Finite Group Databasebeta

← C31×#25945 #21482704

Subgroups — C31×#25945 #21482704

Order 7936.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C2×C31×#491C211yesmaximal, characteristic, complemented
3C2×C31×#461C211yesmaximal, characteristic
4C31×#2568C211yesmaximal, characteristic, complemented
5C31×#2026C212yesmaximal, complemented
6C31×#2024C212yesmaximal, complemented
7C31×#1989C211yesmaximal, characteristic, complemented
8C31×#1985C211yesmaximal, characteristic, complemented
9C31×#1877C212yesmaximal
10C31×#1877C212yesmaximal
11C31×#1762C211yesmaximal, characteristic
12C22×SD16×C31C2212yes—
13C31×#554—24no—
14C31×#554—24no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#125C2211yescharacteristic
18C2×C31×#125C2211yescharacteristic
19C2×C31×C22⋊Q8—21no—
20C2×C31×C22⋊Q8—21no—
21C2×C31×C22⋊Q8—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×D4×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C22×C4⋊C4×C31—21no—
27C31×#491—24no—
28C31×#461—24no—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×#103C2212yes—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#97C2211yescharacteristic
35C31×#400—21no—
36C31×#400—21no—
37C31×#391—21no—
38C31×#391C2212yes—
39C31×#391C2212yes—
40C31×#382—21no—
41C4×C4⋊C4×C31C2211yescharacteristic
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31D411yescharacteristic
49C22×Q8×C31C2312yes—
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C2×SD16×C31—24no—
55C2×SD16×C31—28no—
56C22×C8×C31C2312yes—
57C22×C8×C31C2312yes—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—22no—
62C31×#125—22no—
63C31×#125—22no—
64C31×#125—22no—
65C31×#125D414yes—
66C31×C22⋊Q8—28no—
67C31×C22⋊Q8—28no—
68C31×C22⋊Q8—28no—
69C4×Q8×C31—22no—
70C4×Q8×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31D414yes—
77C2×C4⋊C4×C31—28no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31D411yescharacteristic
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31C2312yes—
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C42×C31—21no—
96C2×C42×C31C2311yescharacteristic
97C2×C42×C31C2311yescharacteristic
98C2×C42×C31C2311yescharacteristic
99C31×C4⋊C8—22no—
100C31×#103—24no—
101C31×#103—24no—
102C31×#103—24no—
103C31×#102—24no—
104C31×#97—22no—
105C31×#95—21no—
106C31×#95—22no—
107C31×#95—22no—
108C31×#95—22no—
109C31×#95—22no—
110C24×C31—21no—
111C2×Q8×C31—24no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—24no—
117C2×Q8×C31—24no—
118C2×Q8×C31—41no—
119C2×Q8×C31—82no—
120C2×D4×C31—21no—
121C2×D4×C31—22no—
122C2×D4×C31C2×D412yes—
123C2×D4×C31#4114yes—
124C22×C4×C31—24no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—24no—
129C22×C4×C31—24no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—82no—
135C22×C4×C31—82no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2411yescharacteristic
143SD16×C31—48no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—24no—
148C2×C8×C31—22no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—42no—
163C4⋊C4×C31—42no—
164C4⋊C4×C31C2×D412yes—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D412yes—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C42×C31—21no—
171C42×C31—21no—
172C42×C31—21no—
173C42×C31—21no—
174C42×C31—42no—
175C42×C31C2×D412yes—
176C42×C31C2×D412yes—
177#25945C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—22no—
179C23×C31—24no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—42no—
183C23×C31C22×D411yescharacteristic
184Q8×C31—44no—
185Q8×C31—42no—
186Q8×C31—42no—
187Q8×C31—44no—
188Q8×C31—84no—
189D4×C31—22no—
190D4×C31—24no—
191D4×C31#12414yes—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—22no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—48no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—44no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—84no—
218C2×C4×C31—84no—
219C2×C4×C31#12012yes—
220C2×C4×C31#12012yes—
221C2×C4×C31#13712yes—
222C2×C4×C31C22×D411yescharacteristic
223C2×C4×C31C22×D411yescharacteristic
224C2×C4×C31C2×#4112yes—
225C2×C4×C312−1+412yes—
226C8×C31—42no—
227C8×C31—42no—
228C2×#554C2×C3111yescharacteristic
229C2×#491C2×C3111yescharacteristic, complemented
230C2×#461C2×C3111yescharacteristic
231#2568C2×C3111yescharacteristic, complemented
232#2026C2×C3112yescomplemented
233#2024C2×C3112yescomplemented
234#1989C2×C3111yescharacteristic, complemented
235#1985C2×C3111yescharacteristic, complemented
236#1877C2×C3112yes—
237#1877C2×C3112yes—
238#1762C2×C3111yescharacteristic
239C22×C31—24no—
240C22×C31—44no—
241C22×C31—44no—
242C22×C31—42no—
243C22×C31—42no—
244C22×C31C2×#12011yescharacteristic
245C22×C31#54812yes—
246C22×C31C2×#13711yescharacteristic
247C22×C31#57511yescharacteristic
248C22×C31#57712yes—
