Finite Group Databasebeta

← C31×#10972 #21467731

Subgroups — C31×#10972 #21467731

Order 7936.

1634 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic
2C22×C31×#99C212yesmaximal, complemented
3C22×C31×#99C211yesmaximal, characteristic, complemented
4C31×#2308C211yesmaximal, characteristic, complemented
5C2×C31×#342C212yesmaximal, complemented
6C23×D4×C31C2211yescharacteristic
7C23×D4×C31C2211yescharacteristic
8C31×#534—24no—
9C31×#534—24no—
10C2×C31×#123—21no—
11C2×C31×#123C412yescomplemented
12C2×C31×#121—22no—
13C2×C31×#120—21no—
14C2×C31×#120C412yescomplemented
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2211yescharacteristic
18C2×C31×#99—22no—
19C2×C31×#99—24no—
20C2×C31×#99—24no—
21C2×C31×#99—24no—
22C2×C31×#99—24no—
23C31×#386—24no—
24C31×#381—21no—
25C31×#379—21no—
26C31×#342—24no—
27C25×C31C2×C411yescharacteristic
28C25×C31D411yescharacteristic
29C25×C31C2311yescharacteristic
30C22×D4×C31—22no—
31C22×D4×C31—22no—
32C22×D4×C31—22no—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31—22no—
36C22×D4×C31—22no—
37C22×D4×C31—22no—
38C22×D4×C31—22no—
39C22×D4×C31—22no—
40C22×D4×C31—22no—
41C22×D4×C31—22no—
42C22×D4×C31—22no—
43C22×D4×C31—22no—
44C22×D4×C31C2×C412yes—
45C22×D4×C31D412yes—
46C22×D4×C31D414yes—
47C23×C4×C31—21no—
48C23×C4×C31—22no—
49C23×C4×C31—22no—
50C23×C4×C31C2×C411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C31×#123—28no—
53C31×#123—24no—
54C31×#123—24no—
55C31×#121—216no—
56C31×#120—28no—
57C31×#120—24no—
58C31×#120—24no—
59C2×C4⋊C4×C31—22no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31C2×C412yes—
76C2×C22⋊C4×C31D414yes—
77C2×C22⋊C4×C31D414yescomplemented
78C2×C22⋊C4×C31D414yescomplemented
79C31×#99—28no—
80C31×#99—216no—
81C31×#95—24no—
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—21no—
90C24×C31—82no—
91C24×C31—82no—
92C24×C31—82no—
93C24×C31C22⋊C412yes—
94C24×C31C22⋊C412yescomplemented
95C24×C31C22×C411yescharacteristic
96C24×C31C2×D411yescharacteristic
97C24×C31C2×D411yescharacteristic
98C24×C31C2×D412yes—
99C24×C31C2×D412yes—
100C24×C31C2×D412yes—
101C24×C31C2×D412yes—
102C24×C31#4112yes—
103C24×C31#4112yes—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—24no—
107C2×D4×C31—24no—
108C2×D4×C31—22no—
109C2×D4×C31—24no—
110C2×D4×C31—24no—
111C2×D4×C31—24no—
112C2×D4×C31—24no—
113C2×D4×C31—24no—
114C2×D4×C31—24no—
115C2×D4×C31—24no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—28no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31—42no—
125C2×D4×C31—42no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31—41no—
131C2×D4×C31—41no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—84no—
138C22×C4×C31—22no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—21no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—21no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31—82no—
155C22×C4×C31—84no—
156C22×C4×C31—84no—
157C22×C4×C31C22⋊C412yes—
158C22×C4×C31C22⋊C412yescomplemented
159C4⋊C4×C31—28no—
160C22⋊C4×C31—28no—
161C22⋊C4×C31—28no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—24no—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—42no—
169C22⋊C4×C31—42no—
170C22⋊C4×C31—42no—
171C22⋊C4×C31—42no—
172C22⋊C4×C31—84no—
173C22⋊C4×C31C22⋊C414yescomplemented
174#10972C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—24no—
180C23×C31—44no—
181C23×C31—42no—
182C23×C31—44no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—44no—
