Finite Group Databasebeta

← #16848373

Subgroups — #16848373

Order 3968.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758459C211yesmaximal, characteristic
2C2×C4×D124C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D31C211yesmaximal, characteristic, complemented
4#14520950C211yesmaximal, characteristic, complemented
5#14520949C211yesmaximal, characteristic, complemented
6#14520942C211yesmaximal, characteristic, complemented
7#14520941C211yesmaximal, characteristic, complemented
8#14520940C211yesmaximal, characteristic, complemented
9#14520927C211yesmaximal, characteristic, complemented
10#14520927C211yesmaximal, characteristic, complemented
11#14520925C211yesmaximal, characteristic, complemented
12#14520920C211yesmaximal, characteristic, complemented
13#14520904C211yesmaximal, characteristic, complemented
14#14520896C211yesmaximal, characteristic, complemented
15C31×#388C211yesmaximal, characteristic, complemented
16C24×D31C2211yescharacteristic
17C22×C22⋊D31C2211yescharacteristic
18C22×D124C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C22×C4×D31C2211yescharacteristic
21C22×C4×D31C2211yescharacteristic
22C22×C4×D31C2211yescharacteristic, complemented
23C2×C4⋊C4×C31C2211yescharacteristic, complemented
24C2×C22⋊C4×C31C2211yescharacteristic
25C2×C22⋊C4×C31C2211yescharacteristic
26C2×C22⋊C4×C31C2211yescharacteristic
27C2×C42×C31C2211yescharacteristic, complemented
28C2×C22⋊Dic31C2211yescharacteristic
29C2×C22⋊Dic31C2211yescharacteristic
30C2×(C2×D31)⋊C4C2211yescharacteristic
31C2×(C2×D31)⋊C4C2211yescharacteristic
32C2×(C2×D31)⋊C4C2211yescharacteristic
33C2×(C2×D31)⋊C4C2211yescharacteristic
34C2×(C2×D31)⋊C4C2211yescharacteristic
35C2×(C2×D31)⋊C4C2211yescharacteristic
36C2×(C2×D31)⋊C4C2211yescharacteristic
37C2×(C2×D31)⋊C4C2211yescharacteristic
38C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D31)⋊C4C2211yescharacteristic, complemented
41C2×C4⋊Dic31C2211yescharacteristic, complemented
42C2×Dic31⋊C4C2211yescharacteristic, complemented
43C2×C4×Dic31C2211yescharacteristic, complemented
44C2×C4×Dic31C2211yescharacteristic, complemented
45#11758459C418yescomplemented
46C22⋊C4×D31—24no—
47C4×D124—24no—
48C31×#95C2211yescharacteristic, complemented
49C31×#95C2211yescharacteristic, complemented
50C31⋊#95C2211yescharacteristic, complemented
51C31⋊#95C2211yescharacteristic, complemented
52C31⋊#95C2211yescharacteristic, complemented
53C31⋊#95C2211yescharacteristic, complemented
54C24×C31C2311yescharacteristic
55C23×D31—22no—
56C23×D31—22no—
57C23×D31—41no—
58C23×D31—41no—
59C23×D31—41no—
60C23×D31—41no—
61C23×D31C2×C414yescomplemented
62C23×D31C2311yescharacteristic
63C23×D31C2311yescharacteristic
64C23×D31C2311yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C22×C4×C31C2311yescharacteristic
67C22×C4×C31C2311yescharacteristic
68C22×C4×C31C2311yescharacteristic
69C22×C4×C31C2311yescharacteristic
70C22×C4×C31C2311yescharacteristic
71C2×C22⋊D31—41no—
72C2×C22⋊D31—41no—
73C2×C22⋊D31C2×C414yescomplemented
74C22×Dic31C2311yescharacteristic
75C22×Dic31C2311yescharacteristic
76C22×Dic31C2311yescharacteristic