249C4×C31—22no—
250C4×C31—22no—
251C4×C31—22no—
252C4×C31—41no—
253C4×C31—42no—
254C4×C31—41no—
255C4×C31—41no—
256C4×C31—42no—
257C4×C31—42no—
258C4×C31—82no—
259C4×C31—82no—
260C4×C31C2×#12411yescharacteristic
261C4×C31#53611yescharacteristic
262C4×C31#54812yes—
263C22×SD16C22×C3112yes—
264#554—24no—
265#554—24no—
266C2×C4⋊Q8C22×C3111yescharacteristic
267C2×C4⋊Q8C22×C3111yescharacteristic
268C2×#125C22×C3111yescharacteristic
269C2×#125C22×C3111yescharacteristic
270C2×C22⋊Q8—21no—
271C2×C22⋊Q8—21no—
272C2×C22⋊Q8—21no—
273C2×C4×Q8C22×C3111yescharacteristic
274C2×C4×D4C22×C3111yescharacteristic
275C2×C4×D4C22×C3111yescharacteristic
276C2×C4×D4C22×C3111yescharacteristic
277C22×C4⋊C4—21no—
278#491—24no—
279#461—24no—
280C2×C4⋊C8C22×C3111yescharacteristic
281C2×#103C22×C3112yes—
282C2×#103C22×C3112yes—
283C2×#103C22×C3112yes—
284C2×#102C22×C3112yes—
285C2×#97C22×C3111yescharacteristic
286#400—21no—
287#400—21no—
288#391—21no—
289#391C22×C3112yes—
290#391C22×C3112yes—
291#382—21no—
292C4×C4⋊C4C22×C3111yescharacteristic
293#340C22×C3112yes—
294#340C22×C3112yes—
295#340C22×C3112yes—
296#336C22×C3112yes—
297#336C22×C3112yes—
298C2×C31—44no—
299C2×C31#243611yescharacteristic
300C2×C31#300711yescharacteristic
301C2×C31#300911yescharacteristic
302C2×C31#337212yes—
303C2×C31#337612yes—
304C22×Q8—21no—
305C22×Q8D4×C3111yescharacteristic
306C22×Q8C23×C3112yes—
307C22×D4C23×C3111yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310C23×C4—21no—
311C2×SD16—24no—
312C2×SD16—28no—
313C22×C8C23×C3112yes—
314C22×C8C23×C3112yes—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319#125—22no—
320#125—22no—
321#125—22no—
322#125D4×C3114yes—
323C22⋊Q8—28no—
324C22⋊Q8—28no—
325C22⋊Q8—28no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4D4×C3114yes—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4D4×C3111yescharacteristic
343C2×C4⋊C4D4×C3111yescharacteristic
344C2×C4⋊C4D4×C3111yescharacteristic
345C2×C4⋊C4C23×C3112yes—
346C2×C4⋊C4C23×C3111yescharacteristic
347C2×C4⋊C4C23×C3111yescharacteristic
348C2×C4⋊C4C23×C3111yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C42—21no—
353C2×C42C23×C3111yescharacteristic
354C2×C42C23×C3111yescharacteristic
355C2×C42C23×C3111yescharacteristic
356C4⋊C8—22no—
357#103—24no—
358#103—24no—
359#103—24no—
360#102—24no—
361#97—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367C31#2594511yescharacteristic, direct product factor, Sylow
368C24—21no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—41no—
377C2×Q8—82no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4C2×D4×C3112yes—
381C2×D4#41×C3114yes—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C2×D4×C3111yescharacteristic
395C22×C4C2×D4×C3111yescharacteristic
396C22×C4C2×D4×C3111yescharacteristic
397C22×C4C2×D4×C3111yescharacteristic
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400C22×C4C24×C3111yescharacteristic
401SD16—48no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4C2×D4×C3112yes—
423C4⋊C4C2×D4×C3112yes—
424C4⋊C4C2×D4×C3112yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42C2×D4×C3112yes—
434C42C2×D4×C3112yes—
435C23—22no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23C22×D4×C3111yescharacteristic
441Q8—44no—
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—84no—
446D4—22no—
447D4—24no—
448D4C31×#12414yes—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4C31×#12012yes—
477C2×C4C31×#12012yes—
478C2×C4C31×#13712yes—
479C2×C4C22×D4×C3111yescharacteristic
480C2×C4C22×D4×C3111yescharacteristic
481C2×C4C2×#41×C3112yes—
482C2×C4C31×2−1+412yes—
483C8—42no—
484C8—42no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22C2×C31×#12011yescharacteristic
491C22C31×#54812yes—
492C22C2×C31×#13711yescharacteristic
493C22C31×#57511yescharacteristic
494C22C31×#57712yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4C2×C31×#12411yescharacteristic
507C4C31×#53611yescharacteristic
508C4C31×#54812yes—
509C2—44no—
510C2C31×#243611yescharacteristic
511C2C31×#300711yescharacteristic
512C2C31×#300911yescharacteristic
513C2C31×#337212yes—
514C2C31×#337612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D4×C31C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#491C2semidirect
4. C31×#2568C2semidirect
5. C31×#2026C2semidirect
6. C31×#2024C2semidirect
7. C31×#1989C2semidirect
8. C31×#1985C2semidirect
177. #25945C31direct (N × H)
229. C2×#491C2×C31semidirect
231. #2568C2×C31semidirect
232. #2026C2×C31semidirect
233. #2024C2×C31semidirect
234. #1989C2×C31semidirect
235. #1985C2×C31semidirect
367. C31#25945direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4912
C2×C31×#4612
C31×#25682
C31×#20262
C31×#20242
C31×#19892
C31×#19852
C31×#18772
C31×#17622
#2594531