Finite Group Databasebeta

← C31×#32718 #21489477

Subgroups — C31×#32718 #21489477

Order 7936.

952 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2856C211yesmaximal, characteristic, complemented
2C31×#2854C211yesmaximal, characteristic, complemented
3C31×#2099C211yesmaximal, characteristic, complemented
4C31×#2093C211yesmaximal, characteristic, complemented
5C31×#2067C212yesmaximal, complemented
6C31×#2054C212yesmaximal, complemented
7C31×#2041C211yesmaximal, characteristic, complemented
8C31×#2033C212yesmaximal, complemented
9C31×#2011C211yesmaximal, characteristic, complemented
10C31×#1999C212yesmaximal, complemented
11C31×#1797C211yesmaximal, characteristic, complemented
12C22×D8×C31C2212yes—
13C2×C31×#127C2211yescharacteristic
14C2×C31×#123—21no—
15C2×C31×#121—21no—
16C2×C31×#121—21no—
17C2×C31×#121C2211yescharacteristic
18C2×C31×#121C2212yes—
19C22×C42×C31C2211yescharacteristic
20C2×C31×#107C2212yes—
21C2×C31×C4⋊C8C2212yes—
22C2×C31×#102C2212yes—
23C2×C31×#102C2212yes—
24C2×C31×#102C2212yes—
25C2×C31×#102C2212yes—
26C2×C31×C22⋊C8C2212yes—
27C31×#390—21no—
28C31×#390—21no—
29C31×#390C2211yescharacteristic
30C31×#390C2212yes—
31C31×#390C2211yescharacteristic
32C31×#389C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#388—21no—
35C31×#388—21no—
36C31×#386—21no—
37C31×#384C2211yescharacteristic
38C31×#380C2211yescharacteristic
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#336C2211yescharacteristic
42C31×#336C2211yescharacteristic
43C22×D4×C31—22no—
44C22×D4×C31—21no—
45C22×D4×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31C2312yes—
48C22×D4×C31C2311yescharacteristic
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C2×D8×C31—42no—
53C2×D8×C31—42no—
54C2×D8×C31—42no—
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×#127—42no—
58C31×#123—44no—
59C31×#121—42no—
60C31×#121—42no—
61C31×#121—42no—
62C31×#121—44no—
63C31×#121—44no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31C2311yescharacteristic
69C2×C4⋊C4×C31C2311yescharacteristic
70C2×C4⋊C4×C31C2311yescharacteristic
71C2×C4⋊C4×C31C2312yes—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—22no—
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—21no—
81C2×C42×C31—42no—
82C2×C42×C31D411yescharacteristic
83C2×C42×C31D411yescharacteristic
84C2×C42×C31C2311yescharacteristic
85C2×C42×C31C2311yescharacteristic
86C31×#107—42no—
87C31×C4⋊C8—42no—
88C31×#102—42no—
89C31×#102—42no—
90C31×#102—42no—
91C31×#102—42no—
92C31×C22⋊C8—42no—
93C31×#95—21no—
94C31×#95—21no—
95C31×#95—21no—
96C24×C31—21no—
97C24×C31—22no—
98C24×C31C2×D411yescharacteristic
99C2×D4×C31—22no—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—21no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—42no—
110C2×D4×C31—42no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31—42no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—42no—
117C2×D4×C31—44no—
118C2×D4×C31—42no—
119C2×D4×C31—42no—
120C2×D4×C31—42no—
121C22×C4×C31—22no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—41no—
131C22×C4×C31—41no—
132C22×C4×C31—41no—
133C22×C4×C31—41no—
134C22×C4×C31D814yescomplemented
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2411yescharacteristic
141D8×C31—48no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—42no—
155C4⋊C4×C31—42no—
156C4⋊C4×C31—42no—
157C4⋊C4×C31—42no—
158C22⋊C4×C31—42no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31—44no—