77C22×Dic31C2311yescharacteristic
78C22×Dic31C2311yescharacteristic
79C2×D124—22no—
80C2×D124—22no—
81C2×D124—41no—
82C2×D124—41no—
83C2×D124C2×C414yes—
84C2×D124C2×C414yescomplemented
85C2×C4×D31—22no—
86C2×C4×D31—22no—
87C2×C4×D31—41no—
88C2×C4×D31—41no—
89C2×C4×D31—41no—
90C2×C4×D31—41no—
91C2×C4×D31—41no—
92C2×C4×D31—41no—
93C2×C4×D31D414yescomplemented
94C4⋊C4×C31—41no—
95C22⋊C4×C31—22no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31C2×C414yescomplemented
98C42×C31—22no—
99C22⋊Dic31—22no—
100C22⋊Dic31—41no—
101(C2×D31)⋊C4—22no—
102(C2×D31)⋊C4—22no—
103(C2×D31)⋊C4—22no—
104(C2×D31)⋊C4—22no—
105(C2×D31)⋊C4—22no—
106(C2×D31)⋊C4—41no—
107(C2×D31)⋊C4—41no—
108(C2×D31)⋊C4—41no—
109(C2×D31)⋊C4—41no—
110(C2×D31)⋊C4C2×C414yescomplemented
111(C2×D31)⋊C4C2×C414yescomplemented
112C4⋊Dic31—22no—
113Dic31⋊C4—41no—
114C4×Dic31—41no—
115C4×Dic31—41no—
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—21no—
119C23×C31—21no—
120C23×C31—21no—
121C23×C31—21no—
122C23×C31C22×C412yes—
123C23×C31C2411yescharacteristic
124C22×D31—21no—
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31—21no—
129C22×D31—21no—
130C22×D31—21no—
131C22×D31—21no—
132C22×D31—21no—
133C22×D31—21no—
134C22×D31—21no—
135C22×D31—21no—
136C22×D31—21no—
137C22×D31—21no—
138C22×D31—21no—
139C22×D31—21no—
140C22×D31—28no—
141C22×D31—44no—
142C22×D31—44no—
143C22×D31—44no—
144C22×D31—44no—
145C22×D31—44no—
146C22×D31—44no—
147C22×D31C22×C412yes—
148C22×D31C22×C412yes—
149C22×D31C22×C412yes—
150C22×D31C2×D412yes—
151C22×D31#4112yes—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—21no—
165C2×C4×C31—21no—
166C2×C4×C31—21no—
167C2×C4×C31C22×C412yes—
168C2×C4×C31C22×C412yes—
169C2×C4×C31C2×D412yes—
170C22⋊D31—44no—
171C2×Dic31—21no—
172C2×Dic31—21no—
173C2×Dic31—21no—
174C2×Dic31—21no—
175C2×Dic31—21no—
176C2×Dic31—21no—
177C2×Dic31—21no—
178C2×Dic31—21no—
179C2×Dic31—21no—
180C2×Dic31—21no—
181C2×Dic31—21no—
182C2×Dic31—21no—
183C2×Dic31—21no—
184C2×Dic31C22×C412yes—
185C2×Dic31C2×D412yes—
186D124—28no—
187D124—44no—
188C4×D31—44no—
189C4×D31—44no—
190C4×D31—44no—
191#2321—311nomaximal, Sylow
192C22×C31—22no—
193C22×C31—22no—
194C22×C31—22no—
195C22×C31—41no—
196C22×C31—41no—
197C22×C31—41no—
198C22×C31—41no—
199C22×C31C23×C411yescharacteristic
200C22×C31C22×D411yescharacteristic
201C22×C31C2×#4111yescharacteristic
202C22×C31H(5,𝔽2)11yescharacteristic
203C22×C31H(5,𝔽2)11yescharacteristic
204C22×C31H(5,𝔽2)11yescharacteristic
205C22×C31H(5,𝔽2)11yescharacteristic
206C2×D31—22no—
207C2×D31—22no—
208C2×D31—22no—
209C2×D31—22no—
210C2×D31—22no—
211C2×D31—22no—
212C2×D31—22no—
213C2×D31—22no—
214C2×D31—22no—
215C2×D31—22no—
216C2×D31—22no—
217C2×D31—22no—
218C2×D31—416no—
219C2×D31—41no—
220C2×D31—41no—
221C2×D31—41no—
222C2×D31—41no—
223C2×D31—41no—
224C2×D31—41no—
225C2×D31—41no—
226C2×D31—41no—
227C2×D31C4×D414yes—
228C4×C31—22no—
229C4×C31—22no—
230C4×C31—22no—
231C4×C31—41no—
232C4×C31—41no—
233C4×C31—41no—
234Dic31—22no—
235Dic31—22no—
236Dic31—41no—
237Dic31—41no—
238Dic31—41no—
239C2×#120—311no—
240C2×C4×D4—311no—
241C22×C22⋊C4—311no—
