Finite Group Databasebeta

← #16857242

Subgroups — #16857242

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×#11758491C211yesmaximal, characteristic, complemented
2C2×#11758493C211yesmaximal, characteristic, complemented
3#14521702C211yesmaximal, characteristic, complemented
4#14521694C211yesmaximal, characteristic, complemented
5#14521441C211yesmaximal, characteristic, complemented
6#14521306C211yesmaximal, characteristic, complemented
7#14521281C211yesmaximal, characteristic, complemented
8#14521265C211yesmaximal, characteristic, complemented
9#14521214C211yesmaximal, characteristic, complemented
10#14521208C211yesmaximal, characteristic, complemented
11#14521198C211yesmaximal, characteristic, complemented
12#14521197C211yesmaximal, characteristic, complemented
13#14521159C211yesmaximal, characteristic, complemented
14#14521119C211yesmaximal, characteristic, complemented
15C31×#461C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92499C2211yescharacteristic, complemented
18C2×#92497C2211yescharacteristic
19C2×D4×D31C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic
22C31×C4⋊Q8C2211yescharacteristic, complemented
23C4×D4×C31C2211yescharacteristic
24#11758523C2211yescharacteristic
25#11758523C2211yescharacteristic, complemented
26C2×#92482C2211yescharacteristic, complemented
27C2×#92482C2211yescharacteristic, complemented
28C2×#92481C2211yescharacteristic
29C2×#92481C2211yescharacteristic
30C2×#92479C2211yescharacteristic
31#11758507—21no—
32C2×(C2×D31)⋊C4—21no—
33#11758493—24no—
34#11758491—24no—
35C2×D248C2211yescharacteristic
36C2×#92470C2211yescharacteristic, complemented
37C2×#92469C2211yescharacteristic
38C2×C8×D31C2211yescharacteristic, complemented
39#11758474C2211yescharacteristic
40#11758472—21no—
41#11758472C2211yescharacteristic
42#11758471—21no—
43#11758471—21no—
44C4⋊C4×D31—21no—
45#11758463—21no—
46#11758459—21no—
47#11758452C2211yescharacteristic
48C4×D124C2211yescharacteristic
49C4×D124C2211yescharacteristic, complemented
50C31×C4⋊C8C2211yescharacteristic, complemented
51C31×#103C2211yescharacteristic, complemented
52C31×#103C2211yescharacteristic, complemented
53#11758420C2211yescharacteristic
54#11758409C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic
56(C2×D31)⋊C8C2211yescharacteristic, complemented
57#11758404C2211yescharacteristic, complemented
58#11758397C2211yescharacteristic, complemented
59#11758396C2211yescharacteristic, complemented
60#11758391C2211yescharacteristic, complemented
61C23×D31—21no—
62C2×Q8×C31C2311yescharacteristic
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31C2311yescharacteristic
65C22×C4×C31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C22×Dic31—21no—
69#92499—22no—
70#92499—22no—
71#92499—41no—
72#92499—41no—
73#92497—22no—
74#92497—41no—
75D4×D31—22no—
76D4×D31—41no—
77C2×D124—21no—
78C2×D124—21no—
79C2×D124—21no—
80C2×D124D411yescharacteristic
81C2×D124C2311yescharacteristic
82C2×D124C2311yescharacteristic
83C2×C4×D31—21no—
84C2×C4×D31—21no—
85C2×C4×D31—21no—
86C2×C4×D31C2311yescharacteristic
87C2×C4×D31C2311yescharacteristic
88C2×Dic62C2311yescharacteristic
89SD16×C31—22no—
90SD16×C31—22no—
91C2×C8×C31C2311yescharacteristic
92C2×C8×C31C2311yescharacteristic
93C4⋊C4×C31—21no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31C2311yescharacteristic
96C22⋊C4×C31—21no—
97C42×C31C2311yescharacteristic
98#92482—22no—
99#92482—41no—
100#92481—22no—
101#92481—41no—
102#92479D414yescomplemented
103(C2×D31)⋊C4—21no—
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)⋊C4—21no—
110(C2×D31)⋊C4D411yescharacteristic
111(C2×D31)⋊C4D411yescharacteristic
112C4⋊Dic31—21no—
113C4⋊Dic31D411yescharacteristic
114C4⋊Dic31C2311yescharacteristic
115Dic31⋊C4—21no—
116Dic31⋊C4—21no—
117C2×C31⋊C8C2311yescharacteristic
118C2×C31⋊C8C2311yescharacteristic
119D248—22no—
120#92470—22no—
121#92469—22no—
122C8×D31—22no—
123C23×C31—21no—
124C22×D31—21no—
125C22×D31—21no—
126C22×D31—42no—
127C22×D31—42no—
128C22×D31—81no—
129C22×D31C2×D411yescharacteristic
130C22×D31C2×D411yescharacteristic
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—21no—
134Q8×C31—21no—
135D4×C31—21no—
136D4×C31C2×D412yes—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—81no—
141C2×C4×C31C2×D411yescharacteristic
142C2×C4×C31C2×D411yescharacteristic
143C2×C4×C31C2411yescharacteristic
144C22⋊D31—42no—
145C22⋊D31—42no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151D124—21no—
152D124—21no—
153D124—21no—
154D124—41no—
155D124—42no—
156D124—42no—
157D124—81no—
158D124C2×D412yes—
