Finite Group Databasebeta

← C2×C31×#1596 #21474855

Subgroups — C2×C31×#1596 #21474855

Order 7936.

1696 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#543C211yesmaximal, characteristic, complemented
2C2×C31×#450C212yesmaximal, complemented
3C2×C31×#392C211yesmaximal, characteristic
4C31×#1596C218yesmaximal, direct product factor
5C2×C31×#328C212yesmaximal, complemented
6C2×C31×#323C211yesmaximal, characteristic, complemented
7C23×D4×C31—21no—
8C23×D4×C31C2211yescharacteristic
9C22×SD16×C31—22no—
10C31×#543—24no—
11C31×#543—22no—
12C31×#543C2214yescomplemented
13C2×C31×C22⋊Q8—22no—
14C2×C31×C22⋊Q8—21no—
15C2×C31×C22⋊Q8—22no—
16C2×C31×C22⋊Q8C2212yes—
17C22×C22⋊C4×C31—21no—
18C22×C22⋊C4×C31—21no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C31×#450—24no—
21C31×#450C2218yescomplemented
22C2×C31×#102—22no—
23C2×C31×C22⋊C8C2212yes—
24C31×#392—22no—
25C31×#392C2214yes—
26C2×C31×#95C2211yescharacteristic
27C31×#328C2218yescomplemented
28C31×#323C2214yescomplemented
29C25×C31—21no—
30C25×C31D411yescharacteristic
31C22×Q8×C31—21no—
32C22×Q8×C31D412yescomplemented
33C22×D4×C31—24no—
34C22×D4×C31—22no—
35C22×D4×C31—22no—
36C22×D4×C31—21no—
37C22×D4×C31—44no—
38C22×D4×C31—44no—
39C22×D4×C31—44no—
40C22×D4×C31—44no—
41C22×D4×C31C2312yes—
42C23×C4×C31—21no—
43C23×C4×C31—21no—
44C23×C4×C31—21no—
45C23×C4×C31D411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C2×SD16×C31—28no—
48C2×SD16×C31—48no—
49C22×C8×C31—22no—
50C31×C22⋊Q8—28no—
51C31×C22⋊Q8—28no—
52C31×C22⋊Q8—24no—
53C31×C22⋊Q8—28no—
54C31×C22⋊Q8—24no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—28no—
58C31×C22⋊Q8D414yescomplemented
59C31×C22⋊Q8C2314yes—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31D412yescomplemented
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—44no—
72C2×C22⋊C4×C31D412yes—
73C2×C22⋊C4×C31C2312yes—
74C31×#102—28no—
75C31×C22⋊C8—22no—
76C31×C22⋊C8C2314yes—
77C31×#95—21no—
78C31×#95C2312yes—
79C24×C31—21no—
80C24×C31—21no—
81C24×C31—22no—
82C24×C31—24no—
83C24×C31—22no—
84C24×C31—42no—
85C24×C31—42no—
86C24×C31—42no—
87C24×C31—42no—
88C24×C31—42no—
89C24×C31—41no—
90C24×C31—41no—
91C24×C31—42no—
92C24×C31—42no—
93C24×C31SD1614yescomplemented
94C24×C31C2×D411yescharacteristic
95C24×C31C2×D412yes—
96C2×Q8×C31—24no—
97C2×Q8×C31—22no—
98C2×Q8×C31—44no—
99C2×Q8×C31—82no—
100C2×Q8×C31C2×D414yescomplemented
101C2×D4×C31—28no—
102C2×D4×C31—24no—
103C2×D4×C31—44no—
104C2×D4×C31—48no—
105C2×D4×C31—44no—
106C2×D4×C31—48no—
107C2×D4×C31—44no—
108C2×D4×C31—48no—
109C2×D4×C31—432no—
110C2×D4×C31—48no—
111C2×D4×C31—42no—
112C22×C4×C31—24no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—24no—
118C22×C4×C31—24no—
119C22×C4×C31—21no—
120C22×C4×C31—42no—
121C22×C4×C31—42no—
122C22×C4×C31—42no—
123C22×C4×C31—42no—
124C22×C4×C31—82no—
125C22×C4×C31—82no—
126C22×C4×C31C2×D412yes—
127C22×C4×C31C2×D412yes—
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D412yes—
130C22×C4×C31C2411yescharacteristic
131SD16×C31—416no—
132C2×C8×C31—24no—
133C2×C8×C31—44no—
134C4⋊C4×C31—24no—
135C4⋊C4×C31—28no—
136C4⋊C4×C31—24no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31C2×D414yescomplemented
140C22⋊C4×C31—28no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—24no—
144C22⋊C4×C31—24no—
145C22⋊C4×C31—48no—
146C2×#1596C3111yesmaximal, characteristic, direct product factor, Sylow
147C23×C31—22no—
148C23×C31—22no—
149C23×C31—22no—
150C23×C31—24no—
151C23×C31—22no—
152C23×C31—24no—
153C23×C31—24no—
154C23×C31—48no—
155C23×C31—42no—