186C23×C31—44no—
187C23×C31—44no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—82no—
194C23×C31—84no—
195C23×C31—82no—
196C23×C31—82no—
197C23×C31—82no—
198C23×C31—82no—
199C23×C31—82no—
200C23×C31—82no—
201C23×C31—84no—
202C23×C31—84no—
203C23×C31—84no—
204C23×C31C2×C22⋊C411yescharacteristic
205C23×C31C2×C22⋊C411yescharacteristic
206C23×C31C2×C22⋊C411yescharacteristic
207C23×C31C4×D412yes—
208C23×C31C4×D412yes—
209C23×C31#12011yescharacteristic
210C23×C31#12012yes—
211C23×C31#12012yes—
212C23×C31#12011yescharacteristic
213C23×C31#12012yes—
214C23×C31#12112yes—
215C23×C31#12112yes—
216C23×C31#12112yes—
217C23×C31#12112yes—
218C23×C31#12112yes—
219C23×C31#12112yes—
220C23×C31#12311yescharacteristic
221C23×C31#12311yescharacteristic
222C23×C31#12312yes—
223C23×C31#12412yes—
224C23×C31#12712yes—
225D4×C31—44no—
226D4×C31—44no—
227D4×C31—44no—
228D4×C31—44no—
229D4×C31—44no—
230D4×C31—44no—
231D4×C31—44no—
232D4×C31—44no—
233D4×C31—88no—
234C2×C4×C31—24no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—22no—
238C2×C4×C31—24no—
239C2×C4×C31—22no—
240C2×C4×C31—44no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—42no—
244C2×C4×C31—42no—
245C2×C4×C31—42no—
246C2×C4×C31—48no—
247C2×C4×C31—44no—
248C2×C4×C31—44no—
249C2×C4×C31—84no—
250C2×C4×C31—84no—
251C2×C4×C31—84no—
252C2×C4×C31—84no—
253C2×C4×C31—84no—
254C2×C4×C31—84no—
255C2×#534C2×C3111yescharacteristic
256C22×#99C2×C3112yescomplemented
257C22×#99C2×C3111yescharacteristic, complemented
258#2308C2×C3111yescharacteristic, complemented
259C2×#342C2×C3112yescomplemented
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—24no—
263C22×C31—24no—
264C22×C31—22no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—41no—
271C22×C31—41no—
272C22×C31—42no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—41no—
277C22×C31—42no—
278C22×C31—41no—
279C22×C31—42no—
280C22×C31—42no—
281C22×C31—44no—
282C22×C31—44no—
283C22×C31—44no—
284C22×C31—42no—
285C22×C31—42no—
286C22×C31—82no—
287C22×C31—84no—
288C22×C31—84no—
289C22×C31—82no—
290C22×C31—84no—
291C22×C31—84no—
292C22×C31—82no—
293C22×C31#37911yescharacteristic
294C22×C31#38611yescharacteristic
295C22×C31#38612yes—
296C22×C31#38611yescharacteristic
297C22×C31#39012yes—
298C22×C31#39212yes—
299C22×C31#39212yes—
300C22×C31#39412yes—
301C22×C31#39412yes—
302C4×C31—44no—
303C4×C31—42no—
304C4×C31—42no—
305C4×C31—84no—
306C4×C31—84no—
307C23×D4C22×C3111yescharacteristic
308C23×D4C22×C3111yescharacteristic
309#534—24no—
310#534—24no—
311C2×#123—21no—
312C2×#123C4×C3112yescomplemented
313C2×#121—22no—
314C2×#120—21no—
315C2×#120C4×C3112yescomplemented
316C22×C22⋊C4C22×C3112yes—
317C22×C22⋊C4C22×C3112yes—
318C22×C22⋊C4C22×C3111yescharacteristic
319C2×#99—22no—
320C2×#99—24no—
321C2×#99—24no—
322C2×#99—24no—
323C2×#99—24no—
324#386—24no—
325#381—21no—
326#379—21no—
327#342—24no—
328C2×C31—21no—
329C2×C31—22no—
330C2×C31—22no—
331C2×C31—21no—
332C2×C31—22no—
333C2×C31—22no—
334C2×C31—22no—
335C2×C31—22no—
336C2×C31—82no—
337C2×C31—82no—
338C2×C31#143811yescharacteristic
339C2×C31#191312yes—
340C25C2×C4×C3111yescharacteristic
341C25D4×C3111yescharacteristic
342C25C23×C3111yescharacteristic
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4C2×C4×C3112yes—
358C22×D4D4×C3112yes—
359C22×D4D4×C3114yes—
360C23×C4—21no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4C2×C4×C3111yescharacteristic
