Finite Group Databasebeta

← C31×#14700 #21471459

Subgroups — C31×#14700 #21471459

Order 7936.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#390C211yesmaximal, characteristic
2C2×C31×#390C211yesmaximal, characteristic
3C2×C31×#390C211yesmaximal, characteristic
4C2×C31×#389C211yesmaximal, characteristic, complemented
5C2×C31×#388C214yesmaximal, complemented
6C2×C43×C31C211yesmaximal, characteristic, complemented
7C31×#1437C214yesmaximal, complemented
8C31×C4⋊#95C211yesmaximal, characteristic, complemented
9C31×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C31C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2214yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2214yes—
19C22×C42×C31C2212yes—
20C22×C42×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C22×C42×C31C2212yes—
24C31×#390—24no—
25C31×#390—24no—
26C31×#390C418yescomplemented
27C31×#389—24no—
28C31×#388—216no—
29C2×C31×#95C2214yes—
30C2×C31×#95C2214yes—
31C2×C31×#95C2212yes—
32C43×C31—24no—
33C25×C31C2312yes—
34C22×D4×C31—22no—
35C22×D4×C31—22no—
36C22×D4×C31—22no—
37C22×D4×C31—22no—
38C22×D4×C31—22no—
39C22×D4×C31—22no—
40C22×D4×C31C2×C414yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2312yes—
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C2×C4⋊C4×C31—22no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—28no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31—22no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31C2×C414yes—
60C2×C22⋊C4×C31—28no—
61C2×C22⋊C4×C31—28no—
62C2×C22⋊C4×C31—28no—
63C2×C22⋊C4×C31—28no—
64C2×C22⋊C4×C31—28no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—28no—
68C2×C22⋊C4×C31—28no—
69C2×C22⋊C4×C31C2×C4116yes—
70C2×C42×C31—24no—
71C2×C42×C31—22no—
72C2×C42×C31—22no—
73C2×C42×C31—24no—
74C2×C42×C31—24no—
75C2×C42×C31—24no—
76C2×C42×C31—22no—
77C2×C42×C31—22no—
78C2×C42×C31—24no—
79C2×C42×C31—22no—
80C2×C42×C31—22no—
81C2×C42×C31C2×C414yes—
82C2×C42×C31D414yes—
83C2×C42×C31D414yes—
84C2×C42×C31D418yescomplemented
85C31×#95—28no—
86C31×#95—28no—
87C31×#95—24no—
88C24×C31—24no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—24no—
93C24×C31—24no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31C22×C414yes—
99C24×C31C2411yescharacteristic
100C2×D4×C31—28no—
101C2×D4×C31—28no—
102C2×D4×C31—28no—
103C2×D4×C31—44no—
104C2×D4×C31—44no—
105C2×D4×C31—44no—
106C2×D4×C31—44no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—24no—
122C22×C4×C31—22no—
123C22×C4×C31—24no—
124C22×C4×C31—22no—
125C22×C4×C31—24no—
126C22×C4×C31—24no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—24no—
131C22×C4×C31—24no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31C22×C414yes—
136C22×C4×C31C22×C414yes—
137C22×C4×C31C22×C412yes—
138C22×C4×C31C2×D414yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D414yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D414yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D412yes—
146C22×C4×C31#4114yes—
147C22×C4×C31#4114yes—
148C22×C4×C31#4114yes—
149C22×C4×C31#4112yes—
150C22×C4×C31#4112yes—
151C22×C4×C31#4112yes—
152C22×C4×C31#4112yes—
153C4⋊C4×C31—28no—
154C4⋊C4×C31—44no—
155C4⋊C4×C31—44no—
156C4⋊C4×C31—48no—
157C22⋊C4×C31—232no—
158C22⋊C4×C31—416no—
159C22⋊C4×C31—48no—
160C22⋊C4×C31—48no—
161C42×C31—216no—
162C42×C31—28no—
163C42×C31—28no—
164C42×C31—216no—
165C42×C31—28no—
166#14700C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—24no—
168C23×C31—28no—
169C23×C31—24no—
170C23×C31—24no—
171C23×C31—28no—
172C23×C31—44no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—44no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—44no—
180C23×C31—44no—
181C23×C31—44no—
182C23×C31—44no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—44no—
186C23×C31—44no—
187C23×C31—44no—
188C23×C31—44no—
189C23×C31—42no—
190C23×C31C23×C411yescharacteristic
191C23×C31C22×D411yescharacteristic
