Finite Group Databasebeta

← #16848408

Subgroups — #16848408

Order 3968.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758461C211yesmaximal, characteristic
2C2×#11758470C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D31C211yesmaximal, characteristic, complemented
4#14520942C211yesmaximal, characteristic, complemented
5#14520941C211yesmaximal, characteristic, complemented
6#14520931C211yesmaximal, characteristic, complemented
7#14520930C211yesmaximal, characteristic, complemented
8#14520928C211yesmaximal, characteristic, complemented
9#14520927C211yesmaximal, characteristic, complemented
10#14520925C211yesmaximal, characteristic, complemented
11#14520925C211yesmaximal, characteristic, complemented
12#14520920C211yesmaximal, characteristic, complemented
13#14520906C211yesmaximal, characteristic, complemented
14#95×D31C211yesmaximal, characteristic, complemented
15C31×#385C211yesmaximal, characteristic, complemented
16C23×C4×C31C2211yescharacteristic
17C22×C22⋊D31C2211yescharacteristic
18C22×C22⋊D31C2211yescharacteristic
19C22×D124C2211yescharacteristic
20C22×C4×D31C2211yescharacteristic
21C22×C4×D31C2211yescharacteristic
22C22×C4×D31C2211yescharacteristic
23C22×C4×D31C2211yescharacteristic
24C2×C4⋊C4×C31C2211yescharacteristic
25C2×C4⋊C4×C31C2211yescharacteristic
26C2×C22⋊C4×C31C2211yescharacteristic
27C2×C22⋊C4×C31C2211yescharacteristic
28C2×C22⋊Dic31C2211yescharacteristic
29C2×C22⋊Dic31C2211yescharacteristic
30C4×C22⋊D31—24no—
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×C4⋊Dic31C2211yescharacteristic
38C2×Dic31⋊C4C2211yescharacteristic
39C2×Dic31⋊C4C2211yescharacteristic
40C2×C4×Dic31C2211yescharacteristic
41C2×C4×Dic31C2211yescharacteristic
42#11758470—24no—
43#11758461C418yescomplemented
44C31×#95C2211yescharacteristic
45C31×#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C31⋊#95C2211yescharacteristic
53C31⋊#95C2211yescharacteristic
54C24×C31C2311yescharacteristic
55C23×D31C2311yescharacteristic
56C23×D31C2311yescharacteristic
57C22×C4×C31—22no—
58C22×C4×C31—41no—
59C22×C4×C31—41no—
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C22×C4×C31C2311yescharacteristic
66C2×C22⋊D31—22no—
67C2×C22⋊D31—22no—
68C2×C22⋊D31—41no—
69C2×C22⋊D31—41no—
70C2×C22⋊D31C2×C414yescomplemented
71C2×C22⋊D31C2×C414yescomplemented
72C22×Dic31C2311yescharacteristic
73C22×Dic31C2311yescharacteristic
74C22×Dic31C2311yescharacteristic
75C22×Dic31C2311yescharacteristic
76C22×Dic31C2311yescharacteristic
77C22×Dic31C2311yescharacteristic
78C2×D124—22no—
79C2×D124—22no—
80C2×D124C2×C414yescomplemented
81C2×C4×D31—22no—
82C2×C4×D31—22no—
83C2×C4×D31—41no—
84C2×C4×D31—41no—
85C2×C4×D31—41no—
86C2×C4×D31—41no—
87C2×C4×D31—41no—
88C2×C4×D31—41no—
89C2×C4×D31—41no—
90C2×C4×D31—41no—
91C2×C4×D31C2×C414yescomplemented
92C2×C4×D31D414yescomplemented
93C4⋊C4×C31—22no—
94C4⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31C2×C414yescomplemented
97C22⋊Dic31—22no—
98C22⋊Dic31—41no—
99(C2×D31)⋊C4—22no—
100(C2×D31)⋊C4—22no—
101(C2×D31)⋊C4—22no—
102(C2×D31)⋊C4—41no—
103(C2×D31)⋊C4—41no—
104(C2×D31)⋊C4C2×C414yescomplemented
105C4⋊Dic31—41no—
106Dic31⋊C4—22no—
107Dic31⋊C4C2×C414yescomplemented
108C4×Dic31—22no—
109C4×Dic31—22no—
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31C22×C412yes—
117C23×C31C2411yescharacteristic
118C22×D31—21no—
119C22×D31—21no—
120C22×D31—21no—
121C22×D31—21no—
122C22×D31—21no—
123C22×D31—21no—
