no—
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C4⋊C4×C31—21no—
93C4⋊C4×C31C2311yescharacteristic
94C4⋊C4×C31C2311yescharacteristic
95C22⋊C4×C31—21no—
96C22⋊C4×C31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31—21no—
99C22⋊Dic31D411yescharacteristic
100#92482—41no—
101#92481—22no—
102#92480—22no—
103#92480—41no—
104(C2×D31)⋊C4—21no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4D411yescharacteristic
108C4⋊Dic31—21no—
109C4⋊Dic31—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
115C4×Dic31C2311yescharacteristic
116C2×C31⋊C8C2311yescharacteristic
117C2×C31⋊C8C2311yescharacteristic
118Dic124—22no—
119Dic124—22no—
120#92470—22no—
121#92470D414yescomplemented
122C8×D31—22no—
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—42no—
127C22×D31—42no—
128C22×D31—81no—
129C22×D31C2×D411yescharacteristic
130Q8×C31—21no—
131Q8×C31—21no—
132D4×C31—21no—
133D4×C31—21no—
134D4×C31—81no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—22no—
143C22⋊D31—42no—
144C22⋊D31—42no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151C2×Dic31C2×D411yescharacteristic
152D124—21no—
153D124C2×D412yes—
154C4×D31—21no—
155C4×D31—21no—
156C4×D31—42no—
157C4×D31—42no—
158C4×D31—81no—
159Dic62—21no—
160Dic62—21no—
161Dic62—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62—42no—
165Dic62—42no—
166Dic62C2×D412yes—
167C8×C31—21no—
168C8×C31C2×D412yes—
169C31⋊C8—21no—
170C31⋊C8—21no—
171#3266—311nomaximal, Sylow
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31C22×D411yescharacteristic
179C2×D31—21no—
180C2×D31—21no—
181C2×D31—22no—
182C2×D31—41no—
183C2×D31—41no—
184C2×D31—84no—
185C2×D31#13712yescomplemented
186C4×C31—21no—
187C4×C31—41no—
188C4×C31—41no—
189C4×C31—41no—
190C4×C31C22×D411yescharacteristic
191C4×C31H(5,𝔽2)11yescharacteristic
192Dic31—22no—
193Dic31—21no—
194Dic31—21no—
195Dic31—21no—
196Dic31—41no—
197Dic31—41no—
198C2×#137—311no—
199C22×SD16—311no—
200D4×Q8—311no—
201#546—311no—
202#497—311no—
203#468D3111yescharacteristic, complemented
204#467—311no—
205#464—311no—
206#461—311no—
207#452—311no—
208#451—311no—
209#450—311no—
210#448—311no—
211#440—311no—
212#435—311no—
213C2×C31—41no—
214C2×C31—81no—
215C2×C31D4211yescharacteristic
216C2×C31C2×#13711yescharacteristic
217C2×C31#57711yescharacteristic
218D31—22no—
219D31—42no—
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—311no—
233C2×SD16—624no—
234C2×SD16C2×D3111yescharacteristic
235C2×M16C2×D3111yescharacteristic, complemented
236C22×C8—311no—
237C4⋊Q8—311no—
238C4⋊Q8—621no—
239#124—311no—
240#123—621no—
241C22⋊Q8—311no—
242C22⋊Q8—311no—
243C22⋊Q8—621no—
244C22⋊Q8—621no—
245C22⋊Q8C2×D3111yescharacteristic, complemented
246#121—621no—
247#121C2×D3111yescharacteristic
248#120—621no—
249C4×Q8—311no—
250C4×D4—311no—
251C4×D4—311no—
252C4×D4—621no—
253C2×C22⋊C4—621no—
254#107C2×D3111yescharacteristic, complemented
255C4⋊C8—311no—
256#103—311no—
257#103—311no—
258#103—311no—
259#103C2×D3111yescharacteristic, complemented
260#102—311no—
261#102C2×D3111yescharacteristic
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—311no—
272C2×Q8—621no—
273C2×Q8—622no—
274C2×Q8—621no—
275C2×Q8—622no—
276C2×Q8—1241no—
277C2×Q8—1241no—
278C2×Q8C22×D3111yescharacteristic
279C2×D4—21no—
280C2×D4—311no—
281C2×D4—311no—
282C2×D4—621no—
