Finite Group Databasebeta

← #16857241

Subgroups — #16857241

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×#11758485C211yesmaximal, characteristic, complemented
2C2×#11758481C211yesmaximal, characteristic, complemented
3#14521704C211yesmaximal, characteristic, complemented
4#14521695C211yesmaximal, characteristic, complemented
5#14521443C211yesmaximal, characteristic, complemented
6#14521307C211yesmaximal, characteristic, complemented
7#14521278C211yesmaximal, characteristic, complemented
8#14521265C211yesmaximal, characteristic, complemented
9#14521255C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521213C211yesmaximal, characteristic, complemented
12#14521202C211yesmaximal, characteristic, complemented
13#14521196C211yesmaximal, characteristic, complemented
14#14521162C211yesmaximal, characteristic, complemented
15C31×#435C211yesmaximal, characteristic, complemented
16C2×#92495C2211yescharacteristic
17C2×#92495C2211yescharacteristic
18C2×#92495C2211yescharacteristic, complemented
19C22×D124C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic
21C22×C8×C31C2211yescharacteristic, complemented
22C4×D4×C31C2211yescharacteristic
23#11758515—21no—
24#11758514—21no—
25#11758507C2211yescharacteristic
26#11758501C2211yescharacteristic
27#11758501C2211yescharacteristic
28#11758501C2211yescharacteristic, complemented
29#11758485—24no—
30C2×Dic124C2211yescharacteristic, complemented
31C2×Dic124C2211yescharacteristic, complemented
32#11758481—24no—
33C2×D248C2211yescharacteristic
34C2×D248C2211yescharacteristic
35C2×#92470C2211yescharacteristic
36C2×#92470C2211yescharacteristic
37C2×#92470C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×#92470C2211yescharacteristic, complemented
40#11758471—21no—
41C4⋊C4×D31—21no—
42#11758461—21no—
43#11758460—21no—
44#11758460—21no—
45C22⋊C4×D31—21no—
46#11758452C2211yescharacteristic
47C4×D124C2211yescharacteristic
48C4×D124C2211yescharacteristic, complemented
49#11758446C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×C22⋊C8C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic, complemented
53C31×#97C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic
55#11758409C2211yescharacteristic
56#11758409C2211yescharacteristic, complemented
57#11758405C2211yescharacteristic, 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—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69#92495—22no—
70#92495—22no—
71#92495—22no—
72#92495—41no—
73#92495—41no—
74#92495—41no—
75C2×D124—22no—
76C2×D124—21no—
77C2×D124—41no—
78C2×D124C2311yescharacteristic
79C2×D124C2311yescharacteristic
80C2×D124C2311yescharacteristic
81C2×C4×D31—21no—
82C2×C4×D31—21no—
83C2×C4×D31—21no—
84C2×C4×D31—21no—
85C2×C4×D31—21no—
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88C2×Dic62C2311yescharacteristic
89M16×C31—22no—
90C2×C8×C31—22no—
91C2×C8×C31C2311yescharacteristic
92C2×C8×C31C2311yescharacteristic
93C2×C8×C31C2311yescharacteristic
94C2×C8×C31C2311yescharacteristic
95C4⋊C4×C31D411yescharacteristic
96C22⋊C4×C31D411yescharacteristic
97C22⋊C4×C31D411yescharacteristic
98C42×C31C2311yescharacteristic
99C22⋊Dic31—21no—
100C22⋊Dic31—21no—
101C22⋊Dic31—21no—
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—
107C4⋊Dic31—21no—
108C4⋊Dic31—21no—
109C4⋊Dic31C2311yescharacteristic
110C4⋊Dic31C2311yescharacteristic
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114Dic124—22no—
115Dic124—41no—
116D248—22no—
117D248D414yescomplemented
118#92470—22no—
119#92470—22no—
120#92470—22no—
121#92470—22no—
122#92470—41no—
123C23×C31C2×D411yescharacteristic
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—42no—
129C22×D31—42no—
130C22×D31—81no—
131D4×C31—41no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31C2×D411yescharacteristic
137C2×C4×C31C2×D411yescharacteristic
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2411yescharacteristic
141C22⋊D31—42no—
142C22⋊D31—42no—
143C22⋊D31—42no—
144C22⋊D31—81no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—21no—
149C2×Dic31—21no—
150D124—21no—
151D124—21no—
152D124—21no—
153D124—21no—
154D124—42no—
155D124C2×D412yes—
156D124C2×D412yes—
157C4×D31—42no—
158C4×D31—42no—
159C4×D31—42no—
160C4×D31—42no—
161C4×D31—81no—
162Dic62—21no—
163Dic62—21no—
164Dic62—21no—
165Dic62—21no—
166Dic62—21no—
167Dic62—21no—
168C8×C31—21no—
169C8×C31—21no—
170C8×C31—21no—
171C8×C31C2×D412yes—
172#3267—311nomaximal, Sylow
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—21no—
179C22×C31C22×D411yescharacteristic
180C2×D31—22no—
181C2×D31—22no—
182C2×D31—41no—
183C2×D31—41no—
184C2×D31—41no—
185C2×D31—41no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C2×D31—84no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—21no—
194C4×C31C22×D411yescharacteristic
195C4×C31H(5,𝔽2)11yescharacteristic
196Dic31—41no—
197Dic31—41no—
198Dic31—41no—
