Finite Group Databasebeta

← #16857238

Subgroups — #16857238

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×#11758488C211yesmaximal, characteristic, complemented
2C2×#11758489C211yesmaximal, characteristic, complemented
3#14521703C211yesmaximal, characteristic, complemented
4#14521694C211yesmaximal, characteristic, complemented
5#14521443C211yesmaximal, characteristic, complemented
6#14521308C211yesmaximal, characteristic, complemented
7#14521279C211yesmaximal, characteristic, complemented
8#14521264C211yesmaximal, characteristic, complemented
9#14521254C211yesmaximal, characteristic, complemented
10#14521212C211yesmaximal, characteristic, complemented
11#14521212C211yesmaximal, characteristic, complemented
12#14521201C211yesmaximal, characteristic, complemented
13#14521195C211yesmaximal, characteristic, complemented
14#14521122C211yesmaximal, characteristic, complemented
15C31×#442C211yesmaximal, characteristic, complemented
16C2×#92497C2211yescharacteristic
17C2×#92497C2211yescharacteristic
18C2×#92497C2211yescharacteristic, complemented
19C2×D4×D31C2211yescharacteristic
20C2×D8×C31C2211yescharacteristic
21C4×D4×C31C2211yescharacteristic
22C4×D4×C31C2211yescharacteristic, complemented
23C2×#92480C2211yescharacteristic
24C2×#92480C2211yescharacteristic
25C2×#92480C2211yescharacteristic, complemented
26C2×#92479C2211yescharacteristic
27#11758507—21no—
28#11758507—21no—
29C2×(C2×D31)⋊C4—21no—
30C2×C4⋊Dic31—21no—
31#11758489—24no—
32#11758488—24no—
33C2×Dic124C2211yescharacteristic, complemented
34C2×Dic124C2211yescharacteristic, complemented
35C2×#92470C2211yescharacteristic
36C2×#92470C2211yescharacteristic
37C2×#92469C2211yescharacteristic
38C2×C8×D31C2211yescharacteristic, complemented
39#11758474C2211yescharacteristic
40#11758474C2211yescharacteristic
41#11758474C2211yescharacteristic, complemented
42#11758472C2211yescharacteristic
43#11758463—21no—
44#11758463—21no—
45#11758463—21no—
46#11758459—21no—
47#11758452C2211yescharacteristic
48C4×D124C2211yescharacteristic
49#11758446C2211yescharacteristic, complemented
50C31×#106C2211yescharacteristic, complemented
51C31×#102C2211yescharacteristic
52C31×#102C2211yescharacteristic, complemented
53C31×#97C2211yescharacteristic, complemented
54(C2×D31)⋊C8C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic, complemented
56#11758398C2211yescharacteristic, complemented
57#11758398C2211yescharacteristic, complemented
58#11758398C2211yescharacteristic, complemented
59#11758397C2211yescharacteristic
60C4⋊(C31⋊C8)C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×D4×C31C2311yescharacteristic
63C2×D4×C31C2311yescharacteristic
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C22×Dic31—21no—
71C22×Dic31—21no—
72C22×Dic31—21no—
73#92497—22no—
74#92497—22no—
75#92497—22no—
76#92497—41no—
77#92497—41no—
78#92497—41no—
79D4×D31—22no—
80D4×D31—41no—
81C2×D124—21no—
82C2×D124D411yescharacteristic
83C2×D124C2311yescharacteristic
84C2×C4×D31C2311yescharacteristic
85C2×C4×D31C2311yescharacteristic
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88C2×Dic62C2311yescharacteristic
89D8×C31D414yescomplemented
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C4⋊C4×C31C2311yescharacteristic
93C4⋊C4×C31C2311yescharacteristic
94C22⋊C4×C31—21no—
95C22⋊C4×C31—21no—
96C42×C31C2311yescharacteristic
97#92480—22no—
98#92480—22no—
99#92480—22no—
100#92479—22no—
101(C2×D31)⋊C4—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—
107(C2×D31)⋊C4D411yescharacteristic
108(C2×D31)⋊C4D411yescharacteristic
109C4⋊Dic31—21no—
110C4⋊Dic31—21no—
111C4⋊Dic31—21no—
112C4⋊Dic31—21no—
113C4⋊Dic31—21no—
114C4⋊Dic31D411yescharacteristic
115C2×C31⋊C8C2311yescharacteristic
116C2×C31⋊C8C2311yescharacteristic
117Dic124—22no—
118Dic124—41no—
119#92470—22no—
120#92470—41no—
121#92469—22no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31—21no—
125C22×D31—21no—
126C22×D31—42no—
127C22×D31—42no—
128C22×D31—81no—
129C22×D31C2×D411yescharacteristic
130C22×D31C2×D411yescharacteristic
131D4×C31—21no—
132D4×C31—21no—
133D4×C31C2×D412yes—
134D4×C31C2×D412yes—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—42no—
138C2×C4×C31—81no—
139C2×C4×C31C2×D411yescharacteristic
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—42no—
143C22⋊D31—42no—
144C22⋊D31—42no—
145C22⋊D31—42no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—21no—
149C2×Dic31—42no—
150C2×Dic31—42no—
151C2×Dic31—42no—
152C2×Dic31C2×D411yescharacteristic
153C2×Dic31C2×D411yescharacteristic
154D124—21no—
155D124—21no—
156D124—41no—
157D124—81no—
158C4×D31—21no—
159C4×D31—21no—
160C4×D31—21no—
161C4×D31—21no—
162Dic62—21no—
163Dic62—21no—
164Dic62—21no—
165Dic62—21no—
166Dic62—21no—
167Dic62—21no—
168C8×C31—21no—
