Finite Group Databasebeta

← C31×#9915 #21466674

Subgroups — C31×#9915 #21466674

Order 7936.

1044 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#447C211yesmaximal, characteristic
2C31×#2618C211yesmaximal, characteristic, complemented
3C2×C31×#394C211yesmaximal, characteristic, complemented
4C31×#1786C211yesmaximal, characteristic, complemented
5C2×C31×#327C211yesmaximal, characteristic, complemented
6C2×C31×#324C211yesmaximal, characteristic, complemented
7C31×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×D8×C31—21no—
10C2×C31×#123—21no—
11C2×C31×#121C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C22⋊C4×C31—21no—
14C22×C22⋊C4×C31—21no—
15C31×#447—22no—
16C31×#447—22no—
17C2×C31×#102—21no—
18C2×C31×#102—22no—
19C2×C31×C22⋊C8C2211yescharacteristic
20C2×C31×C22⋊C8C2211yescharacteristic
21C31×#398—21no—
22C31×#396—22no—
23C31×#394—21no—
24C31×#394—22no—
25C31×#394—22no—
26C31×#386—21no—
27C31×#380C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#327—22no—
30C31×#324—22no—
31C25×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—21no—
38C22×D4×C31—44no—
39C22×D4×C31D412yes—
40C22×D4×C31D411yescharacteristic
41C23×C4×C31—21no—
42C23×C4×C31D411yescharacteristic
43C23×C4×C31D411yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C2×D8×C31—24no—
46C2×D8×C31—44no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C31×#123—42no—
50C31×#123—42no—
51C31×#121—21no—
52C31×#121—21no—
53C31×#121—21no—
54C31×#121—21no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C4⋊C4×C31D412yes—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—44no—
72C2×C22⋊C4×C31—42no—
73C31×#102—24no—
74C31×#102—44no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8D412yes—
79C31×#95—22no—
80C31×#95—21no—
81C31×#95—21no—
82C31×#95D412yes—
83C24×C31—24no—
84C24×C31—22no—
85C24×C31—21no—
86C24×C31—42no—
87C24×C31—42no—
88C24×C31—42no—
89C24×C31—41no—
90C24×C31—41no—
91C24×C31—41no—
92C24×C31—41no—
93C24×C31—41no—
94C24×C31—41no—
95C24×C31C2×D411yescharacteristic
96C2×D4×C31—24no—
97C2×D4×C31—24no—
98C2×D4×C31—22no—
99C2×D4×C31—21no—
100C2×D4×C31—22no—
101C2×D4×C31—21no—
102C2×D4×C31—42no—
103C2×D4×C31—42no—
104C2×D4×C31—42no—
105C2×D4×C31—44no—
106C2×D4×C31—44no—
107C2×D4×C31—44no—
108C2×D4×C31—44no—
109C2×D4×C31—41no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—41no—
116C2×D4×C31—41no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31—41no—
134C22×C4×C31—41no—
135C22×C4×C31D812yescomplemented
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31#4111yescharacteristic
142D8×C31—48no—
143C2×C8×C31—22no—
144C2×C8×C31—42no—
145C2×C8×C31—42no—
146C2×C8×C31—41no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—44no—
150C4⋊C4×C31—42no—
151C4⋊C4×C31—41no—
152C4⋊C4×C31—41no—
153C4⋊C4×C31—41no—
154C4⋊C4×C31—41no—
155C22⋊C4×C31—48no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—41no—
159C22⋊C4×C31—41no—
160C22⋊C4×C31—41no—
161C22⋊C4×C31—41no—
162C22⋊C4×C31—82no—
163#9915C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—24no—
167C23×C31—42no—
168C23×C31—44no—
169C23×C31—44no—
170C23×C31—44no—
171C23×C31—44no—
172C23×C31—44no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—44no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—42no—
183C23×C31—42no—
184C23×C31—82no—
185C23×C31—82no—
186C23×C31—82no—
187C23×C31—82no—
188C23×C31—82no—
189C23×C31—81no—
190C23×C31—81no—
191C23×C31—82no—
192C23×C31—81no—
193C23×C31—81no—
194C23×C31#12011yescharacteristic
195C23×C31#12011yescharacteristic
196C23×C31#12111yescharacteristic
197C23×C31C2×D811yescharacteristic
198C23×C31#13511yescharacteristic
199C23×C31AGL(1,ℤ/8)11yescharacteristic
200C23×C31AGL(1,ℤ/8)11yescharacteristic
201D4×C31—24no—
202D4×C31—24no—
203D4×C31—42no—
204D4×C31—44no—
205D4×C31—42no—
206D4×C31—44no—
207D4×C31—42no—
208D4×C31—81no—
209D4×C31—81no—
210D4×C31—81no—
211D4×C31—81no—
212D4×C31—81no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—82no—
239C2×C4×C31—82no—
240C2×C4×C31—81no—
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12111yescharacteristic
243C2×C4×C31#12111yescharacteristic
244C2×C4×C31#12711yescharacteristic
245C8×C31—42no—
246C8×C31—81no—
247C2×#447C2×C3111yescharacteristic
248#2618C2×C3111yescharacteristic, complemented
249C2×#394C2×C3111yescharacteristic, complemented
250#1786C2×C3111yescharacteristic, complemented
251C2×#327C2×C3111yescharacteristic, complemented
252C2×#324C2×C3111yescharacteristic, complemented
253#1294C2×C3111yescharacteristic, complemented
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—21no—
261C22×C31—21no—
262C22×C31—44no—
