Finite Group Databasebeta

← C2×C31×#1512 #21474312

Subgroups — C2×C31×#1512 #21474312

Order 7936.

1098 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#529C211yesmaximal, characteristic
2C2×C31×#432C211yesmaximal, characteristic, complemented
3C2×C31×#382C211yesmaximal, characteristic, complemented
4C31×#1512C218yesmaximal, direct product factor
5C2×C31×#328C212yesmaximal, complemented
6C2×C31×#327C212yesmaximal
7C31×#529—22no—
8C31×#529—22no—
9C31×#529C414yescomplemented
10C31×#529C2214yes—
11C2×C31×#126—21no—
12C2×C31×#125C411yescharacteristic, complemented
13C2×C31×#124C411yescharacteristic, complemented
14C2×C31×#123—21no—
15C2×C31×C22⋊Q8C2211yescharacteristic
16C2×C31×#121C2211yescharacteristic
17C2×C4×Q8×C31C411yescharacteristic, complemented
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31C411yescharacteristic, complemented
20C2×C31×#117—21no—
21C22×C4⋊C4×C31—21no—
22C22×C42×C31C2211yescharacteristic
23C31×#432—22no—
24C31×#432C2214yescomplemented
25C2×C31×C4⋊C8—21no—
26C2×C31×C22⋊C8C2212yes—
27C2×C31×#97—21no—
28C31×#382—22no—
29C31×#382C2214yescomplemented
30C2×C31×#95—21no—
31C2×C31×#95C2212yes—
32C31×#328C2218yescomplemented
33C31×#327C2218yes—
34C22×Q8×C31C2×C411yescharacteristic
35C22×D4×C31—21no—
36C22×D4×C31C2×C411yescharacteristic
37C23×C4×C31—21no—
38C23×C4×C31—22no—
39C23×C4×C31D411yescharacteristic
40C23×C4×C31D411yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C22×C8×C31—22no—
43C31×#126—24no—
44C31×#126—24no—
45C31×#125—22no—
46C31×#125C2×C414yescomplemented
47C31×#124—22no—
48C31×#124C2×C414yescomplemented
49C31×#123—24no—
50C31×#123—24no—
51C31×C22⋊Q8—21no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8C2×C412yes—
54C31×C22⋊Q8C2312yes—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121C2×C412yes—
58C31×#121C2312yes—
59C4×Q8×C31—22no—
60C4×Q8×C31C2×C414yescomplemented
61C4×D4×C31—24no—
62C4×D4×C31—24no—
63C4×D4×C31—22no—
64C4×D4×C31C2×C414yescomplemented
65C31×#117—24no—
66C31×#117—24no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—24no—
72C2×C4⋊C4×C31—44no—
73C2×C4⋊C4×C31C2×C411yescharacteristic
74C2×C4⋊C4×C31C2×C411yescharacteristic
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—21no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C42×C31—21no—
81C2×C42×C31C2×C411yescharacteristic
82C2×C42×C31C2×C411yescharacteristic
83C2×C42×C31C2×C412yes—
84C2×C42×C31D412yes—
85C2×C42×C31D411yescharacteristic
86C2×C42×C31D411yescharacteristic
87C2×C42×C31D412yes—
88C2×C42×C31C2312yes—
89C31×C4⋊C8—24no—
90C31×C22⋊C8—22no—
91C31×C22⋊C8C2314yes—
92C31×#97—24no—
93C31×#95—24no—
94C31×#95—22no—
95C31×#95C2314yes—
96C24×C31—21no—
97C24×C31C2×D411yescharacteristic
98C2×Q8×C31—21no—
99C2×Q8×C31—42no—
100C2×Q8×C31C22×C412yes—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—21no—
104C2×D4×C31—42no—
105C2×D4×C31—42no—
106C2×D4×C31—42no—
107C2×D4×C31C22×C412yes—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—24no—
114C22×C4×C31—24no—
115C22×C4×C31—42no—
116C22×C4×C31—42no—
117C22×C4×C31—44no—
118C22×C4×C31—44no—
119C22×C4×C31—44no—
120C22×C4×C31—44no—
121C22×C4×C31C22⋊C412yes—
122C22×C4×C31C22⋊C411yescharacteristic
123C22×C4×C31C22⋊C411yescharacteristic
124C22×C4×C31C22⋊C411yescharacteristic
125C22×C4×C31C22⋊C411yescharacteristic
126C22×C4×C31C22⋊C412yes—
127C22×C4×C31C22×C411yescharacteristic
128C22×C4×C31C22×C411yescharacteristic
129C22×C4×C31C2×D412yes—
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2411yescharacteristic
135C2×C8×C31—24no—
136C2×C8×C31—44no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—48no—
145C4⋊C4×C31C22×C412yes—
146C4⋊C4×C31C22×C412yes—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—22no—
154C42×C31—24no—
155C42×C31C22⋊C412yes—
156C42×C31C22⋊C412yes—
157C42×C31C22⋊C412yes—
158C42×C31C22⋊C412yes—
159C42×C31C22×C412yes—
160C42×C31C22×C412yes—
