Finite Group Databasebeta

← #16857239

Subgroups — #16857239

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×#11758526C211yesmaximal, characteristic, complemented
2C2×#11758493C211yesmaximal, characteristic, complemented
3#14521698C211yesmaximal, characteristic, complemented
4#14521690C211yesmaximal, characteristic, complemented
5#14521442C211yesmaximal, characteristic, complemented
6#14521305C211yesmaximal, characteristic, complemented
7#14521281C211yesmaximal, characteristic, complemented
8#14521266C211yesmaximal, characteristic, complemented
9#14521253C211yesmaximal, characteristic, complemented
10#14521210C211yesmaximal, characteristic, complemented
11#14521202C211yesmaximal, characteristic, complemented
12#14521195C211yesmaximal, characteristic, complemented
13#14521158C211yesmaximal, characteristic, complemented
14#14521123C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×#92499C2211yescharacteristic
18C2×#92497C2211yescharacteristic, complemented
19C22×D124C2211yescharacteristic
20C2×Q16×C31C2211yescharacteristic, complemented
21C2×SD16×C31C2211yescharacteristic, complemented
22C31×C22⋊Q8C2211yescharacteristic
23#11758526—24no—
24#11758524C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26#11758521C2211yescharacteristic, complemented
27C2×#92482C2211yescharacteristic, complemented
28C2×#92481C2211yescharacteristic
29C2×#92481C2211yescharacteristic
30#11758516—21no—
31#11758512—21no—
32D4×Dic31C2211yescharacteristic, complemented
33C2×#92479C2211yescharacteristic
34C2×#92479C2211yescharacteristic
35#11758506—21no—
36C4×C22⋊D31C2211yescharacteristic
37#11758501C2211yescharacteristic
38C2×Dic31⋊C4—21no—
39C2×#92474C2211yescharacteristic
40#11758493—24no—
41C2×#92470C2211yescharacteristic
42C2×#92470C2211yescharacteristic
43C2×C8×D31C2211yescharacteristic, complemented
44#11758472—21no—
45#11758472C2211yescharacteristic
46#11758471—21no—
47#11758470C2211yescharacteristic
48#11758459—21no—
49C22⋊C4×D31—21no—
50C31×#103C2211yescharacteristic, complemented
51C31×#102C2211yescharacteristic, complemented
52C31×C22⋊C8C2211yescharacteristic
53#11758423C2211yescharacteristic, complemented
54#11758409C2211yescharacteristic
55#11758409C2211yescharacteristic
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758403C2211yescharacteristic, complemented
58Dic31⋊C8C2211yescharacteristic, complemented
59#11758398C2211yescharacteristic, complemented
60#11758396C2211yescharacteristic, complemented
61C23×D31—21no—
62#41×C31—22no—
63#41×C31—41no—
64C2×Q8×C31C2311yescharacteristic
65C2×Q8×C31C2311yescharacteristic
66C2×D4×C31—21no—
67C2×D4×C31C2311yescharacteristic
68C22×C4×C31—21no—
69C22×C4×C31C2311yescharacteristic
70C2×C22⋊D31—21no—
71C2×C22⋊D31D411yescharacteristic
72C22×Dic31—21no—
73C22×Dic31—21no—
74#92499—22no—
75#92499—41no—
76#92497—22no—
77#92497—41no—
78C2×D124—22no—
79C2×D124—21no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124—41no—
83C2×D124C2311yescharacteristic
84C2×D124C2311yescharacteristic
85C2×C4×D31—21no—
86C2×C4×D31—21no—
87C2×C4×D31C2311yescharacteristic
88C2×Dic62C2311yescharacteristic
89Q16×C31—41no—
90SD16×C31—22no—
91C2×C8×C31C2311yescharacteristic
92C2×C8×C31C2311yescharacteristic
93C4⋊C4×C31—21no—
94C4⋊C4×C31C2311yescharacteristic
95C22⋊C4×C31—21no—
96C22⋊Dic31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31D411yescharacteristic
99#92482—22no—
100#92481—22no—
101#92481—22no—
102#92479—22no—
103#92479D414yescomplemented
104(C2×D31)⋊C4—21no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4—21no—
109(C2×D31)⋊C4D411yescharacteristic
110C4⋊Dic31C2311yescharacteristic
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114Dic31⋊C4—21no—
115Dic31⋊C4D411yescharacteristic
116C4×Dic31C2311yescharacteristic
117#92474—22no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120#92470—22no—
121#92470—41no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—42no—
128C22×D31—42no—
129C22×D31—81no—
130C22×D31C2×D411yescharacteristic
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135D4×C31—21no—
136D4×C31—42no—
137D4×C31C2×D412yes—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—42no—
144C2×C4×C31C2×D411yescharacteristic
145C2×C4×C31C2411yescharacteristic
146C22⋊D31—41no—
147C22⋊D31—42no—
148C2×Dic31—21no—
149C2×Dic31—21no—
150C2×Dic31—42no—
151C2×Dic31—42no—
152C2×Dic31C2×D411yescharacteristic
153C2×Dic31C2×D411yescharacteristic
154C2×Dic31C2×D411yescharacteristic
155D124—21no—
156D124—21no—
157D124—21no—
158D124—42no—
159D124—42no—
160D124—81no—
161D124C2×D412yes—
162C4×D31—21no—
163C4×D31—21no—
164C4×D31—42no—
165C4×D31—81no—
166Dic62—21no—
167Dic62—21no—
168C8×C31—21no—
169C8×C31—21no—
170C31⋊C8—21no—
171C31⋊C8C2×D412yes—
172#3267—311nomaximal, Sylow
173C22×C31—22no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31C22×D411yescharacteristic
180C2×D31—21no—
181C2×D31—22no—
182C2×D31—21no—
183C2×D31—21no—
184C2×D31—41no—
185C2×D31—41no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C2×D31—84no—
190C4×C31—21no—
