Finite Group Databasebeta

← #16854967

Subgroups — #16854967

Order 3968.

754 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D31C211yesmaximal, characteristic
2C2×#11758519C211yesmaximal, characteristic, complemented
3C2×#11758523C211yesmaximal, characteristic, complemented
4#14521210C212yesmaximal, complemented
5#14521208C212yesmaximal, complemented
6#14521203C212yesmaximal, complemented
7#14521198C212yesmaximal, complemented
8#14521051C211yesmaximal, characteristic, complemented
9#14521050C211yesmaximal, characteristic, complemented
10#14521012C211yesmaximal, characteristic, complemented
11C2×C31×#137C211yesmaximal, characteristic, complemented
12C2×#41×C31C2211yescharacteristic
13C22×Q8×C31C2211yescharacteristic, complemented
14#41×D31—24no—
15#41×D31—24no—
16C2×#92499C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×#92497—21no—
19C2×#92497C2211yescharacteristic
20C2×D4×D31—21no—
21C2×D4×D31C2211yescharacteristic
22C2×#92495—21no—
23C2×#92495C2211yescharacteristic
24C22×C4×D31—21no—
25C22×C4×D31C2211yescharacteristic
26C31×#137—24no—
27C2×Q16×C31C2212yescomplemented
28C2×SD16×C31C2212yescomplemented
29C2×M16×C31C2211yescharacteristic, complemented
30#11758523—21no—
31#11758523—21no—
32#11758523C2212yes—
33#11758523C2212yescomplemented
34C2×#92482C2212yescomplemented
35#11758519—24no—
36C2×#92481C2212yescomplemented
37C2×(C2×D31)⋊C4—21no—
38C2×C4⋊Dic31C2211yescharacteristic, complemented
39C2×Dic31⋊C4—21no—
40C2×#92474C2211yescharacteristic, complemented
41#11758423C2212yescomplemented
42#11758420C2212yescomplemented
43#11758409C2212yescomplemented
44(C2×D31)⋊C8C2212yes—
45(C2×D31)⋊C8C2212yescomplemented
46#11758404C2212yescomplemented
47C23×D31—21no—
48C23×D31D411yescharacteristic
49#41×C31—22no—
50#41×C31D414yes—
51C2×Q8×C31—21no—
52C2×Q8×C31—22no—
53C2×Q8×C31—22no—
54C2×Q8×C31C2312yes—
55C2×Q8×C31C2311yescharacteristic
56C2×D4×C31—21no—
57C2×D4×C31C2311yescharacteristic
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31C2311yescharacteristic
61C2×C22⋊D31—21no—
62C2×C22⋊D31—21no—
63C2×C22⋊D31—21no—
64C22×Dic31—21no—
65C22×Dic31—21no—
66C22×Dic31D411yescharacteristic
67#92499—22no—
68#92499—22no—
69Q8×D31—22no—
70Q8×D31—22no—
71#92497—28no—
72#92497—22no—
73#92497—22no—
74D4×D31—28no—
75D4×D31—22no—
76D4×D31—22no—
77#92495—28no—
78#92495—22no—
79#92495D414yes—
80C2×D124—21no—
81C2×D124C2311yescharacteristic
82C2×C4×D31—28no—
83C2×C4×D31—21no—
84C2×C4×D31—21no—
85C2×C4×D31—22no—
86C2×C4×D31—22no—
87C2×C4×D31—21no—
88C2×C4×D31D411yescharacteristic
89C2×C4×D31D411yescharacteristic
90C2×C4×D31C2311yescharacteristic
91C2×C4×D31C2311yescharacteristic
92C2×Dic62—21no—
93C2×Dic62C2311yescharacteristic
94Q16×C31—24no—
95SD16×C31—24no—
96M16×C31—22no—
97C2×C8×C31C2312yes—
98#92482—24no—
99#92481—24no—
100(C2×D31)⋊C4—22no—
101(C2×D31)⋊C4—22no—
102C4⋊Dic31—21no—
103C4⋊Dic31C2312yes—
104Dic31⋊C4—22no—
105Dic31⋊C4—22no—
106#92474—22no—
107C2×C31⋊C8C2312yes—
108C23×C31—21no—
109C23×C31C2×D411yescharacteristic
110C22×D31—24no—
111C22×D31—22no—
112C22×D31—21no—
113C22×D31—22no—
114C22×D31—41no—
115C22×D31—42no—
116C22×D31—42no—