242#390—311no—
243#390—311no—
244#388—311no—
245#388—311no—
246#388—311no—
247#388D3111yescharacteristic, complemented
248#386—311no—
249#386—311no—
250#386—311no—
251#385—311no—
252#379—311no—
253C4×C22⋊C4—311no—
254C2×C31—42no—
255C2×C31C2×C4×D411yescharacteristic
256C2×C31#51811yescharacteristic
257C2×C31#51811yescharacteristic
258C2×C31#53411yescharacteristic
259C2×C31#53511yescharacteristic
260C2×C31#53811yescharacteristic
261C2×C31#53811yescharacteristic
262D31—24no—
263D31—42no—
264D31—42no—
265C25—311no—
266C22×D4—311no—
267C22×D4—311no—
268C22×D4—311no—
269C23×C4—311no—
270C23×C4—311no—
271C23×C4—311no—
272#120—318no—
273C4×D4—624no—
274C2×C4⋊C4—311no—
275C2×C4⋊C4—311no—
276C2×C4⋊C4C2×D3111yescharacteristic, complemented
277C2×C22⋊C4—311no—
278C2×C22⋊C4—311no—
279C2×C22⋊C4—311no—
280C2×C22⋊C4—311no—
281C2×C22⋊C4—311no—
282C2×C22⋊C4—311no—
283C2×C22⋊C4—311no—
284C2×C22⋊C4—311no—
285C2×C22⋊C4—311no—
286C2×C22⋊C4—311no—
287C2×C22⋊C4—311no—
288C2×C22⋊C4—311no—
289C2×C22⋊C4—311no—
290C2×C22⋊C4—624no—
291C2×C22⋊C4C2×D3111yescharacteristic
292C2×C22⋊C4C2×D3111yescharacteristic
293C2×C22⋊C4C2×D3111yescharacteristic
294C2×C42—311no—
295C2×C42—311no—
296C2×C42C2×D3111yescharacteristic, complemented
297#95—311no—
298#95—311no—
299#95—311no—
300#95—311no—
301#95C2×D3111yescharacteristic, complemented
302#95C2×D3111yescharacteristic, complemented
303C31#232111yescharacteristic, complemented, Sylow
304C24—311no—
305C24—311no—
306C24—311no—
307C24—314no—
308C24—622no—
309C24—622no—
310C24—1241no—
311C24—1241no—
312C24—1241no—
313C24—1241no—
314C24C22×D3111yescharacteristic
315C2×D4—314no—
316C2×D4—314no—
317C2×D4—314no—
318C2×D4—622no—
319C2×D4—622no—
320C2×D4—1241no—
321C2×D4—1241no—
322C2×D4—1241no—
323C2×D4—1241no—
324C22×C4—311no—
325C22×C4—311no—
326C22×C4—314no—
327C22×C4—311no—
328C22×C4—311no—
329C22×C4—311no—
330C22×C4—622no—
331C22×C4—622no—
332C22×C4—1241no—
333C22×C4—1241no—
334C22×C4—1241no—
335C22×C4—1241no—
336C22×C4—1241no—
337C22×C4—1241no—
338C22×C4C22×D3111yescharacteristic
339C22×C4C22×D3111yescharacteristic
340C22×C4C22×D3111yescharacteristic
341C22×C4C22×D3111yescharacteristic
342C22×C4C22×D3111yescharacteristic
343C22×C4C22×D3111yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—622no—
346C4⋊C4—1241no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—314no—
350C22⋊C4—314no—
351C22⋊C4—622no—
352C22⋊C4—622no—
353C22⋊C4—622no—
354C22⋊C4—622no—
355C22⋊C4—622no—
356C22⋊C4—622no—
357C22⋊C4—1241no—
358C22⋊C4—1241no—
359C22⋊C4—1241no—
360C22⋊C4—1241no—
361C22⋊C4—1241no—
362C22⋊C4C4×D3114yescomplemented
363C42—22no—
364C42—1241no—
365C42—1241no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—312no—
373C23—312no—
374C23—312no—
375C23—312no—
376C23—312no—
377C23—628no—
378C23—621no—
379C23—621no—
380C23—621no—
381C23—621no—
382C23—621no—
383C23—621no—
384C23—621no—
385C23—621no—
386C23—621no—
387C23—621no—
388C23—621no—
389C23—621no—
390C23—621no—
391C23—621no—
392C23—621no—
393C23—621no—
394C23—1244no—
395C23—1244no—
396C23—1244no—
397C23—1244no—
398C23—1244no—
399C23—1244no—
400C23C2×C4×D3112yes—