192C23×C31C22×D411yescharacteristic
193C23×C31C22×D412yes—
194C23×C31C2×#4111yescharacteristic
195C23×C31C2×#4111yescharacteristic
196C23×C31C2×#4111yescharacteristic
197C23×C31C2×#4112yes—
198C23×C31C2×#4111yescharacteristic
199C23×C31C2×#4112yes—
200C23×C31C2×#4112yes—
201D4×C31—416no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—24no—
214C2×C4×C31—24no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—24no—
218C2×C4×C31—24no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—24no—
222C2×C4×C31—28no—
223C2×C4×C31—24no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—44no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—44no—
234C2×C4×C31—42no—
235C2×C4×C31—44no—
236C2×C4×C31—44no—
237C2×C4×C31#11714yes—
238C2×C4×C31C4×D418yes—
239C2×C4×C31C4×D414yes—
240C2×C4×C31C4×D414yes—
241C2×C4×C31#12118yes—
242C2×C4×C31#12114yes—
243C2×C4×C31#12314yes—
244C2×C4×C31#12414yes—
245C2×C4×C31#12414yes—
246C2×C4×C31#12614yes—
247C2×C4×C31#12714yes—
248C2×#390C2×C3111yescharacteristic
249C2×#390C2×C3111yescharacteristic
250C2×#390C2×C3111yescharacteristic
251C2×#389C2×C3111yescharacteristic, complemented
252C2×#388C2×C3114yescomplemented
253C2×C43C2×C3111yescharacteristic, complemented
254#1437C2×C3114yescomplemented
255C4⋊#95C2×C3111yescharacteristic, complemented
256C4⋊#95C2×C3111yescharacteristic, complemented
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—48no—
260C22×C31—48no—
261C22×C31—44no—
262C22×C31—82no—
263C22×C31—84no—
264C22×C31—82no—
265C22×C31—82no—
266C22×C31—84no—
267C22×C31—82no—
268C22×C31C2×#11711yescharacteristic
269C22×C31C2×C4×D411yescharacteristic
270C22×C31C2×C4×D412yes—
271C22×C31C2×C4×D411yescharacteristic
272C22×C31C4×#4112yes—
273C22×C31C2×#12111yescharacteristic
274C22×C31C2×#12112yes—
275C22×C31C2×#12311yescharacteristic
276C22×C31#52512yes—
277C22×C31C2×#12411yescharacteristic
278C22×C31C2×#12411yescharacteristic
279C22×C31C2×#12611yescharacteristic
280C22×C31#52912yes—
281C22×C31#52912yes—
282C22×C31#52912yes—
283C22×C31#52912yes—
284C22×C31C2×#12711yescharacteristic
285C22×C31#53211yescharacteristic
286C22×C31#53212yes—
287C22×C31#53212yes—
288C22×C31#53212yes—
289C22×C31#53211yescharacteristic
290C22×C31#53212yes—
291C4×C31—24no—
292C4×C31—28no—
293C4×C31—24no—
294C4×C31—28no—
295C4×C31—44no—
296C4×C31—82no—
297C4×C31—82no—
298C4×C31#38818yescomplemented
299C23×D4C22×C3111yescharacteristic
300C22×C4⋊C4C22×C3111yescharacteristic
301C22×C4⋊C4C22×C3111yescharacteristic
302C22×C4⋊C4C22×C3112yes—
303C22×C4⋊C4C22×C3111yescharacteristic
304C22×C22⋊C4C22×C3112yes—
305C22×C22⋊C4C22×C3114yes—
306C22×C22⋊C4C22×C3112yes—
307C22×C22⋊C4C22×C3114yes—
308C22×C42C22×C3112yes—
309C22×C42C22×C3111yescharacteristic
310C22×C42C22×C3111yescharacteristic
311C22×C42C22×C3111yescharacteristic
312C22×C42C22×C3112yes—
313#390—24no—
314#390—24no—
315#390C4×C3118yescomplemented
316#389—24no—
317#388—216no—
318C2×#95C22×C3114yes—
319C2×#95C22×C3114yes—
320C2×#95C22×C3112yes—
321C43—24no—
322C2×C31—84no—
323C2×C31C2×#38811yescharacteristic
324C2×C31#229312yes—
325C2×C31C4×#12112yes—
326C2×C31C4×#12411yescharacteristic
327C2×C31C4×#12711yescharacteristic
328C2×C31#285011yescharacteristic
329C2×C31#285412yes—
330C2×C31#285611yescharacteristic
331C2×C31#285811yescharacteristic
332C2×C31#286012yes—
333C2×C31#286111yescharacteristic
334C25C23×C3112yes—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4C2×C4×C3114yes—
342C23×C4C23×C3112yes—
343C23×C4C23×C3112yes—
344C23×C4C23×C3112yes—
345C23×C4C23×C3112yes—
346C23×C4C23×C3111yescharacteristic
347C23×C4C23×C3112yes—
348C23×C4C23×C3111yescharacteristic
349C23×C4C23×C3111yescharacteristic
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4C2×C4×C3114yes—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—28no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4C2×C4×C31116yes—