159C4×D31—21no—
160C4×D31—21no—
161C4×D31—21no—
162C4×D31—21no—
163C4×D31—42no—
164C4×D31—42no—
165C4×D31—42no—
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—41no—
175C22×C31—41no—
176C22×C31C22×D411yescharacteristic
177C2×D31—21no—
178C2×D31—21no—
179C2×D31—22no—
180C2×D31—21no—
181C2×D31—21no—
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—21no—
192C4×C31—41no—
193C4×C31—41no—
194C4×C31—41no—
195C4×C31C22×D411yescharacteristic
196C4×C31H(5,𝔽2)11yescharacteristic
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#461D3111yescharacteristic, complemented
210#452—311no—
211#452—311no—
212#450—311no—
213#449—311no—
214#442—311no—
215#435—311no—
216C2×C31—42no—
217C2×C31D4211yescharacteristic
218C2×C31#57511yescharacteristic
219C2×C31#57711yescharacteristic
220D31—41no—
221D31—41no—
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×SD16C2×D3111yescharacteristic
236C2×SD16C2×D3111yescharacteristic
237C2×D8—311no—
238C2×D8—311no—
239C2×M16—311no—
240C22×C8—311no—
241C4⋊Q8C2×D3111yescharacteristic, complemented
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#103C2×D3111yescharacteristic, complemented
262#103C2×D3111yescharacteristic, complemented
263#102—311no—
264#102—311no—
265#102—311no—
266C22⋊C8—311no—
267C22⋊C8—311no—
268#97—311no—
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×Q8C22×D3111yescharacteristic
279C2×Q8C22×D3111yescharacteristic
280C2×D4—311no—
281C2×D4—311no—
282C2×D4—311no—
283C2×D4—621no—
284C2×D4—621no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—622no—
288C2×D4—621no—
289C2×D4—1241no—
290C2×D4C22×D3111yescharacteristic
291C22×C4—21no—
292C22×C4—311no—
293C22×C4—311no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4—621no—
298Q16—622no—
299Q16—1241no—
300SD16—22no—
301SD16—22no—
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—21no—
315C4⋊C4—311no—
316C4⋊C4—311no—
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—
331C42C22×D3111yescharacteristic
332C23—21no—
333C23—311no—
334C23—311no—
335C23—621no—
336C23—621no—
337C23—1242no—
338C23—1242no—
339C23—2481no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—621no—
345Q8—621no—
346D4—21no—
347D4—312no—
348D4—621no—
349D4—621no—
350D4—621no—
351D4—1241no—
352D4—1242no—
353D4—1242no—
354D4—1242no—
355D4—1242no—
356D4—2481no—
357D4D4×D3112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
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—1242no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4D4×D3111yescharacteristic
375C2×C4D4×D3111yescharacteristic
376C2×C4C23×D3111yescharacteristic
377C8—21no—
378C8—21no—
379C8—312no—
380C8—621no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—621no—
385C22—621no—
386C22—622no—
387C22—621no—
388C22—621no—
389C22—621no—
390C22—621no—
391C22—1241no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22—1241no—
396C22—2484no—
397C22C2×D4×D3111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—621no—
404C4—621no—
405C4—1241no—
406C4—1241no—
407C4C2×D4×D3111yescharacteristic
408C4#1175856611yescharacteristic
409C2—42no—
410C2—1241no—
411C2—1241no—
412C2—1242no—
413C2—2481no—
414C2#1452168311yescharacteristic
415C2#1452184811yescharacteristic
416C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#461C2
socleC22×C31C22×D4
hypercenter#461D31
hypocenterC31#3267
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#11758491C2semidirect
2. C2×#11758493C2semidirect
3. #14521702C2semidirect
4. #14521694C2semidirect
5. #14521441C2semidirect
6. #14521306C2semidirect
7. #14521281C2semidirect
8. #14521265C2semidirect
9. #14521214C2semidirect
10. #14521208C2semidirect
11. #14521198C2semidirect
12. #14521197C2semidirect
13. #14521159C2semidirect
14. #14521119C2semidirect
15. C31×#461C2semidirect
17. C2×#92499C22semidirect
22. C31×C4⋊Q8C22semidirect
25. #11758523C22semidirect
26. C2×#92482C22semidirect
27. C2×#92482C22semidirect
36. C2×#92470C22semidirect
38. C2×C8×D31C22semidirect
49. C4×D124C22semidirect
50. C31×C4⋊C8C22semidirect
51. C31×#103C22semidirect
52. C31×#103C22semidirect
56. (C2×D31)⋊C8C22semidirect
57. #11758404C22semidirect
58. #11758397C22semidirect
59. #11758396C22semidirect
60. #11758391C22semidirect
102. #92479D4semidirect
209. #461D31semidirect
241. C4⋊Q8C2×D31semidirect
259. C4⋊C8C2×D31semidirect
261. #103C2×D31semidirect
262. #103C2×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×#117584912
C2×#117584932
#145217022
#145216942
#145214412
#145213062
#145212812
#145212652
#145212142
#145212082
#145211982
#145211972
#145211592
#145211192
C31×#4612
#326731