124C22×D31—21no—
125C22×D31—21no—
126C22×D31—21no—
127C22×D31—21no—
128C22×D31C22×C412yes—
129C22×D31C22×C412yes—
130C22×D31C2×D412yes—
131C22×D31#4112yes—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—44no—
148C2×C4×C31C22×C412yes—
149C2×C4×C31C22×C412yes—
150C2×C4×C31C2×D412yes—
151C22⋊D31—28no—
152C22⋊D31—44no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—21no—
156C2×Dic31—21no—
157C2×Dic31—21no—
158C2×Dic31—21no—
159C2×Dic31—21no—
160C2×Dic31—21no—
161C2×Dic31—21no—
162C2×Dic31—21no—
163C2×Dic31—21no—
164C2×Dic31—21no—
165C2×Dic31—21no—
166C2×Dic31—21no—
167C2×Dic31—21no—
168C2×Dic31C22×C412yes—
169C2×Dic31C22×C412yes—
170C2×Dic31C2×D412yes—
171D124—28no—
172C4×D31—44no—
173C4×D31—44no—
174C4×D31—44no—
175C4×D31—44no—
176#2325—311nomaximal, Sylow
177C22×C31—22no—
178C22×C31—22no—
179C22×C31—22no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31C23×C411yescharacteristic
185C22×C31C22×D411yescharacteristic
186C22×C31C2×#4111yescharacteristic
187C22×C31H(5,𝔽2)11yescharacteristic
188C22×C31H(5,𝔽2)11yescharacteristic
189C22×C31H(5,𝔽2)11yescharacteristic
190C22×C312−1+411yescharacteristic
191C2×D31—22no—
192C2×D31—22no—
193C2×D31—22no—
194C2×D31—22no—
195C2×D31—22no—
196C2×D31—22no—
197C2×D31—22no—
198C2×D31—22no—
199C2×D31—22no—
200C2×D31—41no—
201C2×D31—41no—
202C2×D31—41no—
203C2×D31—41no—
204C2×D31C4×D414yes—
205C4×C31—22no—
206C4×C31—22no—
207C4×C31—22no—
208C4×C31—41no—
209C4×C31—41no—
210C4×C31—41no—
211Dic31—22no—
212Dic31—22no—
213Dic31—22no—
214Dic31—41no—
215Dic31—41no—
216Dic31—41no—
217C2×#121—311no—
218C2×C4×D4—311no—
219C2×C4×D4—311no—
220#388—311no—
221#388—311no—
222#387—311no—
223#387—311no—
224#386—311no—
225#386—311no—
226#386—311no—
227#386—311no—
228#385—311no—
229#385D3111yescharacteristic, complemented
230#380—311no—
231C2×#95—311no—
232C2×C31—42no—
233C2×C31C2×C4×D411yescharacteristic
234C2×C31#51811yescharacteristic
235C2×C31#52011yescharacteristic
236C2×C31#53411yescharacteristic
237C2×C31#53711yescharacteristic
238C2×C31#53811yescharacteristic
239C2×C31#53911yescharacteristic
240D31—24no—
241D31—42no—
242C22×D4—311no—
243C22×D4—311no—
244C22×D4—311no—
245C23×C4—311no—
246C23×C4—311no—
247C23×C4—311no—
248C23×C4—311no—
249C23×C4C2×D3111yescharacteristic
250#121—318no—
251C4×D4—624no—
252C4×D4—624no—
253C2×C4⋊C4—311no—
254C2×C4⋊C4—311no—
255C2×C4⋊C4—311no—
256C2×C4⋊C4C2×D3111yescharacteristic
257C2×C4⋊C4C2×D3111yescharacteristic
258C2×C22⋊C4—311no—
259C2×C22⋊C4—311no—
260C2×C22⋊C4—311no—
261C2×C22⋊C4—311no—
262C2×C22⋊C4—311no—
263C2×C22⋊C4—311no—
264C2×C22⋊C4—311no—
265C2×C22⋊C4—311no—
266C2×C22⋊C4C2×D3111yescharacteristic
267C2×C22⋊C4C2×D3111yescharacteristic
268C2×C42—311no—
269C2×C42—311no—
270#95—311no—
271#95—311no—
272#95—311no—
273#95—311no—
274#95—311no—
275#95—311no—
276#95—311no—
277#95—311no—
278#95C2×D3111yescharacteristic
279#95C2×D3111yescharacteristic
280C31#232511yescharacteristic, complemented, Sylow
281C24—311no—
282C24—311no—
283C24C22×D3111yescharacteristic
284C2×D4—314no—
285C2×D4—314no—
286C2×D4—314no—
287C2×D4—622no—
288C2×D4—622no—
289C2×D4—622no—
290C2×D4—622no—
291C2×D4—1241no—
292C2×D4—1241no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—311no—
297C22×C4—311no—
298C22×C4—311no—
299C22×C4—314no—
300C22×C4—314no—
301C22×C4—311no—
302C22×C4—311no—
303C22×C4—311no—
304C22×C4—622no—
305C22×C4—622no—
306C22×C4—1241no—
307C22×C4—1241no—
