Finite Group Databasebeta

← #16857226

Subgroups — #16857226

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
2C22×#92470C211yesmaximal, characteristic, complemented
3D4×Dic62C211yesmaximal, characteristic, complemented
4#14521695C211yesmaximal, characteristic
5#14521443C211yesmaximal, characteristic, complemented
6#14521295C211yesmaximal, characteristic, complemented
7#14521290C211yesmaximal, characteristic, complemented
8#14521272C211yesmaximal, characteristic, complemented
9#14521255C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521207C211yesmaximal, characteristic, complemented
12#14521202C211yesmaximal, characteristic, complemented
13#14521190C211yesmaximal, characteristic, complemented
14#14521151C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C22×Dic62C2211yescharacteristic
19C22×Dic62C2211yescharacteristic, complemented
20C2×M16×C31C2211yescharacteristic, complemented
21C22×C8×C31C2211yescharacteristic
22C4×D4×C31C2211yescharacteristic
23#11758514—21no—
24D4×Dic31—21no—
25#11758507C2211yescharacteristic
26#11758501C2211yescharacteristic
27#11758501C2211yescharacteristic
28#11758501C2211yescharacteristic, complemented
29#11758486—24no—
30C2×Dic124C2211yescharacteristic, complemented
31C2×Dic124C2211yescharacteristic, complemented
32C2×Dic124C2211yescharacteristic, complemented
33C2×#92470—24no—
34C2×#92470C2211yescharacteristic
35C2×#92470C2211yescharacteristic
36C2×#92470C2211yescharacteristic
37C2×#92470C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×#92470C2211yescharacteristic
40#11758471—21no—
41#11758465—21no—
42#11758461—21no—
43C22⋊C4×D31—21no—
44#11758455—21no—
45#11758455—21no—
46#11758452C2211yescharacteristic
47C4×D124C2211yescharacteristic
48#11758446C2211yescharacteristic, complemented
49C4×Dic62C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×C22⋊C8C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic, complemented
53C31×#97C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic
55#11758409C2211yescharacteristic
56#11758406C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758404C2211yescharacteristic, complemented
59#11758404C2211yescharacteristic, complemented
60#11758404C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×D4×C31D411yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C22×Dic31—21no—
68C22×Dic31—21no—
69#92495—22no—
70#92495—41no—
71C2×D124—22no—
72C2×D124—21no—
73C2×D124—41no—
74C2×D124C2311yescharacteristic
75C2×D124C2311yescharacteristic
76C2×C4×D31—21no—
77C2×C4×D31—21no—
78C2×Dic62—22no—
79C2×Dic62—22no—
80C2×Dic62—21no—
81C2×Dic62—21no—
82C2×Dic62—41no—
83C2×Dic62—41no—
84C2×Dic62C2311yescharacteristic
85C2×Dic62C2311yescharacteristic
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88M16×C31—22no—
89C2×C8×C31—22no—
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C2×C8×C31C2311yescharacteristic
93C2×C8×C31C2311yescharacteristic
94C4⋊C4×C31D411yescharacteristic
95C22⋊C4×C31D411yescharacteristic
96C22⋊C4×C31D411yescharacteristic
97C42×C31C2311yescharacteristic
98C22⋊Dic31—21no—
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101(C2×D31)⋊C4—21no—
102(C2×D31)⋊C4—21no—
103(C2×D31)⋊C4—21no—
104(C2×D31)⋊C4—21no—
105C4⋊Dic31—21no—
106C4⋊Dic31—21no—
107C4⋊Dic31C2311yescharacteristic
108C4⋊Dic31C2311yescharacteristic
109Dic31⋊C4—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113C4×Dic31—21no—
114Dic124—22no—
115Dic124—22no—
116Dic124—41no—
117#92470—22no—
118#92470—22no—
119#92470—22no—
120#92470—22no—
121#92470—41no—
122#92470D414yescomplemented
123C23×C31C2×D411yescharacteristic
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—42no—
128C22×D31—42no—
129C22×D31—81no—
130D4×C31—22no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31C2×D411yescharacteristic
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2411yescharacteristic
140C22⋊D31—42no—
141C22⋊D31—81no—
142C2×Dic31—21no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31—42no—
150D124—21no—
151D124—21no—
152D124—21no—
153D124—42no—
154D124C2×D412yes—
155C4×D31—42no—
156C4×D31—81no—
157Dic62—21no—
158Dic62—21no—
159Dic62—21no—
160Dic62—21no—
161Dic62—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62—42no—
165Dic62—42no—
166Dic62C2×D412yes—
167C8×C31—21no—
168C8×C31—21no—
169C8×C31—21no—
170C8×C31C2×D412yes—
171#3266—311nomaximal, Sylow
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31#13712yescomplemented
178C22×C31C22×D411yescharacteristic
179C2×D31—22no—
180C2×D31—41no—
181C2×D31—41no—
182C2×D31—41no—
183C2×D31—41no—
184C2×D31—41no—
185C2×D31—84no—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—21no—
190C4×C31C22×D411yescharacteristic
191C4×C31H(5,𝔽2)11yescharacteristic
192Dic31—22no—
193Dic31—41no—
194Dic31—41no—
195Dic31—41no—
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#461—311no—