156C23×C31—42no—
157C23×C31—42no—
158C23×C31—48no—
159C23×C31—42no—
160C23×C31—42no—
161C23×C31—44no—
162C23×C31—48no—
163C23×C31—42no—
164C23×C31—42no—
165C23×C31—42no—
166C23×C31—42no—
167C23×C31—42no—
168C23×C31—44no—
169C23×C31—84no—
170C23×C31—81no—
171C23×C31—81no—
172C23×C31—84no—
173C23×C31—84no—
174C23×C31—82no—
175C23×C31—82no—
176C23×C31—82no—
177C23×C31—82no—
178C23×C31#12011yescharacteristic
179C23×C31#12011yescharacteristic
180C23×C31C2×SD1612yes—
181C23×C31C2×SD1614yescomplemented
182C23×C31#13712yes—
183C23×C31C22×D411yescharacteristic
184Q8×C31—44no—
185Q8×C31—82no—
186Q8×C31—84no—
187D4×C31—28no—
188D4×C31—44no—
189D4×C31—48no—
190D4×C31—432no—
191C2×C4×C31—24no—
192C2×C4×C31—24no—
193C2×C4×C31—24no—
194C2×C4×C31—22no—
195C2×C4×C31—24no—
196C2×C4×C31—42no—
197C2×C4×C31—41no—
198C2×C4×C31—41no—
199C2×C4×C31—48no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—44no—
204C2×C4×C31—44no—
205C2×C4×C31—42no—
206C2×C4×C31—81no—
207C2×C4×C31—84no—
208C2×C4×C31—84no—
209C2×C4×C31#12012yes—
210C2×C4×C31C22×D412yes—
211C8×C31—44no—
212C2×#543C2×C3111yescharacteristic, complemented
213C2×#450C2×C3112yescomplemented
214C2×#392C2×C3111yescharacteristic
215#1596C2×C3118yesdirect product factor
216C2×#328C2×C3112yescomplemented
217C2×#323C2×C3111yescharacteristic, complemented
218C22×C31—24no—
219C22×C31—24no—
220C22×C31—21no—
221C22×C31—21no—
222C22×C31—21no—
223C22×C31—21no—
224C22×C31—24no—
225C22×C31—44no—
226C22×C31—44no—
227C22×C31—44no—
228C22×C31—48no—
229C22×C31—42no—
230C22×C31—42no—
231C22×C31—44no—
232C22×C31—82no—
233C22×C31—84no—
234C22×C31—82no—
235C22×C31—82no—
236C22×C31—88no—
237C22×C31—88no—
238C22×C31—82no—
239C22×C31—82no—
240C22×C31#44812yes—
241C22×C31#44812yes—
242C22×C31#45312yes—
243C22×C31B(4,𝔽2)12yes—
244C22×C31C2×#12011yescharacteristic
245C22×C31C22×SD1611yescharacteristic
246C22×C31C2×#13711yescharacteristic
247C4×C31—22no—
248C4×C31—22no—
249C4×C31—44no—
250C4×C31—42no—
251C4×C31—44no—
252C4×C31—44no—
253C23×D4—21no—
254C23×D4C22×C3111yescharacteristic
255C22×SD16—22no—
256#543—24no—
257#543—22no—
258#543C22×C3114yescomplemented
259C2×C22⋊Q8—22no—
260C2×C22⋊Q8—21no—
261C2×C22⋊Q8—22no—
262C2×C22⋊Q8C22×C3112yes—
263C22×C22⋊C4—21no—
264C22×C22⋊C4—21no—
265C22×C22⋊C4C22×C3111yescharacteristic
266#450—24no—
267#450C22×C3118yescomplemented
268C2×#102—22no—
269C2×C22⋊C8C22×C3112yes—
270#392—22no—
271#392C22×C3114yes—
272C2×#95C22×C3111yescharacteristic
273#328C22×C3118yescomplemented
274#323C22×C3114yescomplemented
275C2×C31—21no—
276C2×C31—21no—
277C2×C31—22no—
278C2×C31—44no—
279C2×C31—84no—
280C2×C31#159614yesdirect product factor
281C2×C31C2×#44811yescharacteristic
282C2×C31C2×#45311yescharacteristic
283C2×C31C2×B(4,𝔽2)11yescharacteristic
284C25—21no—
285C25D4×C3111yescharacteristic
286C22×Q8—21no—
287C22×Q8D4×C3112yescomplemented
288C22×D4—24no—
289C22×D4—22no—
290C22×D4—22no—
291C22×D4—21no—
292C22×D4—44no—
293C22×D4—44no—
294C22×D4—44no—
295C22×D4—44no—
296C22×D4C23×C3112yes—
297C23×C4—21no—
298C23×C4—21no—
299C23×C4—21no—
300C23×C4D4×C3111yescharacteristic
301C23×C4C23×C3111yescharacteristic
302C2×SD16—28no—
303C2×SD16—48no—
304C22×C8—22no—
305C22⋊Q8—28no—
306C22⋊Q8—28no—
307C22⋊Q8—24no—
308C22⋊Q8—28no—
309C22⋊Q8—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—28no—
313C22⋊Q8D4×C3114yescomplemented
314C22⋊Q8C23×C3114yes—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4D4×C3112yescomplemented
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4D4×C3112yes—
