Finite Group Databasebeta

← C31×#20898 #21477657

Subgroups — C31×#20898 #21477657

Order 7936.

2230 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C31C211yesmaximal, characteristic, complemented
2C22×C4×D4×C31C211yesmaximal, characteristic
3C2×C31×#407C211yesmaximal, characteristic, complemented
4C2×C31×#407C212yesmaximal, complemented
5C31×#1870C214yesmaximal, complemented
6C31×#1869C214yesmaximal, complemented
7C31×#1728C212yesmaximal, complemented
8C23×D4×C31C411yescharacteristic
9C24×C4×C31C2211yescharacteristic
10C24×C4×C31C2211yescharacteristic
11C22×M16×C31—24no—
12C22×M16×C31—28no—
13C22×M16×C31C2212yes—
14C22×M16×C31C2212yes—
15C22×M16×C31C2212yescomplemented
16C23×C8×C31C2212yescomplemented
17C2×C4×D4×C31—24no—
18C2×C4×D4×C31—24no—
19C2×C4×D4×C31C412yes—
20C2×C4×D4×C31C412yes—
21C2×C4×D4×C31C2212yes—
22C2×C4×D4×C31C2212yes—
23C22×C4⋊C4×C31C411yescharacteristic
24C22×C22⋊C4×C31C411yescharacteristic
25C22×C22⋊C4×C31C411yescharacteristic
26C22×C42×C31C2211yescharacteristic
27C31×#407—24no—
28C31×#407—28no—
29C2×C31×C22⋊C8C2212yescomplemented
30C2×C31×C22⋊C8C2218yescomplemented
31C2×C31×C22⋊C8C2212yescomplemented
32C31×#336C2218yescomplemented
33C25×C31C2×C411yescharacteristic
34C25×C31C2×C411yescharacteristic
35C22×D4×C31—24no—
36C22×D4×C31—28no—
37C22×D4×C31C2×C412yes—
38C22×D4×C31C2×C412yes—
39C23×C4×C31—24no—
40C23×C4×C31—24no—
41C23×C4×C31—24no—
42C23×C4×C31—22no—
43C23×C4×C31—22no—
44C23×C4×C31—24no—
45C23×C4×C31C2×C411yescharacteristic
46C23×C4×C31C2×C411yescharacteristic
47C23×C4×C31C2×C412yes—
48C23×C4×C31C2×C411yescharacteristic
49C23×C4×C31C2×C411yescharacteristic
50C23×C4×C31C2×C411yescharacteristic
51C23×C4×C31C2×C411yescharacteristic
52C23×C4×C31D414yes—
53C23×C4×C31D414yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C2×M16×C31—232no—
59C2×M16×C31—24no—
60C2×M16×C31—24no—
61C2×M16×C31—24no—
62C2×M16×C31—24no—
63C2×M16×C31—24no—
64C2×M16×C31—24no—
65C2×M16×C31—28no—
66C2×M16×C31D418yescomplemented
67C22×C8×C31—24no—
68C22×C8×C31—28no—
69C22×C8×C31C2314yes—
70C22×C8×C31C2314yes—
71C4×D4×C31—28no—
72C4×D4×C31—28no—
73C4×D4×C31—24no—
74C4×D4×C31—24no—
75C4×D4×C31—24no—
76C4×D4×C31—24no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31C2×C412yes—
79C2×C4⋊C4×C31C2×C412yes—
80C2×C22⋊C4×C31—24no—
81C2×C22⋊C4×C31—24no—
82C2×C22⋊C4×C31C2×C412yes—
83C2×C22⋊C4×C31C2×C412yes—
84C2×C22⋊C4×C31C2×C412yes—
85C2×C22⋊C4×C31C2×C412yes—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31C2×C412yes—
89C2×C42×C31C2312yes—
90C31×C22⋊C8—24no—
91C31×C22⋊C8—24no—
92C31×C22⋊C8—216no—
93C24×C31—22no—
94C24×C31—24no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—24no—
98C24×C31—42no—
99C24×C31—42no—
100C24×C31C22⋊C414yes—
101C24×C31C22×C411yescharacteristic
102C24×C31C22×C412yes—
103C24×C31C22×C411yescharacteristic
104C24×C31C22×C411yescharacteristic
105C2×D4×C31—232no—
106C2×D4×C31—24no—
107C2×D4×C31—24no—
108C2×D4×C31—24no—
109C2×D4×C31—24no—
110C2×D4×C31—24no—
111C2×D4×C31M1618yescomplemented
112C22×C4×C31—22no—
113C22×C4×C31—24no—
114C22×C4×C31—24no—
115C22×C4×C31—24no—
116C22×C4×C31—28no—
117C22×C4×C31—216no—
118C22×C4×C31—28no—
119C22×C4×C31—24no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—24no—