117C22×D31C2×D411yescharacteristic
118C22×D31C2×D411yescharacteristic
119Q8×C31—21no—
120Q8×C31—22no—
121Q8×C31—22no—
122Q8×C31—42no—
123Q8×C31C2×D412yes—
124D4×C31—24no—
125D4×C31—21no—
126D4×C31C2×D412yes—
127C2×C4×C31—21no—
128C2×C4×C31—24no—
129C2×C4×C31—22no—
130C2×C4×C31—42no—
131C2×C4×C31C2×D411yescharacteristic
132C2×C4×C31C2×D412yes—
133C2×C4×C31C2×D412yes—
134C2×C4×C31C2411yescharacteristic
135C22⋊D31—24no—
136C22⋊D31—24no—
137C22⋊D31—24no—
138C2×Dic31—24no—
139C2×Dic31—22no—
140C2×Dic31—21no—
141C2×Dic31—22no—
142C2×Dic31—42no—
143C2×Dic31C2×D411yescharacteristic
144C2×Dic31C2×D411yescharacteristic
145D124—24no—
146D124—21no—
147D124C2×D412yes—
148C4×D31—24no—
149C4×D31—24no—
150C4×D31—22no—
151C4×D31—24no—
152C4×D31—22no—
153C4×D31—21no—
154C4×D31—21no—
155C4×D31—21no—
156C4×D31—21no—
157Dic62—24no—
158Dic62—21no—
159Dic62C2×D412yes—
160C8×C31—22no—
161C31⋊C8—22no—
162#3007—311nomaximal, Sylow
163C22×C31—21no—
164C22×C31—21no—
165C22×C31—22no—
166C22×C31—41no—
167C22×C31—41no—
168C22×C31#12012yes—
169C22×C31C22×D411yescharacteristic
170C2×D31—21no—
171C2×D31—21no—
172C2×D31—21no—
173C2×D31—22no—
174C2×D31—21no—
175C2×D31—21no—
176C2×D31—44no—
177C2×D31—48no—
178C2×D31—41no—
179C2×D31—41no—
180C2×D31#13712yescomplemented
181C4×C31—22no—
182C4×C31—42no—
183C4×C31#12012yes—
184C4×C31C22×D411yescharacteristic
185C4×C31C22×D411yescharacteristic
186Dic31—21no—
187Dic31—22no—
188Dic31—21no—
189Dic31—42no—
190C22×#41—311no—
191C2×#137—311no—
192C2×#137D3111yescharacteristic, complemented
193C2×C22⋊Q8—311no—
194#452—312no—
195#452—312no—
196#451—312no—
197#450—312no—
198#417—311no—
199#417—311no—
200#407—311no—
201C2×C31—22no—
202C2×C31—42no—
203C2×C31C2×#12011yescharacteristic
204C2×C31C2×#13711yescharacteristic
205C2×C31#57511yescharacteristic
206D31—22no—
207D31—41no—
208D31—42no—
209C2×#41—311no—
210C2×#41—311no—
211C2×#41—311no—
212C2×#41—621no—
213C2×#41—621no—
214C2×#41—624no—
215C2×#41—624no—
216C2×#41C2×D3111yescharacteristic
217C22×Q8—311no—
218C22×Q8C2×D3111yescharacteristic, complemented
219C22×D4—311no—
220C22×D4—621no—
221C23×C4—311no—
222C23×C4—621no—
223#137—24no—
224#137—624no—
225C2×Q16—312no—
226C2×Q16C2×D3112yescomplemented
227C2×SD16—312no—
228C2×SD16C2×D3112yescomplemented
229C2×M16—311no—
230C2×M16C2×D3111yescharacteristic, complemented
231C22⋊Q8—312no—
232C22⋊Q8—312no—
233C22⋊Q8—621no—
234C22⋊Q8—621no—
235C2×C4⋊C4—311no—
236C2×C4⋊C4—621no—
237C2×C22⋊C4—621no—
238#103—312no—
239#103—312no—
240#102—312no—
241#102—312no—
242C22⋊C8—312no—
243C22⋊C8—312no—
244C31#300711yescharacteristic, complemented, Sylow
245C24—311no—
246C24—621no—
247#41—22no—
248#41—314no—
249#41—628no—
250#41—622no—
251#41—628no—
252#41—622no—
253#41—622no—
254#41—622no—
255#41—622no—
256#41C22⋊D3114yes—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—311no—
261C2×Q8—621no—
262C2×Q8—622no—
263C2×Q8—622no—
264C2×Q8C22×D3112yes—
265C2×Q8C22×D3111yescharacteristic
266C2×D4—21no—
267C2×D4—311no—
268C2×D4—621no—
269C2×D4—621no—
270C2×D4—622no—
271C2×D4—628no—
272C2×D4—621no—
273C2×D4—621no—
274C2×D4—622no—
275C2×D4C22×D3111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—311no—