169C8×C31C2×D412yes—
170C31⋊C8—21no—
171C31⋊C8—21no—
172#3267—311nomaximal, Sylow
173C22×C31—22no—
174C22×C31—22no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31C22×D411yescharacteristic
180C2×D31—21no—
181C2×D31—21no—
182C2×D31—21no—
183C2×D31—21no—
184C2×D31—21no—
185C2×D31—21no—
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×C31C22×D411yescharacteristic
195C4×C31H(5,𝔽2)11yescharacteristic
196Dic31—21no—
197Dic31—21no—
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#442D3111yescharacteristic, complemented
215#435—311no—
216C2×C31—42no—
217C2×C31—42no—
218C2×C31D4211yescharacteristic
219C2×C31#57511yescharacteristic
220C2×C31#57711yescharacteristic
221D31—41no—
222D31—41no—
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×D8C2×D3111yescharacteristic
239C2×M16—311no—
240C22×C8—311no—
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×D4C2×D3111yescharacteristic
255C4×D4C2×D3111yescharacteristic, complemented
256C2×C4⋊C4—621no—
257C2×C22⋊C4—621no—
258#106C2×D3111yescharacteristic, complemented
259C4⋊C8—311no—
260#103—311no—
261#103—311no—
262#103—311no—
263#102—311no—
264#102C2×D3111yescharacteristic
265#102C2×D3111yescharacteristic, complemented
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×Q8—311no—
278C2×Q8—311no—
279C2×Q8—311no—
280C2×D4—311no—
281C2×D4—311no—
282C2×D4—621no—
283C2×D4—621no—
284C2×D4—622no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—1241no—
289C2×D4C22×D3111yescharacteristic
290C2×D4C22×D3111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—311no—
294C22×C4—311no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4—621no—
298Q16—622no—
299Q16—1241no—
300SD16—622no—
301SD16—622no—
302SD16—622no—
303SD16—622no—
304SD16—1241no—
305D8—622no—
306D8D12414yescomplemented
307M16—622no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—622no—
311C2×C8C22×D3111yescharacteristic
312C2×C8C22×D3111yescharacteristic
313C4⋊C4—311no—
314C4⋊C4—621no—
315C4⋊C4—621no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4C22×D3111yescharacteristic
320C4⋊C4C22×D3111yescharacteristic
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—21no—
334C23—311no—
335C23—311no—
336C23—621no—
337C23—1242no—
338C23—1242no—
339C23—2481no—
340Q8—621no—
341Q8—621no—
342Q8—621no—
343Q8—621no—
344Q8—621no—
345Q8—621no—
346D4—21no—
347D4—21no—
348D4—621no—
349D4—621no—
350D4—1241no—
351D4—1242no—
352D4—1242no—
353D4—1242no—
354D4—1242no—
355D4—2481no—
356D4C2×D12412yes—
357D4C2×D12412yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—311no—
363C2×C4—311no—
364C2×C4—621no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—621no—
370C2×C4—621no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4D4×D3111yescharacteristic
375C2×C4D4×D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—621no—
379C8—621no—
380C8C2×D12412yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—621no—
388C22—621no—
389C22—621no—
390C22—621no—
391C22—621no—
392C22—621no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C2×D4×D3111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—621no—
403C4—621no—
404C4—1241no—
405C4—1241no—
406C4—1241no—
407C4C22×D12411yescharacteristic
408C4#1175856611yescharacteristic
409C2—42no—
410C2—42no—
411C2—1241no—
412C2—1241no—
413C2—2481no—
414C2D4×D12411yescharacteristic
415C2#1452184511yescharacteristic
416C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#442C2
socleC22×C31C22×D4
hypercenter#442D31
hypocenterC31#3267
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#11758488C2semidirect
2. C2×#11758489C2semidirect
3. #14521703C2semidirect
4. #14521694C2semidirect
5. #14521443C2semidirect
6. #14521308C2semidirect
7. #14521279C2semidirect
8. #14521264C2semidirect
9. #14521254C2semidirect
10. #14521212C2semidirect
11. #14521212C2semidirect
12. #14521201C2semidirect
13. #14521195C2semidirect
14. #14521122C2semidirect
15. C31×#442C2semidirect
18. C2×#92497C22semidirect
22. C4×D4×C31C22semidirect
25. C2×#92480C22semidirect
33. C2×Dic124C22semidirect
34. C2×Dic124C22semidirect
38. C2×C8×D31C22semidirect
41. #11758474C22semidirect
49. #11758446C22semidirect
50. C31×#106C22semidirect
52. C31×#102C22semidirect
53. C31×#97C22semidirect
55. (C2×D31)⋊C8C22semidirect
56. #11758398C22semidirect
57. #11758398C22semidirect
58. #11758398C22semidirect
60. C4⋊(C31⋊C8)C22semidirect
89. D8×C31D4semidirect
214. #442D31semidirect
255. C4×D4C2×D31semidirect
258. #106C2×D31semidirect
265. #102C2×D31semidirect
268. #97C2×D31semidirect
269. C31#3267semidirect
306. D8D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584882
C2×#117584892
#145217032
#145216942
#145214432
#145213082
#145212792
#145212642
#145212542
#145212122
#145212012
#145211952
#145211222
C31×#4422
#326731