308C22×C4—1241no—
309C22×C4—1241no—
310C22×C4—1241no—
311C22×C4—1241no—
312C22×C4—1241no—
313C22×C4—1241no—
314C22×C4C22×D3111yescharacteristic
315C22×C4C22×D3111yescharacteristic
316C22×C4C22×D3111yescharacteristic
317C22×C4C22×D3111yescharacteristic
318C22×C4C22×D3111yescharacteristic
319C22×C4C22×D3111yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—314no—
323C4⋊C4—622no—
324C4⋊C4—1241no—
325C22⋊C4—41no—
326C22⋊C4—314no—
327C22⋊C4—622no—
328C22⋊C4—622no—
329C22⋊C4—622no—
330C22⋊C4—622no—
331C22⋊C4—1241no—
332C22⋊C4—1241no—
333C22⋊C4—1241no—
334C22⋊C4C4×D3114yescomplemented
335C42—622no—
336C42—622no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—312no—
344C23—312no—
345C23—312no—
346C23—312no—
347C23—621no—
348C23—621no—
349C23—621no—
350C23—621no—
351C23—621no—
352C23—621no—
353C23—621no—
354C23—621no—
355C23—621no—
356C23—621no—
357C23C2×C4×D3112yes—
358C23C23×D3111yescharacteristic
359D4—628no—
360D4—628no—
361D4—1244no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—312no—
379C2×C4—312no—
380C2×C4—312no—
381C2×C4—621no—
382C2×C4—621no—
383C2×C4—621no—
384C2×C4—621no—
385C2×C4—621no—
386C2×C4—621no—
387C2×C4—621no—
388C2×C4—621no—
389C2×C4—621no—
390C2×C4—621no—
391C2×C4—621no—
392C2×C4—621no—
393C2×C4—621no—
394C2×C4—621no—
395C2×C4—621no—
396C2×C4—1244no—
397C2×C4—1244no—
398C2×C4—1244no—
399C2×C4—1244no—
400C2×C4C2×C4×D3112yes—
401C2×C4C2×C4×D3112yes—
402C2×C4D4×D3112yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—314no—
411C22—622no—
412C22—622no—
413C22—622no—
414C22—622no—
415C22—622no—
416C22—622no—
417C22—622no—
418C22—622no—
419C22—622no—
420C22—1241no—
421C22—1241no—
422C22—1241no—
423C22—1241no—
424C22C22×C4×D3111yescharacteristic
425C22C2×D4×D3111yescharacteristic
426C22#1175856111yescharacteristic
427C22#1175856111yescharacteristic
428C22#1175856411yescharacteristic
429C22#41×D3111yescharacteristic
430C22#1175856611yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—622no—
438C4—622no—
439C4—622no—
440C4—1241no—
441C4—1241no—
442C4—1241no—
443C2—42no—
444C2—624no—
445C2—1242no—
446C2C4×D4×D3111yescharacteristic
447C2#1452152111yescharacteristic
448C2#1452153011yescharacteristic
449C2#1452160911yescharacteristic
450C2#1452162911yescharacteristic
451C2#1452163511yescharacteristic
452C2#1452164511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#385C2
socle, cosocleC23×C31C24
prime radicalC2×#11758461C2
hypercenter#385D31
hypocenterC31#2325
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#11758470C2semidirect
3. C2×C4×C22⋊D31C2semidirect
4. #14520942C2semidirect
5. #14520941C2semidirect
6. #14520931C2semidirect
7. #14520930C2semidirect
8. #14520928C2semidirect
9. #14520927C2semidirect
10. #14520925C2semidirect
11. #14520925C2semidirect
12. #14520920C2semidirect
13. #14520906C2semidirect
14. #95×D31C2semidirect
15. C31×#385C2semidirect
43. #11758461C4semidirect
70. C2×C22⋊D31C2×C4semidirect
71. C2×C22⋊D31C2×C4semidirect
80. C2×D124C2×C4semidirect
91. C2×C4×D31C2×C4semidirect
92. C2×C4×D31D4semidirect
96. C22⋊C4×C31C2×C4semidirect
104. (C2×D31)⋊C4C2×C4semidirect
107. Dic31⋊C4C2×C4semidirect
229. #385D31semidirect
280. C31#2325semidirect
334. C22⋊C4C4×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117584612
C2×#117584702
C2×C4×C22⋊D312
#145209422
#145209412
#145209312
#145209302
#145209282
#145209272
#145209252
#145209202
#145209062
#95×D312
C31×#3852
#232531