127C22×C4×C31—24no—
128C22×C4×C31—28no—
129C22×C4×C31—24no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—24no—
133C22×C4×C31—42no—
134C22×C4×C31—44no—
135C22×C4×C31—48no—
136C22×C4×C31—44no—
137C22×C4×C31C22⋊C414yes—
138C22×C4×C31C22⋊C414yes—
139C22×C4×C31C22⋊C414yes—
140C22×C4×C31C22×C412yes—
141C22×C4×C31C22×C412yes—
142C22×C4×C31C22×C411yescharacteristic
143C22×C4×C31C22×C411yescharacteristic
144C22×C4×C31C22×C412yes—
145C22×C4×C31C22×C411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D414yes—
153C22×C4×C31#4114yes—
154C22×C4×C31C2411yescharacteristic
155M16×C31—216no—
156M16×C31—216no—
157M16×C31—48no—
158C2×C8×C31—216no—
159C2×C8×C31—24no—
160C2×C8×C31—24no—
161C2×C8×C31—24no—
162C2×C8×C31—24no—
163C2×C8×C31—24no—
164C2×C8×C31C2×D418yescomplemented
165C4⋊C4×C31—24no—
166C4⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—24no—
171C42×C31—24no—
172C42×C31—22no—
173C42×C31—22no—
174#20898C3111yesmaximal, characteristic, direct product factor, Sylow
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—28no—
180C23×C31—22no—
181C23×C31—24no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—24no—
187C23×C31—28no—
188C23×C31—28no—
189C23×C31—44no—
190C23×C31—416no—
191C23×C31—416no—
192C23×C31—44no—
193C23×C31—44no—
194C23×C31C2×C22⋊C412yes—
195C23×C31C2×C22⋊C412yes—
196C23×C31C2×C22⋊C412yes—
197C23×C31C4×D414yes—
198C23×C31C2×M1614yes—
199C23×C31C2×M1614yescomplemented
200C23×C31C23×C411yescharacteristic
201D4×C31—216no—
202D4×C31—216no—
203C2×C4×C31—28no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—28no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—28no—
216C2×C4×C31—28no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—44no—
222C2×C4×C31—44no—
223C2×C4×C31—42no—
224C2×C4×C31—44no—
225C2×C4×C31—42no—
226C2×C4×C31—48no—
227C2×C4×C31C2×C22⋊C412yes—
228C2×C4×C31C2×C22⋊C412yes—
229C2×C4×C31C2×C22⋊C412yes—
230C2×C4×C31C4×D414yes—
231C2×C4×C31#12014yes—
232C2×C4×C31#12118yes—
233C2×C4×C31#12314yes—
234C2×C4×C31C2×M1614yescomplemented
235C2×C4×C31C22×D412yes—
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C2×#4112yes—
239C8×C31—28no—
240C8×C31—44no—
241C23×M16C2×C3111yescharacteristic, complemented
242C22×C4×D4C2×C3111yescharacteristic
243C2×#407C2×C3111yescharacteristic, complemented
244C2×#407C2×C3112yescomplemented
245#1870C2×C3114yescomplemented
246#1869C2×C3114yescomplemented
247#1728C2×C3112yescomplemented
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—22no—
251C22×C31—28no—
252C22×C31—28no—
253C22×C31—22no—
254C22×C31—22no—
255C22×C31—24no—
256C22×C31—22no—
257C22×C31—28no—
258C22×C31—22no—
259C22×C31—416no—
260C22×C31—416no—
261C22×C31—42no—
262C22×C31—44no—
263C22×C31—44no—
264C22×C31—42no—
265C22×C31#38614yes—
266C22×C31#40714yescomplemented
267C22×C31C22×C22⋊C411yescharacteristic
268C22×C31C2×C4×D412yes—
269C22×C31C22×M1612yes—
270C22×C31#56812yes—
271C4×C31—24no—
272C4×C31—22no—
273C4×C31—22no—
274C4×C31—42no—
275C4×C31—42no—
276C4×C31#38614yes—
277C4×C31C2×#12011yescharacteristic
278C4×C31C2×#12112yes—
279C4×C31C2×#12311yescharacteristic
280C23×D4C4×C3111yescharacteristic
281C24×C4C22×C3111yescharacteristic