364C23×C4D4×C3111yescharacteristic
365#123—28no—
366#123—24no—
367#123—24no—
368#121—216no—
369#120—28no—
370#120—24no—
371#120—24no—
372C2×C4⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4C2×C4×C3112yes—
389C2×C22⋊C4D4×C3114yes—
390C2×C22⋊C4D4×C3114yescomplemented
391C2×C22⋊C4D4×C3114yescomplemented
392#99—216no—
393#99—28no—
394#95—24no—
395C31#1097211yescharacteristic, direct product factor, Sylow
396C24—21no—
397C24—21no—
398C24—21no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—82no—
405C24—82no—
406C24—82no—
407C24C22⋊C4×C3112yes—
408C24C22⋊C4×C3112yescomplemented
409C24C22×C4×C3111yescharacteristic
410C24C2×D4×C3111yescharacteristic
411C24C2×D4×C3111yescharacteristic
412C24C2×D4×C3112yes—
413C24C2×D4×C3112yes—
414C24C2×D4×C3112yes—
415C24C2×D4×C3112yes—
416C24#41×C3112yes—
417C24#41×C3112yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—84no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—82no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4C22⋊C4×C3112yes—
472C22×C4C22⋊C4×C3112yescomplemented
473C4⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4C22⋊C4×C3114yescomplemented
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—82no—
507C23—84no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23C2×C22⋊C4×C3111yescharacteristic
518C23C2×C22⋊C4×C3111yescharacteristic
519C23C2×C22⋊C4×C3111yescharacteristic
520C23C4×D4×C3112yes—
521C23C4×D4×C3112yes—
522C23C31×#12011yescharacteristic
523C23C31×#12012yes—
524C23C31×#12012yes—
525C23C31×#12011yescharacteristic
526C23C31×#12012yes—
527C23C31×#12112yes—
528C23C31×#12112yes—
529C23C31×#12112yes—
530C23C31×#12112yes—
531C23C31×#12112yes—
532C23C31×#12112yes—
533C23C31×#12311yescharacteristic
534C23C31×#12311yescharacteristic
535C23C31×#12312yes—
536C23C31×#12412yes—
537C23C31×#12712yes—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—88no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—84no—
596C22—84no—
597C22—82no—
598C22—84no—
599C22—84no—
600C22—82no—
601C22C31×#37911yescharacteristic
602C22C31×#38611yescharacteristic
603C22C31×#38612yes—
604C22C31×#38611yescharacteristic
605C22C31×#39012yes—
606C22C31×#39212yes—
607C22C31×#39212yes—
608C22C31×#39412yes—
609C22C31×#39412yes—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—84no—
614C4—84no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—22no—
622C2—22no—
623C2—82no—
624C2—82no—
625C2C31×#143811yescharacteristic
626C2C31×#191312yes—
subgroupG/H
center, socleC22×C31#386
commutatorC24C22×C4×C31
Frattini, cosocleC25C23×C31
prime radicalC2×C31×#534C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#99C2semidirect
3. C22×C31×#99C2semidirect
4. C31×#2308C2semidirect
5. C2×C31×#342C2semidirect
11. C2×C31×#123C4semidirect
14. C2×C31×#120C4semidirect
77. C2×C22⋊C4×C31D4semidirect
78. C2×C22⋊C4×C31D4semidirect
94. C24×C31C22⋊C4semidirect
158. C22×C4×C31C22⋊C4semidirect
173. C22⋊C4×C31C22⋊C4semidirect
174. #10972C31direct (N × H)
256. C22×#99C2×C31semidirect
257. C22×#99C2×C31semidirect
258. #2308C2×C31semidirect
259. C2×#342C2×C31semidirect
312. C2×#123C4×C31semidirect
315. C2×#120C4×C31semidirect
390. C2×C22⋊C4D4×C31semidirect
391. C2×C22⋊C4D4×C31semidirect
395. C31#10972direct (N × H)
408. C24C22⋊C4×C31semidirect
472. C22×C4C22⋊C4×C31semidirect
487. C22⋊C4C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5342
C22×C31×#992
C31×#23082
C2×C31×#3422
#1097231