199Dic31—41no—
200Dic31—41no—
201C2×AGL(1,ℤ/8)—311no—
202C2×#135—311no—
203#547—311no—
204#546—311no—
205#497—311no—
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#435D3111yescharacteristic, complemented
216C2×C31—41no—
217C2×C31—41no—
218C2×C31D4211yescharacteristic
219C2×C31#57511yescharacteristic
220C2×C31#57711yescharacteristic
221D31—42no—
222D31—42no—
223D31—81no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41—311no—
227C22×D4—311no—
228AGL(1,ℤ/8)—624no—
229#135—624no—
230C2×Q16—311no—
231C2×Q16—311no—
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16—311no—
236C2×SD16—311no—
237C2×D8—311no—
238C2×D8—311no—
239C2×M16C2×D3111yescharacteristic
240C22×C8C2×D3111yescharacteristic, complemented
241C4⋊Q8—311no—
242#124—311no—
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×D4C2×D3111yescharacteristic
256C2×C4⋊C4—621no—
257C2×C22⋊C4—621no—
258#106—311no—
259C4⋊C8C2×D3111yescharacteristic, complemented
260#103—311no—
261#103—311no—
262#103—311no—
263#102—311no—
264#102—311no—
265#102—311no—
266C22⋊C8C2×D3111yescharacteristic
267C22⋊C8C2×D3111yescharacteristic, complemented
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×Q8—311no—
278C2×Q8—311no—
279C2×Q8—311no—
280C2×D4—311no—
281C2×D4—311no—
282C2×D4—311no—
283C2×D4—622no—
284C2×D4—621no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—621no—
289C2×D4—1241no—
290C2×D4D12411yescharacteristic
291C22×C4—621no—
292C22×C4—621no—
293C22×C4—621no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4C22×D3111yescharacteristic
297C22×C4C22×D3111yescharacteristic
298Q16—622no—
299Q16—1241no—
300SD16—622no—
301SD16—622no—
302SD16—622no—
303SD16—622no—
304SD16—1241no—
305D8—314no—
306D8—622no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D3111yescharacteristic
310C2×C8C22×D3111yescharacteristic
311C2×C8C22×D3111yescharacteristic
312C2×C8C22×D3111yescharacteristic
313C4⋊C4—311no—
314C4⋊C4—311no—
315C4⋊C4—621no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4D12411yescharacteristic
321C22⋊C4—621no—
322C22⋊C4—621no—
323C22⋊C4—621no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C22⋊C4D12411yescharacteristic
330C22⋊C4D12411yescharacteristic
331C42C22×D3111yescharacteristic
332C23—621no—
333C23—621no—
334C23—621no—
335C23—1242no—
336C23—1242no—
337C23—2481no—
338C23C2×D12411yescharacteristic
339C23C2×D12411yescharacteristic
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—621no—
344Q8—621no—
345Q8—621no—
346D4—41no—
347D4—312no—
348D4—312no—
349D4—621no—
350D4—621no—
351D4—621no—
352D4—621no—
353D4—1242no—
354D4—1242no—
355D4—1242no—
356D4—1242no—
357D4—2481no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—621no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—1242no—
368C2×C4—1242no—
369C2×C4—1242no—
370C2×C4—1242no—
371C2×C4—2481no—
372C2×C4C2×D12411yescharacteristic
373C2×C4C2×D12411yescharacteristic
374C2×C4C2×D12411yescharacteristic
375C2×C4C2×D12411yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D3112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—622no—
388C22—622no—
389C22—1241no—
390C22—1241no—
391C22—1241no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C22×D12411yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—1241no—
403C4—1241no—
404C4—1241no—
405C4—1241no—
406C4—1241no—
407C4C2×D4×D3111yescharacteristic
408C4#1175856611yescharacteristic
409C2—41no—
410C2—41no—
411C2—1242no—
412C2—1242no—
413C2—2481no—
414C2D4×D12411yescharacteristic
415C2#1452184911yescharacteristic
416C2#1452186111yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#435C2
socleC22×C31C22×D4
hypercenter#435D31
hypocenterC31#3267
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×#11758485C2semidirect
2. C2×#11758481C2semidirect
3. #14521704C2semidirect
4. #14521695C2semidirect
5. #14521443C2semidirect
6. #14521307C2semidirect
7. #14521278C2semidirect
8. #14521265C2semidirect
9. #14521255C2semidirect
10. #14521213C2semidirect
11. #14521213C2semidirect
12. #14521202C2semidirect
13. #14521196C2semidirect
14. #14521162C2semidirect
15. C31×#435C2semidirect
18. C2×#92495C22semidirect
21. C22×C8×C31C22semidirect
28. #11758501C22semidirect
30. C2×Dic124C22semidirect
31. C2×Dic124C22semidirect
39. C2×#92470C22semidirect
48. C4×D124C22semidirect
49. #11758446C22semidirect
50. C31×C4⋊C8C22semidirect
52. C31×C22⋊C8C22semidirect
53. C31×#97C22semidirect
56. #11758409C22semidirect
57. #11758405C22semidirect
58. #11758404C22semidirect
59. #11758404C22semidirect
60. #11758404C22semidirect
117. D248D4semidirect
215. #435D31semidirect
240. C22×C8C2×D31semidirect
259. C4⋊C8C2×D31semidirect
267. C22⋊C8C2×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×#117584852
C2×#117584812
#145217042
#145216952
#145214432
#145213072
#145212782
#145212652
#145212552
#145212132
#145212022
#145211962
#145211622
C31×#4352
#326731