282C24×C4C22×C3111yescharacteristic
283C22×M16—24no—
284C22×M16—28no—
285C22×M16C22×C3112yes—
286C22×M16C22×C3112yes—
287C22×M16C22×C3112yescomplemented
288C23×C8C22×C3112yescomplemented
289C2×C4×D4—24no—
290C2×C4×D4—24no—
291C2×C4×D4C4×C3112yes—
292C2×C4×D4C4×C3112yes—
293C2×C4×D4C22×C3112yes—
294C2×C4×D4C22×C3112yes—
295C22×C4⋊C4C4×C3111yescharacteristic
296C22×C22⋊C4C4×C3111yescharacteristic
297C22×C22⋊C4C4×C3111yescharacteristic
298C22×C42C22×C3111yescharacteristic
299#407—24no—
300#407—28no—
301C2×C22⋊C8C22×C3112yescomplemented
302C2×C22⋊C8C22×C3118yescomplemented
303C2×C22⋊C8C22×C3112yescomplemented
304#336C22×C3118yescomplemented
305C2×C31—24no—
306C2×C31—24no—
307C2×C31—42no—
308C2×C31—44no—
309C2×C31C2×#38611yescharacteristic
310C2×C31C2×#40711yescharacteristic
311C2×C31#287911yescharacteristic
312C2×C31D4×M1614yes—
313C25C2×C4×C3111yescharacteristic
314C25C2×C4×C3111yescharacteristic
315C22×D4—24no—
316C22×D4—28no—
317C22×D4C2×C4×C3112yes—
318C22×D4C2×C4×C3112yes—
319C23×C4—24no—
320C23×C4—24no—
321C23×C4—22no—
322C23×C4—22no—
323C23×C4—24no—
324C23×C4—24no—
325C23×C4C2×C4×C3111yescharacteristic
326C23×C4C2×C4×C3111yescharacteristic
327C23×C4C2×C4×C3112yes—
328C23×C4C2×C4×C3111yescharacteristic
329C23×C4C2×C4×C3111yescharacteristic
330C23×C4C2×C4×C3111yescharacteristic
331C23×C4C2×C4×C3111yescharacteristic
332C23×C4D4×C3114yes—
333C23×C4D4×C3114yes—
334C23×C4C23×C3112yes—
335C23×C4C23×C3111yescharacteristic
336C23×C4C23×C3111yescharacteristic
337C23×C4C23×C3111yescharacteristic
338C2×M16—232no—
339C2×M16—24no—
340C2×M16—24no—
341C2×M16—24no—
342C2×M16—24no—
343C2×M16—24no—
344C2×M16—24no—
345C2×M16—28no—
346C2×M16D4×C3118yescomplemented
347C22×C8—24no—
348C22×C8—28no—
349C22×C8C23×C3114yes—
350C22×C8C23×C3114yes—
351C4×D4—28no—
352C4×D4—28no—
353C4×D4—24no—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4C2×C4×C3112yes—
359C2×C4⋊C4C2×C4×C3112yes—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4C2×C4×C3112yes—
363C2×C22⋊C4C2×C4×C3112yes—
364C2×C22⋊C4C2×C4×C3112yes—
365C2×C22⋊C4C2×C4×C3112yes—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42C2×C4×C3112yes—
369C2×C42C23×C3112yes—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—216no—
373C31#2089811yescharacteristic, direct product factor, Sylow
374C24—22no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—42no—
380C24—42no—
381C24C22⋊C4×C3114yes—
382C24C22×C4×C3111yescharacteristic
383C24C22×C4×C3112yes—
384C24C22×C4×C3111yescharacteristic
385C24C22×C4×C3111yescharacteristic
386C2×D4—232no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4M16×C3118yescomplemented
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4C22⋊C4×C3114yes—
419C22×C4C22⋊C4×C3114yes—
420C22×C4C22⋊C4×C3114yes—
421C22×C4C22×C4×C3112yes—
422C22×C4C22×C4×C3112yes—
423C22×C4C22×C4×C3111yescharacteristic
424C22×C4C22×C4×C3111yescharacteristic
425C22×C4C22×C4×C3112yes—
426C22×C4C22×C4×C3111yescharacteristic
427C22×C4C2×D4×C3112yes—
428C22×C4C2×D4×C3112yes—
429C22×C4C2×D4×C3112yes—
430C22×C4C2×D4×C3112yes—
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3112yes—
433C22×C4C2×D4×C3114yes—
434C22×C4#41×C3114yes—