161C42×C31C2×D412yes—
162C42×C31C2×D412yes—
163C2×#1512C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—22no—
167C23×C31—42no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31C2×C22⋊C411yescharacteristic
171C23×C31C2×C22⋊C411yescharacteristic
172C23×C31AGL(1,ℤ/8)12yes—
173C23×C31#13712yes—
174C23×C31C22×D411yescharacteristic
175Q8×C31—42no—
176Q8×C31—42no—
177D4×C31—42no—
178D4×C31—42no—
179D4×C31—42no—
180D4×C31—42no—
181D4×C31—44no—
182C2×C4×C31—22no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—24no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—42no—
195C2×C4×C31—41no—
196C2×C4×C31—41no—
197C2×C4×C31—41no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—48no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—44no—
205C2×C4×C31—44no—
206C2×C4×C31—44no—
207C2×C4×C31—88no—
208C2×C4×C31C2×C22⋊C412yes—
209C2×C4×C31C2×C22⋊C411yescharacteristic
210C2×C4×C31C2×C22⋊C411yescharacteristic
211C2×C4×C31C2×C22⋊C412yes—
212C2×C4×C31C2×C22⋊C412yes—
213C2×C4×C31C2×C22⋊C412yes—
214C2×C4×C31C23×C411yescharacteristic
215C2×C4×C31C22×D411yescharacteristic
216C8×C31—44no—
217C2×#529C2×C3111yescharacteristic
218C2×#432C2×C3111yescharacteristic, complemented
219C2×#382C2×C3111yescharacteristic, complemented
220#1512C2×C3118yesdirect product factor
221C2×#328C2×C3112yescomplemented
222C2×#327C2×C3112yes—
223C22×C31—21no—
224C22×C31—21no—
225C22×C31—21no—
226C22×C31—21no—
227C22×C31—24no—
228C22×C31—24no—
229C22×C31—42no—
230C22×C31—42no—
231C22×C31—42no—
232C22×C31—84no—
233C22×C31#41012yes—
234C22×C31#41712yes—
235C22×C31#41712yes—
236C22×C31#42112yes—
237C22×C31C22×C22⋊C411yescharacteristic
238C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
239C22×C31C2×#13711yescharacteristic
240C4×C31—21no—
241C4×C31—21no—
242C4×C31—21no—
243C4×C31—21no—
244C4×C31—22no—
245C4×C31—22no—
246C4×C31—22no—
247C4×C31—22no—
248C4×C31—42no—
249C4×C31—42no—
250C4×C31—42no—
251C4×C31—84no—
252#529—22no—
253#529—22no—
254#529C4×C3114yescomplemented
255#529C22×C3114yes—
256C2×#126—21no—
257C2×#125C4×C3111yescharacteristic, complemented
258C2×#124C4×C3111yescharacteristic, complemented
259C2×#123—21no—
260C2×C22⋊Q8C22×C3111yescharacteristic
261C2×#121C22×C3111yescharacteristic
262C2×C4×Q8C4×C3111yescharacteristic, complemented
263C2×C4×D4—21no—
264C2×C4×D4C4×C3111yescharacteristic, complemented
265C2×#117—21no—
266C22×C4⋊C4—21no—
267C22×C42C22×C3111yescharacteristic
268#432—22no—
269#432C22×C3114yescomplemented
270C2×C4⋊C8—21no—
271C2×C22⋊C8C22×C3112yes—
272C2×#97—21no—
273#382—22no—
274#382C22×C3114yescomplemented
275C2×#95—21no—
276C2×#95C22×C3112yes—
277#328C22×C3118yescomplemented
278#327C22×C3118yes—
279C2×C31—21no—
280C2×C31—21no—
281C2×C31—22no—
282C2×C31—82no—
283C2×C31#151214yesdirect product factor
284C2×C31C2×#41011yescharacteristic
285C2×C31C2×#41711yescharacteristic
286C2×C31C2×#42111yescharacteristic
287C22×Q8C2×C4×C3111yescharacteristic
288C22×D4—21no—
289C22×D4C2×C4×C3111yescharacteristic
290C23×C4—21no—
291C23×C4—22no—
292C23×C4D4×C3111yescharacteristic
293C23×C4D4×C3111yescharacteristic
294C23×C4C23×C3111yescharacteristic
295C22×C8—22no—
296#126—24no—
297#126—24no—
298#125—22no—
299#125C2×C4×C3114yescomplemented
300#124—22no—
301#124C2×C4×C3114yescomplemented
302#123—24no—
303#123—24no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C22⋊Q8C2×C4×C3112yes—
307C22⋊Q8C23×C3112yes—
308#121—21no—
309#121—21no—
310#121C2×C4×C3112yes—
311#121C23×C3112yes—
312C4×Q8—22no—
313C4×Q8C2×C4×C3114yescomplemented
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—22no—
317C4×D4C2×C4×C3114yescomplemented
318#117—24no—
319#117—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×C4×C3111yescharacteristic
327C2×C4⋊C4C2×C4×C3111yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42C2×C4×C3111yescharacteristic
335C2×C42C2×C4×C3111yescharacteristic