191C4×C31—41no—
192C4×C31—41no—
193C4×C31—41no—
194C4×C31C22×D411yescharacteristic
195C4×C31H(5,𝔽2)11yescharacteristic
196Dic31—21no—
197Dic31—21no—
198Dic31—21no—
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#452D3111yescharacteristic, complemented
212#450—311no—
213#449—311no—
214#442—311no—
215#435—311no—
216C2×C31—41no—
217C2×C31—42no—
218C2×C31D4211yescharacteristic
219C2×C31#57511yescharacteristic
220C2×C31#57711yescharacteristic
221D31—41no—
222D31—42no—
223D31—81no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41C2×D3111yescharacteristic
227C22×D4—311no—
228AGL(1,ℤ/8)—624no—
229#135—624no—
230C2×Q16—311no—
231C2×Q16C2×D3111yescharacteristic, complemented
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16—311no—
236C2×SD16C2×D3111yescharacteristic, complemented
237C2×D8—311no—
238C2×D8—311no—
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⋊Q8C2×D3111yescharacteristic
249#121—311no—
250#121—621no—
251#121—621no—
252#120—621no—
253C4×D4—311no—
254C4×D4—311no—
255C4×D4—311no—
256C2×C4⋊C4—621no—
257C2×C22⋊C4—621no—
258#106—311no—
259C4⋊C8—311no—
260#103—311no—
261#103—311no—
262#103C2×D3111yescharacteristic, complemented
263#102—311no—
264#102—311no—
265#102C2×D3111yescharacteristic, complemented
266C22⋊C8—311no—
267C22⋊C8C2×D3111yescharacteristic
268#97—311no—
269C31#326711yescharacteristic, complemented, Sylow
270C24—621no—
271#41—22no—
272#41—41no—
273#41—622no—
274#41—622no—
275#41—1241no—
276#41—1241no—
277C2×Q8—311no—
278C2×Q8C22×D3111yescharacteristic
279C2×Q8C22×D3111yescharacteristic
280C2×D4—21no—
281C2×D4—311no—
282C2×D4—311no—
283C2×D4—311no—
284C2×D4—622no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—621no—
289C2×D4—1241no—
290C2×D4C22×D3111yescharacteristic
291C22×C4—21no—
292C22×C4—311no—
293C22×C4—621no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4C22×D3111yescharacteristic
298Q16—41no—
299Q16—622no—
300SD16—22no—
301SD16—622no—
302SD16—622no—
303SD16—622no—
304SD16—1241no—
305D8—314no—
306D8—622no—
307M16—622no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—622no—
311C2×C8C22×D3111yescharacteristic
312C2×C8C22×D3111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—311no—
315C4⋊C4—311no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4C22×D3111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—311no—
323C22⋊C4—311no—
324C22⋊C4—621no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C22⋊C4—621no—
330C22⋊C4—621no—
331C42—311no—
332C23—21no—
333C23—311no—
334C23—621no—
335C23—621no—
336C23—1242no—
337C23—1242no—
338C23—2481no—
339C23D4×D3111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—621no—
345Q8—621no—
346D4—21no—
347D4—42no—
348D4—312no—
349D4—621no—
350D4—621no—
351D4—621no—
352D4—1242no—
353D4—1242no—
354D4—1241no—
355D4—1242no—
356D4—2481no—
357D4D4×D3112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—311no—
365C2×C4—311no—
366C2×C4—311no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—621no—
370C2×C4—621no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4—2481no—
375C2×C4D4×D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—21no—
379C8—312no—
380C8—621no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—621no—
388C22—621no—
389C22—622no—
390C22—621no—
391C22—1241no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C2×D4×D3111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—621no—
403C4—621no—
404C4—621no—
405C4—1241no—
406C4—1241no—
407C4C2×D4×D3111yescharacteristic
408C4#1175856111yescharacteristic
409C2—41no—
410C2—42no—
411C2—1241no—
412C2—1242no—
413C2—2481no—
414C2#1452168111yescharacteristic
415C2#1452184811yescharacteristic
416C2#1452185611yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
hypercenter#452D31
hypocenterC31#3267
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#11758526C2semidirect
2. C2×#11758493C2semidirect
3. #14521698C2semidirect
4. #14521690C2semidirect
5. #14521442C2semidirect
6. #14521305C2semidirect
7. #14521281C2semidirect
8. #14521266C2semidirect
9. #14521253C2semidirect
10. #14521210C2semidirect
11. #14521202C2semidirect
12. #14521195C2semidirect
13. #14521158C2semidirect
14. #14521123C2semidirect
15. C31×#452C2semidirect
18. C2×#92497C22semidirect
20. C2×Q16×C31C22semidirect
21. C2×SD16×C31C22semidirect
25. #11758523C22semidirect
26. #11758521C22semidirect
27. C2×#92482C22semidirect
32. D4×Dic31C22semidirect
43. C2×C8×D31C22semidirect
50. C31×#103C22semidirect
51. C31×#102C22semidirect
53. #11758423C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758403C22semidirect
58. Dic31⋊C8C22semidirect
59. #11758398C22semidirect
60. #11758396C22semidirect
103. #92479D4semidirect
211. #452D31semidirect
231. C2×Q16C2×D31semidirect
236. C2×SD16C2×D31semidirect
262. #103C2×D31semidirect
265. #102C2×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×#117585262
C2×#117584932
#145216982
#145216902
#145214422
#145213052
#145212812
#145212662
#145212532
#145212102
#145212022
#145211952
#145211582
#145211232
C31×#4522
#326731