435C22×C4C24×C3111yescharacteristic
436M16—216no—
437M16—216no—
438M16—48no—
439C2×C8—216no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8C2×D4×C3118yescomplemented
446C4⋊C4—24no—
447C4⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C42—24no—
453C42—22no—
454C42—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—28no—
469C23—44no—
470C23—416no—
471C23—416no—
472C23—44no—
473C23—44no—
474C23C2×C22⋊C4×C3112yes—
475C23C2×C22⋊C4×C3112yes—
476C23C2×C22⋊C4×C3112yes—
477C23C4×D4×C3114yes—
478C23C2×M16×C3114yes—
479C23C2×M16×C3114yescomplemented
480C23C23×C4×C3111yescharacteristic
481D4—216no—
482D4—216no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4C2×C22⋊C4×C3112yes—
508C2×C4C2×C22⋊C4×C3112yes—
509C2×C4C2×C22⋊C4×C3112yes—
510C2×C4C4×D4×C3114yes—
511C2×C4C31×#12014yes—
512C2×C4C31×#12118yes—
513C2×C4C31×#12314yes—
514C2×C4C2×M16×C3114yescomplemented
515C2×C4C22×D4×C3112yes—
516C2×C4C22×D4×C3111yescharacteristic
517C2×C4C22×D4×C3111yescharacteristic
518C2×C4C2×#41×C3112yes—
519C8—28no—
520C8—44no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—28no—
525C22—28no—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—22no—
530C22—28no—
531C22—22no—
532C22—416no—
533C22—416no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22C31×#38614yes—
539C22C31×#40714yescomplemented
540C22C22×C22⋊C4×C3111yescharacteristic
541C22C2×C4×D4×C3112yes—
542C22C22×M16×C3112yes—
543C22C31×#56812yes—
544C4—24no—
545C4—22no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4C31×#38614yes—
550C4C2×C31×#12011yescharacteristic
551C4C2×C31×#12112yes—
552C4C2×C31×#12311yescharacteristic
553C2—24no—
554C2—24no—
555C2—42no—
556C2—44no—
557C2C2×C31×#38611yescharacteristic
558C2C2×C31×#40711yescharacteristic
559C2C31×#287911yescharacteristic
560C2D4×M16×C3114yes—
subgroupG/H
centerC22×C4×C31C24
commutatorC23C23×C4×C31
Frattini, cosocleC22×C4C24×C31
socleC23×C31C23×C4
prime radicalC23×D4×C31C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×M16×C31C2semidirect
3. C2×C31×#407C2semidirect
4. C2×C31×#407C2semidirect
5. C31×#1870C2semidirect
6. C31×#1869C2semidirect
7. C31×#1728C2semidirect
15. C22×M16×C31C22semidirect
16. C23×C8×C31C22semidirect
29. C2×C31×C22⋊C8C22semidirect
30. C2×C31×C22⋊C8C22semidirect
31. C2×C31×C22⋊C8C22semidirect
32. C31×#336C22semidirect
66. C2×M16×C31D4semidirect
111. C2×D4×C31M16semidirect
164. C2×C8×C31C2×D4semidirect
174. #20898C31direct (N × H)
199. C23×C31C2×M16semidirect
234. C2×C4×C31C2×M16semidirect
241. C23×M16C2×C31semidirect
243. C2×#407C2×C31semidirect
244. C2×#407C2×C31semidirect
245. #1870C2×C31semidirect
246. #1869C2×C31semidirect
247. #1728C2×C31semidirect
266. C22×C31#407semidirect
287. C22×M16C22×C31semidirect
288. C23×C8C22×C31semidirect
301. C2×C22⋊C8C22×C31semidirect
302. C2×C22⋊C8C22×C31semidirect
303. C2×C22⋊C8C22×C31semidirect
304. #336C22×C31semidirect
346. C2×M16D4×C31semidirect
373. C31#20898direct (N × H)
392. C2×D4M16×C31semidirect
445. C2×C8C2×D4×C31semidirect
479. C23C2×M16×C31semidirect
514. C2×C4C2×M16×C31semidirect
539. C22C31×#407semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C312
C22×C4×D4×C312
C2×C31×#4072
C31×#18702
C31×#18692
C31×#17282
#2089831