328C2×C22⋊C4C23×C3112yes—
329#102—28no—
330C22⋊C8—22no—
331C22⋊C8C23×C3114yes—
332#95—21no—
333#95C23×C3112yes—
334C31C2×#159611yescharacteristic, direct product factor, Sylow
335C24—21no—
336C24—21no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—42no—
349C24SD16×C3114yescomplemented
350C24C2×D4×C3111yescharacteristic
351C24C2×D4×C3112yes—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—44no—
355C2×Q8—82no—
356C2×Q8C2×D4×C3114yescomplemented
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—432no—
366C2×D4—48no—
367C2×D4—42no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C2×D4×C3112yes—
383C22×C4C2×D4×C3112yes—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3112yes—
386C22×C4C24×C3111yescharacteristic
387SD16—416no—
388C2×C8—24no—
389C2×C8—44no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4C2×D4×C3114yescomplemented
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—48no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—48no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—48no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—84no—
425C23—81no—
426C23—81no—
427C23—84no—
428C23—84no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C31×#12011yescharacteristic
434C23C31×#12011yescharacteristic
435C23C2×SD16×C3112yes—
436C23C2×SD16×C3114yescomplemented
437C23C31×#13712yes—
438C23C22×D4×C3111yescharacteristic
439Q8—44no—
440Q8—82no—
441Q8—84no—
442D4—28no—
443D4—44no—
444D4—48no—
445D4—432no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4C31×#12012yes—
465C2×C4C22×D4×C3112yes—
466C8—44no—
467C22—24no—
468C22—24no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—24no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—82no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—88no—
486C22—88no—
487C22—82no—
488C22—82no—
489C22C31×#44812yes—
490C22C31×#44812yes—
491C22C31×#45312yes—
492C22C31×B(4,𝔽2)12yes—
493C22C2×C31×#12011yescharacteristic
494C22C22×SD16×C3111yescharacteristic
495C22C2×C31×#13711yescharacteristic
496C4—22no—
497C4—22no—
498C4—44no—
499C4—42no—
500C4—44no—
501C4—44no—
502C2—21no—
503C2—21no—
504C2—22no—
505C2—44no—
506C2—84no—
507C2C31×#159614yesdirect product factor
508C2C2×C31×#44811yescharacteristic
509C2C2×C31×#45311yescharacteristic
510C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#392C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#543C2semidirect
2. C2×C31×#450C2semidirect
4. C31×#1596C2direct (N × H)
5. C2×C31×#328C2semidirect
6. C2×C31×#323C2semidirect
12. C31×#543C22semidirect
21. C31×#450C22semidirect
27. C31×#328C22semidirect
28. C31×#323C22semidirect
32. C22×Q8×C31D4semidirect
58. C31×C22⋊Q8D4semidirect
63. C2×C4⋊C4×C31D4semidirect
93. C24×C31SD16semidirect
100. C2×Q8×C31C2×D4semidirect
139. C4⋊C4×C31C2×D4semidirect
146. C2×#1596C31direct (N × H)
181. C23×C31C2×SD16semidirect
212. C2×#543C2×C31semidirect
213. C2×#450C2×C31semidirect
215. #1596C2×C31direct (N × H)
216. C2×#328C2×C31semidirect
217. C2×#323C2×C31semidirect
258. #543C22×C31semidirect
267. #450C22×C31semidirect
273. #328C22×C31semidirect
274. #323C22×C31semidirect
280. C2×C31#1596direct (N × H)
287. C22×Q8D4×C31semidirect
313. C22⋊Q8D4×C31semidirect
318. C2×C4⋊C4D4×C31semidirect
334. C31C2×#1596direct (N × H)
349. C24SD16×C31semidirect
356. C2×Q8C2×D4×C31semidirect
395. C4⋊C4C2×D4×C31semidirect
436. C23C2×SD16×C31semidirect
507. C2C31×#1596direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C2×C31×#4502
C2×C31×#3922
C31×#15962
C2×C31×#3282
C2×C31×#3232
C2×#159631