161C22⋊C4×C31—44no—
162C22⋊C4×C31—42no—
163C22⋊C4×C31—44no—
164C22⋊C4×C31—42no—
165C22⋊C4×C31—42no—
166C22⋊C4×C31—42no—
167C42×C31—22no—
168C42×C31—22no—
169C42×C31—41no—
170C42×C31—41no—
171#32718C3111yesmaximal, characteristic, direct product factor, Sylow
172C23×C31—21no—
173C23×C31—21no—
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—41no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—41no—
182C23×C31—42no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—41no—
191C23×C31—42no—
192C23×C31C2×D812yes—
193C23×C31C22×D411yescharacteristic
194D4×C31—42no—
195D4×C31—42no—
196D4×C31—42no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—84no—
201D4×C31—84no—
202D4×C31—84no—
203D4×C31—88no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—44no—
217C2×C4×C31—44no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31C2×D812yes—
233C2×C4×C31C2×D812yes—
234C2×C4×C31AGL(1,ℤ/8)12yes—
235C2×C4×C31H(5,𝔽2)11yescharacteristic
236C2×C4×C31H(5,𝔽2)12yes—
237C2×C4×C31H(5,𝔽2)12yes—
238C2×C4×C312−1+411yescharacteristic
239C8×C31—42no—
240C8×C31—42no—
241#2856C2×C3111yescharacteristic, complemented
242#2854C2×C3111yescharacteristic, complemented
243#2099C2×C3111yescharacteristic, complemented
244#2093C2×C3111yescharacteristic, complemented
245#2067C2×C3112yescomplemented
246#2054C2×C3112yescomplemented
247#2041C2×C3111yescharacteristic, complemented
248#2033C2×C3112yescomplemented
249#2011C2×C3111yescharacteristic, complemented
250#1999C2×C3112yescomplemented
251#1797C2×C3111yescharacteristic, complemented
252C22×C31—22no—
253C22×C31—22no—
254C22×C31—22no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31—82no—
260C22×C31—81no—
261C22×C31—82no—
262C22×C31—81no—
263C22×C31—82no—
264C22×C31—81no—
265C22×C31—82no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—81no—
271C22×C31—81no—
272C22×C31—82no—
273C22×C31#53411yescharacteristic
274C22×C31#53511yescharacteristic
275C22×C31#53711yescharacteristic
276C22×C31C22×D811yescharacteristic
277C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C31#57511yescharacteristic
279C22×C31#57511yescharacteristic
280C4×C31—22no—
281C4×C31—22no—
282C4×C31—22no—
283C4×C31—41no—
284C4×C31—41no—
285C4×C31—81no—
286C4×C31—81no—
287C4×C31—82no—
288C4×C31—81no—
289C4×C31#56012yes—
290C4×C31#56212yes—
291C22×D8C22×C3112yes—
292C2×#127C22×C3111yescharacteristic
293C2×#123—21no—
294C2×#121—21no—
295C2×#121—21no—
296C2×#121C22×C3111yescharacteristic
297C2×#121C22×C3112yes—
298C22×C42C22×C3111yescharacteristic
299C2×#107C22×C3112yes—
300C2×C4⋊C8C22×C3112yes—
301C2×#102C22×C3112yes—
302C2×#102C22×C3112yes—
303C2×#102C22×C3112yes—
304C2×#102C22×C3112yes—
305C2×C22⋊C8C22×C3112yes—
306#390—21no—
307#390—21no—
308#390C22×C3111yescharacteristic
309#390C22×C3112yes—
310#390C22×C3111yescharacteristic
311#389C22×C3111yescharacteristic
312#389C22×C3111yescharacteristic
313#388—21no—
314#388—21no—
315#386—21no—
316#384C22×C3111yescharacteristic
317#380C22×C3111yescharacteristic
318#340C22×C3112yes—
319#340C22×C3112yes—
320#336C22×C3111yescharacteristic
321#336C22×C3111yescharacteristic
322C2×C31—42no—
323C2×C31—162no—
324C2×C31—161no—
325C2×C31#281311yescharacteristic
326C2×C31#318611yescharacteristic
327C2×C31#319411yescharacteristic
328C2×C31#320511yescharacteristic
329C2×C31#321911yescharacteristic
330C2×C31#324511yescharacteristic
331C2×C31#325111yescharacteristic
332C22×D4—22no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C23×C3112yes—