283C2×D4—622no—
284C2×D4—621no—
285C2×D4—1241no—
286C2×D4C22×D3111yescharacteristic
287C22×C4—311no—
288C22×C4—621no—
289C22×C4—621no—
290C22×C4—621no—
291C22×C4—621no—
292C22×C4C22×D3111yescharacteristic
293Q16—622no—
294Q16—622no—
295Q16—1241no—
296SD16—22no—
297SD16—314no—
298SD16—622no—
299SD16—622no—
300SD16—622no—
301SD16—1241no—
302M16—22no—
303C2×C8—311no—
304C2×C8—311no—
305C2×C8—622no—
306C2×C8C22×D3111yescharacteristic
307C2×C8C22×D3111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—311no—
310C4⋊C4—621no—
311C4⋊C4—621no—
312C4⋊C4—621no—
313C4⋊C4—621no—
314C4⋊C4—621no—
315C4⋊C4C22×D3111yescharacteristic
316C4⋊C4C22×D3111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—311no—
320C22⋊C4—311no—
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C42—311no—
327C42—621no—
328C23—21no—
329C23—311no—
330C23—621no—
331C23—1242no—
332C23—1242no—
333C23—2481no—
334C23D4×D3111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—312no—
338Q8—621no—
339Q8—621no—
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—1242no—
344Q8—1242no—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—312no—
349D4—622no—
350D4—621no—
351D4—1242no—
352D4—1242no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—311no—
359C2×C4—311no—
360C2×C4—311no—
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×C4—2481no—
370C2×C4D4×D3111yescharacteristic
371C2×C4C23×D3111yescharacteristic
372C8—21no—
373C8—621no—
374C8—621no—
375C8D4×D3112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—312no—
383C22—621no—
384C22—621no—
385C22—622no—
386C22—1241no—
387C22—1241no—
388C22—2484no—
389C22C2×D4×D3111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—621no—
395C4—621no—
396C4—622no—
397C4—621no—
398C4—1241no—
399C4—1241no—
400C4C2×D4×D3111yescharacteristic
401C4#1175856611yescharacteristic
402C2—41no—
403C2—81no—
404C2—622no—
405C2—1242no—
406C2#1452168311yescharacteristic
407C2#137×D3111yescharacteristic
408C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#468C2
socleC22×C31C22×D4
prime radical#14521696C2
hypercenter#468D31
hypocenterC31#3266
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
2. C2×SD16×D31C2semidirect
3. #14521712C2semidirect
5. #14521440C2semidirect
6. #14521291C2semidirect
7. #14521270C2semidirect
8. #14521253C2semidirect
9. #14521209C2semidirect
10. #14521206C2semidirect
11. #14521201C2semidirect
12. #14521187C2semidirect
13. #14521148C2semidirect
14. #14521118C2semidirect
15. C31×#468C2semidirect
19. C22×Dic62C22semidirect
21. C2×M16×C31C22semidirect
22. C31×C22⋊Q8C22semidirect
24. #11758521C22semidirect
25. C2×#92482C22semidirect
35. C2×Dic124C22semidirect
36. C2×Dic124C22semidirect
45. #11758464C22semidirect
50. C31×#107C22semidirect
51. C31×#103C22semidirect
53. C22⋊(C31⋊C8)C22semidirect
55. #11758403C22semidirect
56. Dic31⋊C8C22semidirect
57. #11758398C22semidirect
58. #11758398C22semidirect
59. #11758398C22semidirect
121. #92470D4semidirect
185. C2×D31#137semidirect
203. #468D31semidirect
235. C2×M16C2×D31semidirect
245. C22⋊Q8C2×D31semidirect
254. #107C2×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×#117584862
C2×SD16×D312
#145217122
#145216962
#145214402
#145212912
#145212702
#145212532
#145212092
#145212062
#145212012
#145211872
#145211482
#145211182
C31×#4682
#326631