207#452—311no—
208#451—311no—
209#450—311no—
210#448—311no—
211#440—311no—
212#435D3111yescharacteristic, complemented
213C2×C31—22no—
214C2×C31—41no—
215C2×C31D4211yescharacteristic
216C2×C31C2×#13711yescharacteristic
217C2×C31#57711yescharacteristic
218D31—42no—
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—311no—
233C2×SD16—311no—
234C2×SD16—624no—
235C2×M16C2×D3111yescharacteristic, complemented
236C22×C8C2×D3111yescharacteristic
237C4⋊Q8—311no—
238C4⋊Q8—621no—
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#103—311no—
259#103—311no—
260#102—311no—
261#102—311no—
262C22⋊C8C2×D3111yescharacteristic
263C22⋊C8C2×D3111yescharacteristic, complemented
264#97C2×D3111yescharacteristic, complemented
265C31#326611yescharacteristic, complemented, Sylow
266C24—621no—
267#41—622no—
268#41—1241no—
269C2×Q8—311no—
270C2×Q8—311no—
271C2×Q8—311no—
272C2×Q8—311no—
273C2×Q8—622no—
274C2×Q8—622no—
275C2×Q8—621no—
276C2×Q8—621no—
277C2×Q8—1241no—
278C2×Q8—1241no—
279C2×D4—311no—
280C2×D4—311no—
281C2×D4—622no—
282C2×D4—621no—
283C2×D4—621no—
284C2×D4—621no—
285C2×D4—1241no—
286C2×D4D12411yescharacteristic
287C22×C4—621no—
288C22×C4—621no—
289C22×C4—621no—
290C22×C4—621no—
291C22×C4C22×D3111yescharacteristic
292C22×C4C22×D3111yescharacteristic
293Q16—622no—
294Q16—622no—
295Q16—1241no—
296SD16—314no—
297SD16—622no—
298SD16—622no—
299SD16—622no—
300SD16—622no—
301SD16—1241no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D3111yescharacteristic
305C2×C8C22×D3111yescharacteristic
306C2×C8C22×D3111yescharacteristic
307C2×C8C22×D3111yescharacteristic
308C4⋊C4—311no—
309C4⋊C4—311no—
310C4⋊C4—621no—
311C4⋊C4—621no—
312C4⋊C4—621no—
313C4⋊C4—621no—
314C4⋊C4—621no—
315C4⋊C4—621no—
316C4⋊C4D12411yescharacteristic
317C22⋊C4—621no—
318C22⋊C4—621no—
319C22⋊C4—621no—
320C22⋊C4—621no—
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4D12411yescharacteristic
325C22⋊C4D12411yescharacteristic
326C42—621no—
327C42C22×D3111yescharacteristic
328C23—621no—
329C23—621no—
330C23—1242no—
331C23—1242no—
332C23—2481no—
333C23C2×D12411yescharacteristic
334C23C2×D12411yescharacteristic
335Q8—312no—
336Q8—621no—
337Q8—621no—
338Q8—621no—
339Q8—621no—
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—1242no—
344Q8—1242no—
345D4—22no—
346D4—312no—
347D4—621no—
348D4—621no—
349D4—621no—
350D4—1242no—
351D4—1242no—
352D4—2481no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—621no—
358C2×C4—621no—
359C2×C4—621no—
360C2×C4—621no—
361C2×C4—621no—
362C2×C4—621no—
363C2×C4—1242no—
364C2×C4—1242no—
365C2×C4—1242no—
366C2×C4—2481no—
367C2×C4C2×D12411yescharacteristic
368C2×C4C2×D12411yescharacteristic
369C2×C4C2×D12411yescharacteristic
370C2×C4C2×D12411yescharacteristic
371C2×C4C23×D3111yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D3112yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—622no—
382C22—1241no—
383C22—1241no—
384C22—1241no—
385C22—1241no—
386C22—1241no—
387C22—2484no—
388C22#1175848612yescomplemented
389C22C22×D12411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—622no—
395C4—1241no—
396C4—1241no—
397C4—1241no—
398C4—1241no—
399C4—1241no—
400C4C2×D4×D3111yescharacteristic
401C4#1175856611yescharacteristic
402C2—22no—
403C2—41no—
404C2—1242no—
405C2—2481no—
406C2D4×D12411yescharacteristic
407C2C2×#1175848611yescharacteristic
408C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
prime radical#14521695C2
hypercenter#435D31
hypocenterC31#3266
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
2. C22×#92470C2semidirect
3. D4×Dic62C2semidirect
5. #14521443C2semidirect
6. #14521295C2semidirect
7. #14521290C2semidirect
8. #14521272C2semidirect
9. #14521255C2semidirect
10. #14521213C2semidirect
11. #14521207C2semidirect
12. #14521202C2semidirect
13. #14521190C2semidirect
14. #14521151C2semidirect
15. C31×#435C2semidirect
19. C22×Dic62C22semidirect
20. C2×M16×C31C22semidirect
28. #11758501C22semidirect
30. C2×Dic124C22semidirect
31. C2×Dic124C22semidirect
32. C2×Dic124C22semidirect
48. #11758446C22semidirect
49. C4×Dic62C22semidirect
50. C31×C4⋊C8C22semidirect
52. C31×C22⋊C8C22semidirect
53. C31×#97C22semidirect
56. #11758406C22semidirect
57. #11758404C22semidirect
58. #11758404C22semidirect
59. #11758404C22semidirect
60. #11758404C22semidirect
122. #92470D4semidirect
177. C22×C31#137semidirect
212. #435D31semidirect
235. C2×M16C2×D31semidirect
255. C4⋊C8C2×D31semidirect
263. C22⋊C8C2×D31semidirect
264. #97C2×D31semidirect
265. C31#3266semidirect
388. C22#11758486semidirect

Maximal subgroups

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

subgroupindex
C2×#117584862
C22×#924702
D4×Dic622
#145216952
#145214432
#145212952
#145212902
#145212722
#145212552
#145212132
#145212072
#145212022
#145211902
#145211512
C31×#4352
#326631