337C22×D4C23×C3111yescharacteristic
338C23×C4D4×C3111yescharacteristic
339C23×C4D4×C3111yescharacteristic
340C23×C4C23×C3111yescharacteristic
341C2×D8—42no—
342C2×D8—42no—
343C2×D8—42no—
344C22×C8C23×C3112yes—
345C22×C8C23×C3112yes—
346#127—42no—
347#123—44no—
348#121—42no—
349#121—42no—
350#121—42no—
351#121—44no—
352#121—44no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4C23×C3112yes—
358C2×C4⋊C4C23×C3111yescharacteristic
359C2×C4⋊C4C23×C3111yescharacteristic
360C2×C4⋊C4C23×C3111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C42—42no—
371C2×C42D4×C3111yescharacteristic
372C2×C42D4×C3111yescharacteristic
373C2×C42C23×C3111yescharacteristic
374C2×C42C23×C3111yescharacteristic
375#107—42no—
376C4⋊C8—42no—
377#102—42no—
378#102—42no—
379#102—42no—
380#102—42no—
381C22⋊C8—42no—
382#95—21no—
383#95—21no—
384#95—21no—
385C31#3271811yescharacteristic, direct product factor, Sylow
386C24—21no—
387C24—22no—
388C24C2×D4×C3111yescharacteristic
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4D8×C3114yescomplemented
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C24×C3111yescharacteristic
431D8—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42—41no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23C2×D8×C3112yes—
482C23C22×D4×C3111yescharacteristic
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—84no—
490D4—84no—
491D4—84no—
492D4—88no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4C2×D8×C3112yes—
522C2×C4C2×D8×C3112yes—
523C2×C4C31×AGL(1,ℤ/8)12yes—
524C2×C4C31×H(5,𝔽2)11yescharacteristic
525C2×C4C31×H(5,𝔽2)12yes—
526C2×C4C31×H(5,𝔽2)12yes—
527C2×C4C31×2−1+411yescharacteristic
528C8—42no—
529C8—42no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—82no—
538C22—81no—
539C22—82no—
540C22—81no—
541C22—82no—
542C22—81no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—82no—
551C22C31×#53411yescharacteristic
552C22C31×#53511yescharacteristic
553C22C31×#53711yescharacteristic
554C22C22×D8×C3111yescharacteristic
555C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
556C22C31×#57511yescharacteristic
557C22C31×#57511yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—82no—
565C4—81no—
566C4—81no—
567C4C31×#56012yes—
568C4C31×#56212yes—
569C2—42no—
570C2—162no—
571C2—161no—
572C2C31×#281311yescharacteristic
573C2C31×#318611yescharacteristic
574C2C31×#319411yescharacteristic
575C2C31×#320511yescharacteristic
576C2C31×#321911yescharacteristic
577C2C31×#324511yescharacteristic
578C2C31×#325111yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2856C2semidirect
2. C31×#2854C2semidirect
3. C31×#2099C2semidirect
4. C31×#2093C2semidirect
5. C31×#2067C2semidirect
6. C31×#2054C2semidirect
7. C31×#2041C2semidirect
8. C31×#2033C2semidirect
9. C31×#2011C2semidirect
10. C31×#1999C2semidirect
11. C31×#1797C2semidirect
134. C22×C4×C31D8semidirect
171. #32718C31direct (N × H)
241. #2856C2×C31semidirect
242. #2854C2×C31semidirect
243. #2099C2×C31semidirect
244. #2093C2×C31semidirect
245. #2067C2×C31semidirect
246. #2054C2×C31semidirect
247. #2041C2×C31semidirect
248. #2033C2×C31semidirect
249. #2011C2×C31semidirect
250. #1999C2×C31semidirect
251. #1797C2×C31semidirect
385. C31#32718direct (N × H)
424. C22×C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28562
C31×#28542
C31×#20992
C31×#20932
C31×#20672
C31×#20542
C31×#20412
C31×#20332
C31×#20112
C31×#19992
C31×#17972
#3271831