371C2×C42—24no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42C2×C4×C3114yes—
383C2×C42D4×C3114yes—
384C2×C42D4×C3114yes—
385C2×C42D4×C3118yescomplemented
386#95—28no—
387#95—28no—
388#95—24no—
389C31#1470011yescharacteristic, direct product factor, Sylow
390C24—24no—
391C24—22no—
392C24—22no—
393C24—24no—
394C24—24no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C22×C4×C3114yes—
401C24C24×C3111yescharacteristic
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4C22×C4×C3114yes—
438C22×C4C22×C4×C3114yes—
439C22×C4C22×C4×C3112yes—
440C22×C4C2×D4×C3114yes—
441C22×C4C2×D4×C3114yes—
442C22×C4C2×D4×C3114yes—
443C22×C4C2×D4×C3112yes—
444C22×C4C2×D4×C3112yes—
445C22×C4C2×D4×C3114yes—
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3112yes—
448C22×C4#41×C3114yes—
449C22×C4#41×C3114yes—
450C22×C4#41×C3114yes—
451C22×C4#41×C3112yes—
452C22×C4#41×C3112yes—
453C22×C4#41×C3112yes—
454C22×C4#41×C3112yes—
455C4⋊C4—28no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C22⋊C4—232no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—216no—
464C42—28no—
465C42—28no—
466C42—216no—
467C42—28no—
468C23—24no—
469C23—28no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23C23×C4×C3111yescharacteristic
492C23C22×D4×C3111yescharacteristic
493C23C22×D4×C3111yescharacteristic
494C23C22×D4×C3112yes—
495C23C2×#41×C3111yescharacteristic
496C23C2×#41×C3111yescharacteristic
497C23C2×#41×C3111yescharacteristic
498C23C2×#41×C3112yes—
499C23C2×#41×C3111yescharacteristic
500C23C2×#41×C3112yes—
501C23C2×#41×C3112yes—
502D4—416no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4C31×#11714yes—
539C2×C4C4×D4×C3118yes—
540C2×C4C4×D4×C3114yes—
541C2×C4C4×D4×C3114yes—
542C2×C4C31×#12118yes—
543C2×C4C31×#12114yes—
544C2×C4C31×#12314yes—
545C2×C4C31×#12414yes—
546C2×C4C31×#12414yes—
547C2×C4C31×#12614yes—
548C2×C4C31×#12714yes—
549C22—44no—
550C22—48no—
551C22—48no—
552C22—44no—
553C22—44no—
554C22—82no—
555C22—84no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22—82no—
560C22C2×C31×#11711yescharacteristic
561C22C2×C4×D4×C3111yescharacteristic
562C22C2×C4×D4×C3111yescharacteristic
563C22C2×C4×D4×C3112yes—
564C22C4×#41×C3112yes—
565C22C2×C31×#12111yescharacteristic
566C22C2×C31×#12112yes—
567C22C2×C31×#12311yescharacteristic
568C22C31×#52512yes—
569C22C2×C31×#12411yescharacteristic
570C22C2×C31×#12411yescharacteristic
571C22C2×C31×#12611yescharacteristic
572C22C31×#52912yes—
573C22C31×#52912yes—
574C22C31×#52912yes—
575C22C31×#52912yes—
576C22C2×C31×#12711yescharacteristic
577C22C31×#53211yescharacteristic
578C22C31×#53212yes—
579C22C31×#53212yes—
580C22C31×#53212yes—
581C22C31×#53211yescharacteristic
582C22C31×#53212yes—
583C4—24no—
584C4—28no—
585C4—24no—
586C4—28no—
587C4—44no—
588C4—82no—
589C4—82no—
590C4C31×#38818yescomplemented
591C2—84no—
592C2C2×C31×#38811yescharacteristic
593C2C31×#229312yes—
594C2C4×C31×#12112yes—
595C2C4×C31×#12411yescharacteristic
596C2C4×C31×#12711yescharacteristic
597C2C31×#285011yescharacteristic
598C2C31×#285412yes—
599C2C31×#285611yescharacteristic
600C2C31×#285811yescharacteristic
601C2C31×#286012yes—
602C2C31×#286111yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C31×#389C2semidirect
5. C2×C31×#388C2semidirect
6. C2×C43×C31C2semidirect
7. C31×#1437C2semidirect
8. C31×C4⋊#95C2semidirect
9. C31×C4⋊#95C2semidirect
26. C31×#390C4semidirect
84. C2×C42×C31D4semidirect
166. #14700C31direct (N × H)
251. C2×#389C2×C31semidirect
252. C2×#388C2×C31semidirect
253. C2×C43C2×C31semidirect
254. #1437C2×C31semidirect
255. C4⋊#95C2×C31semidirect
256. C4⋊#95C2×C31semidirect
298. C4×C31#388semidirect
315. #390C4×C31semidirect
385. C2×C42D4×C31semidirect
389. C31#14700direct (N × H)
590. C4C31×#388semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#3902
C2×C31×#3892
C2×C31×#3882
C2×C43×C312
C31×#14372
C31×C4⋊#952
#1470031