263C22×C31—44no—
264C22×C31—44no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—84no—
268C22×C31—84no—
269C22×C31—84no—
270C22×C31—84no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31—82no—
274C22×C31—82no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31#39211yescharacteristic
279C22×C31#44711yescharacteristic
280C22×C31#44711yescharacteristic
281C22×C31#44911yescharacteristic
282C22×C31#44911yescharacteristic
283C22×C31#45412yes—
284C22×C31B(4,𝔽2)12yes—
285C22×C31#46611yescharacteristic
286C22×C31#46911yescharacteristic
287C4×C31—21no—
288C4×C31—21no—
289C4×C31—21no—
290C4×C31—21no—
291C4×C31—42no—
292C4×C31—82no—
293C4×C31—82no—
294C4×C31—81no—
295C4×C31—81no—
296C23×D4C22×C3111yescharacteristic
297C22×D8—21no—
298C2×#123—21no—
299C2×#121C22×C3111yescharacteristic
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C22⋊C4—21no—
302C22×C22⋊C4—21no—
303#447—22no—
304#447—22no—
305C2×#102—21no—
306C2×#102—22no—
307C2×C22⋊C8C22×C3111yescharacteristic
308C2×C22⋊C8C22×C3111yescharacteristic
309#398—21no—
310#396—22no—
311#394—21no—
312#394—22no—
313#394—22no—
314#386—21no—
315#380C22×C3111yescharacteristic
316C2×#95C22×C3111yescharacteristic
317#327—22no—
318#324—22no—
319C2×C31—21no—
320C2×C31—21no—
321C2×C31—21no—
322C2×C31—21no—
323C2×C31—84no—
324C2×C31—161no—
325C2×C31#159712yes—
326C2×C31#159812yes—
327C2×C31#199911yescharacteristic
328C2×C31#200311yescharacteristic
329C2×C31#202111yescharacteristic
330C25—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—21no—
337C22×D4—44no—
338C22×D4D4×C3111yescharacteristic
339C22×D4D4×C3112yes—
340C23×C4—21no—
341C23×C4D4×C3111yescharacteristic
342C23×C4D4×C3111yescharacteristic
343C23×C4C23×C3111yescharacteristic
344C2×D8—24no—
345C2×D8—44no—
346C22×C8—21no—
347C22×C8—21no—
348#123—42no—
349#123—42no—
350#121—21no—
351#121—21no—
352#121—21no—
353#121—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4D4×C3111yescharacteristic
362C2×C4⋊C4D4×C3112yes—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—42no—
372#102—24no—
373#102—44no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8D4×C3112yes—
378#95—22no—
379#95—21no—
380#95—21no—
381#95D4×C3112yes—
382C31#991511yescharacteristic, direct product factor, Sylow
383C24—24no—
384C24—22no—
385C24—21no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24C2×D4×C3111yescharacteristic
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4D8×C3112yescomplemented
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4C2×D4×C3111yescharacteristic
441C22×C4#41×C3111yescharacteristic
442D8—48no—
443C2×C8—22no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—41no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—82no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23—81no—
492C23—81no—
493C23C31×#12011yescharacteristic
494C23C31×#12011yescharacteristic
495C23C31×#12111yescharacteristic
496C23C2×D8×C3111yescharacteristic
497C23C31×#13511yescharacteristic
498C23C31×AGL(1,ℤ/8)11yescharacteristic
499C23C31×AGL(1,ℤ/8)11yescharacteristic
500D4—24no—
501D4—24no—
502D4—42no—
503D4—44no—
504D4—42no—
505D4—44no—
506D4—42no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4C31×#12011yescharacteristic
541C2×C4C31×#12111yescharacteristic
542C2×C4C31×#12111yescharacteristic
543C2×C4C31×#12711yescharacteristic
544C8—42no—
545C8—81no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22C31×#39211yescharacteristic
571C22C31×#44711yescharacteristic
572C22C31×#44711yescharacteristic
573C22C31×#44911yescharacteristic
574C22C31×#44911yescharacteristic
575C22C31×#45412yes—
576C22C31×B(4,𝔽2)12yes—
577C22C31×#46611yescharacteristic
578C22C31×#46911yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—81no—
587C4—81no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—84no—
593C2—161no—
594C2C31×#159712yes—
595C2C31×#159812yes—
596C2C31×#199911yescharacteristic
597C2C31×#200311yescharacteristic
598C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#447C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2618C2semidirect
3. C2×C31×#394C2semidirect
4. C31×#1786C2semidirect
5. C2×C31×#327C2semidirect
6. C2×C31×#324C2semidirect
7. C31×#1294C2semidirect
135. C22×C4×C31D8semidirect
163. #9915C31direct (N × H)
248. #2618C2×C31semidirect
249. C2×#394C2×C31semidirect
250. #1786C2×C31semidirect
251. C2×#327C2×C31semidirect
252. C2×#324C2×C31semidirect
253. #1294C2×C31semidirect
382. C31#9915direct (N × H)
435. C22×C4D8×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4472
C31×#26182
C2×C31×#3942
C31×#17862
C2×C31×#3272
C2×C31×#3242
C31×#12942
#991531