279C22×C4—311no—
280C22×C4—311no—
281C22×C4—311no—
282C22×C4—311no—
283C22×C4—621no—
284C22×C4—621no—
285C22×C4—628no—
286C22×C4—621no—
287C22×C4—622no—
288C22×C4—621no—
289C22×C4—622no—
290C22×C4—621no—
291C22×C4C22×D3111yescharacteristic
292Q16—24no—
293Q16—624no—
294SD16—24no—
295SD16—624no—
296M16—22no—
297M16—622no—
298C2×C8—312no—
299C2×C8C22×D3112yes—
300C4⋊C4—312no—
301C4⋊C4—622no—
302C4⋊C4—622no—
303C4⋊C4—621no—
304C22⋊C4—622no—
305C22⋊C4—622no—
306C23—21no—
307C23—311no—
308C23—311no—
309C23—622no—
310C23—624no—
311C23—622no—
312C23—621no—
313C23—1241no—
314C23—1242no—
315C23—1242no—
316C23D4×D3111yescharacteristic
317Q8—21no—
318Q8—22no—
319Q8—22no—
320Q8—42no—
321Q8—312no—
322Q8—624no—
323Q8—621no—
324Q8C2×C22⋊D3112yes—
325D4—24no—
326D4—21no—
327D4—312no—
328D4—624no—
329D4—624no—
330D4—624no—
331D4—624no—
332D4—621no—
333D4C2×C22⋊D3112yes—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—311no—
339C2×C4—311no—
340C2×C4—624no—
341C2×C4—624no—
342C2×C4—624no—
343C2×C4—622no—
344C2×C4—621no—
345C2×C4—622no—
346C2×C4—622no—
347C2×C4—624no—
348C2×C4—621no—
349C2×C4—621no—
350C2×C4—621no—
351C2×C4—621no—
352C2×C4—622no—
353C2×C4—1242no—
354C2×C4D4×D3111yescharacteristic
355C2×C4D4×D3112yes—
356C2×C4C2×C22⋊D3112yes—
357C2×C4C23×D3111yescharacteristic
358C8—22no—
359C8—622no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—312no—
366C22—621no—
367C22—621no—
368C22—622no—
369C22—621no—
370C22—621no—
371C22—621no—
372C22—1244no—
373C22—1248no—
374C22—1241no—
375C22—1241no—
376C22#1175851412yes—
377C22C2×D4×D3111yescharacteristic
378C4—22no—
379C4—42no—
380C4—621no—
381C4—621no—
382C4—622no—
383C4—1242no—
384C4#1175851412yes—
385C4C2×D4×D3111yescharacteristic
386C4C22×C22⋊D3111yescharacteristic
387C2—22no—
388C2—42no—
389C2—622no—
390C2—1241no—
391C2—1242no—
392C2C2×#1175851411yescharacteristic
393C2#137×D3111yescharacteristic
394C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#137C2
socleC22×C31C22×D4
prime radicalC2×#41×D31C2
hypercenterC2×#137D31
hypocenterC31#3007
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. C2×#11758523C2semidirect
4. #14521210C2semidirect
5. #14521208C2semidirect
6. #14521203C2semidirect
7. #14521198C2semidirect
8. #14521051C2semidirect
9. #14521050C2semidirect
10. #14521012C2semidirect
11. C2×C31×#137C2semidirect
13. C22×Q8×C31C22semidirect
27. C2×Q16×C31C22semidirect
28. C2×SD16×C31C22semidirect
29. C2×M16×C31C22semidirect
33. #11758523C22semidirect
34. C2×#92482C22semidirect
36. C2×#92481C22semidirect
38. C2×C4⋊Dic31C22semidirect
40. C2×#92474C22semidirect
41. #11758423C22semidirect
42. #11758420C22semidirect
43. #11758409C22semidirect
45. (C2×D31)⋊C8C22semidirect
46. #11758404C22semidirect
180. C2×D31#137semidirect
192. C2×#137D31semidirect
218. C22×Q8C2×D31semidirect
226. C2×Q16C2×D31semidirect
228. C2×SD16C2×D31semidirect
230. C2×M16C2×D31semidirect
244. C31#3007semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D312
C2×#117585192
C2×#117585232
#145212102
#145212082
#145212032
#145211982
#145210512
#145210502
#145210122
C2×C31×#1372
#300731