336C2×C42C2×C4×C3112yes—
337C2×C42D4×C3112yes—
338C2×C42D4×C3111yescharacteristic
339C2×C42D4×C3111yescharacteristic
340C2×C42D4×C3112yes—
341C2×C42C23×C3112yes—
342C4⋊C8—24no—
343C22⋊C8—22no—
344C22⋊C8C23×C3114yes—
345#97—24no—
346#95—24no—
347#95—22no—
348#95C23×C3114yes—
349C31C2×#151211yescharacteristic, direct product factor, Sylow
350C24—21no—
351C24C2×D4×C3111yescharacteristic
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8C22×C4×C3112yes—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4C22×C4×C3112yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C22⋊C4×C3111yescharacteristic
376C22×C4C22⋊C4×C3111yescharacteristic
377C22×C4C22⋊C4×C3111yescharacteristic
378C22×C4C22⋊C4×C3111yescharacteristic
379C22×C4C22⋊C4×C3112yes—
380C22×C4C22⋊C4×C3112yes—
381C22×C4C22×C4×C3111yescharacteristic
382C22×C4C22×C4×C3111yescharacteristic
383C22×C4C2×D4×C3112yes—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3112yes—
387C22×C4C2×D4×C3111yescharacteristic
388C22×C4C24×C3111yescharacteristic
389C2×C8—24no—
390C2×C8—44no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—48no—
399C4⋊C4C22×C4×C3112yes—
400C4⋊C4C22×C4×C3112yes—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C42—24no—
409C42C22⋊C4×C3112yes—
410C42C22⋊C4×C3112yes—
411C42C22⋊C4×C3112yes—
412C42C22⋊C4×C3112yes—
413C42C22×C4×C3112yes—
414C42C22×C4×C3112yes—
415C42C2×D4×C3112yes—
416C42C2×D4×C3112yes—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C2×C22⋊C4×C3111yescharacteristic
424C23C2×C22⋊C4×C3111yescharacteristic
425C23C31×AGL(1,ℤ/8)12yes—
426C23C31×#13712yes—
427C23C22×D4×C3111yescharacteristic
428Q8—42no—
429Q8—42no—
430D4—42no—
431D4—42no—
432D4—44no—
433D4—42no—
434D4—42no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—88no—
461C2×C4C2×C22⋊C4×C3112yes—
462C2×C4C2×C22⋊C4×C3111yescharacteristic
463C2×C4C2×C22⋊C4×C3111yescharacteristic
464C2×C4C2×C22⋊C4×C3112yes—
465C2×C4C2×C22⋊C4×C3112yes—
466C2×C4C2×C22⋊C4×C3112yes—
467C2×C4C23×C4×C3111yescharacteristic
468C2×C4C22×D4×C3111yescharacteristic
469C8—44no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—24no—
475C22—24no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—84no—
480C22C31×#41012yes—
481C22C31×#41712yes—
482C22C31×#41712yes—
483C22C31×#42112yes—
484C22C22×C22⋊C4×C3111yescharacteristic
485C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
486C22C2×C31×#13711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—84no—
499C2—21no—
500C2—22no—
501C2—21no—
502C2—82no—
503C2C31×#151214yesdirect product factor
504C2C2×C31×#41011yescharacteristic
505C2C2×C31×#41711yescharacteristic
506C2C2×C31×#42111yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#432C2semidirect
3. C2×C31×#382C2semidirect
4. C31×#1512C2direct (N × H)
5. C2×C31×#328C2semidirect
9. C31×#529C4semidirect
12. C2×C31×#125C4semidirect
13. C2×C31×#124C4semidirect
17. C2×C4×Q8×C31C4semidirect
19. C2×C4×D4×C31C4semidirect
24. C31×#432C22semidirect
29. C31×#382C22semidirect
32. C31×#328C22semidirect
46. C31×#125C2×C4semidirect
48. C31×#124C2×C4semidirect
60. C4×Q8×C31C2×C4semidirect
64. C4×D4×C31C2×C4semidirect
163. C2×#1512C31direct (N × H)
218. C2×#432C2×C31semidirect
219. C2×#382C2×C31semidirect
220. #1512C2×C31direct (N × H)
221. C2×#328C2×C31semidirect
254. #529C4×C31semidirect
257. C2×#125C4×C31semidirect
258. C2×#124C4×C31semidirect
262. C2×C4×Q8C4×C31semidirect
264. C2×C4×D4C4×C31semidirect
269. #432C22×C31semidirect
274. #382C22×C31semidirect
277. #328C22×C31semidirect
283. C2×C31#1512direct (N × H)
299. #125C2×C4×C31semidirect
301. #124C2×C4×C31semidirect
313. C4×Q8C2×C4×C31semidirect
317. C4×D4C2×C4×C31semidirect
349. C31C2×#1512direct (N × H)
503. C2C31×#1512direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5292
C2×C31×#4322
C2×C31×#3822
C31×#15122
C2×C31×#3282
C2×C31×#3272
C2×#151231