401C23C23×D3111yescharacteristic
402D4—628no—
403D4—1244no—
404D4—1244no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—312no—
421C2×C4—312no—
422C2×C4—621no—
423C2×C4—621no—
424C2×C4—621no—
425C2×C4—621no—
426C2×C4—621no—
427C2×C4—621no—
428C2×C4—621no—
429C2×C4—621no—
430C2×C4—621no—
431C2×C4—621no—
432C2×C4—621no—
433C2×C4—621no—
434C2×C4—621no—
435C2×C4—1244no—
436C2×C4—1244no—
437C2×C4—1244no—
438C2×C4C2×C4×D3112yes—
439C2×C4C2×C4×D3112yes—
440C2×C4D4×D3112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—314no—
449C22—622no—
450C22—622no—
451C22—622no—
452C22—622no—
453C22—622no—
454C22—622no—
455C22—622no—
456C22—622no—
457C22—622no—
458C22—622no—
459C22—622no—
460C22—622no—
461C22—12416no—
462C22—1241no—
463C22—1241no—
464C22—1241no—
465C22—1241no—
466C22—1241no—
467C22—1241no—
468C22—1241no—
469C22—1241no—
470C22C22×C4×D3111yescharacteristic
471C22C2×D4×D3111yescharacteristic
472C22#1175856111yescharacteristic
473C22#41×D3111yescharacteristic
474C22#1175856611yescharacteristic
475C22#1175856611yescharacteristic
476C22#1175856611yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—622no—
484C4—622no—
485C4—1241no—
486C4—1241no—
487C4—1241no—
488C2—42no—
489C2—624no—
490C2—1242no—
491C2—1242no—
492C2C4×D4×D3111yescharacteristic
493C2#1452152111yescharacteristic
494C2#1452152311yescharacteristic
495C2#1452161011yescharacteristic
496C2#1452161511yescharacteristic
497C2#1452163311yescharacteristic
498C2#1452163811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#388C2
socle, cosocleC23×C31C24
prime radicalC2×#11758459C2
hypercenter#388D31
hypocenterC31#2321
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D124C2semidirect
3. C2×C22⋊C4×D31C2semidirect
4. #14520950C2semidirect
5. #14520949C2semidirect
6. #14520942C2semidirect
7. #14520941C2semidirect
8. #14520940C2semidirect
9. #14520927C2semidirect
10. #14520927C2semidirect
11. #14520925C2semidirect
12. #14520920C2semidirect
13. #14520904C2semidirect
14. #14520896C2semidirect
15. C31×#388C2semidirect
22. C22×C4×D31C22semidirect
23. C2×C4⋊C4×C31C22semidirect
27. C2×C42×C31C22semidirect
38. C2×(C2×D31)⋊C4C22semidirect
39. C2×(C2×D31)⋊C4C22semidirect
40. C2×(C2×D31)⋊C4C22semidirect
41. C2×C4⋊Dic31C22semidirect
42. C2×Dic31⋊C4C22semidirect
43. C2×C4×Dic31C22semidirect
44. C2×C4×Dic31C22semidirect
45. #11758459C4semidirect
48. C31×#95C22semidirect
49. C31×#95C22semidirect
50. C31⋊#95C22semidirect
51. C31⋊#95C22semidirect
52. C31⋊#95C22semidirect
53. C31⋊#95C22semidirect
61. C23×D31C2×C4semidirect
73. C2×C22⋊D31C2×C4semidirect
84. C2×D124C2×C4semidirect
93. C2×C4×D31D4semidirect
97. C22⋊C4×C31C2×C4semidirect
110. (C2×D31)⋊C4C2×C4semidirect
111. (C2×D31)⋊C4C2×C4semidirect
247. #388D31semidirect
276. C2×C4⋊C4C2×D31semidirect
296. C2×C42C2×D31semidirect
301. #95C2×D31semidirect
302. #95C2×D31semidirect
303. C31#2321semidirect
362. C22⋊C4C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584592
C2×C4×D1242
C2×C22⋊C4×D312
#145209502
#145209492
#145209422
#145209412
#145209402
#145209272
#145209252
#145209202
#145209042
#145208962
C31×#3882
#232131