Finite Group Databasebeta

← C31×#24326 #21481085

Subgroups — C31×#24326 #21481085

Order 7936.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8×C31C211yesmaximal, characteristic, complemented
2C2×C31×#474C211yesmaximal, characteristic, complemented
3C2×C4×D8×C31C211yesmaximal, characteristic
4C31×#2538C211yesmaximal, characteristic, complemented
5C31×#2057C212yesmaximal, complemented
6C31×#2023C212yesmaximal, complemented
7C31×#1990C211yesmaximal, characteristic, complemented
8C31×#1984C211yesmaximal, characteristic, complemented
9C31×#1879C212yesmaximal
10C31×#1862C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C23×Q8×C31—21no—
13C22×D8×C31C2211yescharacteristic
14D4×Q8×C31—24no—
15D4×Q8×C31—24no—
16C2×C31×C4⋊Q8C2212yes—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8—22no—
20C2×C31×C22⋊Q8—21no—
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×D4×C31C2212yes—
23C2×C4×D4×C31C2211yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C31×#474—24no—
26C4×D8×C31—24no—
27C2×C31×#107C2211yescharacteristic
28C2×C31×#107C2212yes—
29C2×C31×C4⋊C8C2211yescharacteristic
30C2×C31×#102C2212yes—
31C2×C31×#102C2211yescharacteristic
32C2×C31×#102C2212yes—
33C2×C31×#102C2211yescharacteristic
34C2×C4×C8×C31C2211yescharacteristic
35C31×#396—22no—
36C31×#390C2212yes—
37C31×#389C2212yes—
38C31×#384C2211yescharacteristic
39C31×#380—21no—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#336C2212yes—
44C31×#336C2212yes—
45C22×Q8×C31—28no—
46C22×Q8×C31—22no—
47C22×Q8×C31—21no—
48C22×Q8×C31—44no—
49C22×Q8×C31—44no—
50C22×Q8×C31D411yescharacteristic
51C22×D4×C31C2311yescharacteristic
52C22×D4×C31C2311yescharacteristic
53C23×C4×C31—22no—
54C23×C4×C31—21no—
55C23×C4×C31—21no—
56C2×D8×C31—22no—
57C2×D8×C31—22no—
58C2×D8×C31Q814yescomplemented
59C22×C8×C31C2311yescharacteristic
60C22×C8×C31C2312yes—
61C22×C8×C31C2311yescharacteristic
62C31×C4⋊Q8—24no—
63C31×C4⋊Q8—24no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8D414yes—
66C31×C22⋊Q8—28no—
67C31×C22⋊Q8—216no—
68C31×C22⋊Q8—44no—
69C4×Q8×C31—22no—
70C4×Q8×C31—22no—
71C4×D4×C31—24no—
72C4×D4×C31—24no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31D414yes—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31D412yes—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2312yes—
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31—22no—
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C42×C31C2312yes—
94C2×C42×C31C2311yescharacteristic
95C31×#107—22no—
96C31×#107—24no—
97C31×C4⋊C8—22no—
98C31×#102—22no—
99C31×#102—24no—
100C31×#102—22no—
101C31×#102—24no—
102C4×C8×C31—22no—
103C31×#95—22no—
104C24×C31—21no—
105C24×C31—21no—
106C2×Q8×C31—28no—
107C2×Q8×C31—22no—
108C2×Q8×C31—24no—
109C2×Q8×C31—22no—
110C2×Q8×C31—44no—
111C2×Q8×C31—432no—
112C2×Q8×C31—44no—
113C2×Q8×C31—42no—
114C2×Q8×C31—42no—
115C2×Q8×C31D814yescomplemented
116C2×D4×C31—21no—
117C2×D4×C31—21no—
118C2×D4×C31—21no—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—21no—
123C2×D4×C31—21no—
124C2×D4×C31C2×D412yes—
125C2×D4×C31C2×Q812yes—
126C2×D4×C31C2×Q812yescomplemented
127C2×D4×C31#4112yes—
128C22×C4×C31—28no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—24no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—44no—
135C22×C4×C31—44no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2411yescharacteristic
146D8×C31—28no—
147C2×C8×C31—21no—
148C2×C8×C31—21no—
149C2×C8×C31—21no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—22no—
155C2×C8×C31C2×Q812yescomplemented
156C4⋊C4×C31—28no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—42no—
167C4⋊C4×C31—44no—
168C4⋊C4×C31—81no—
169C4⋊C4×C31C2×D414yes—
170C4⋊C4×C31C2×D412yes—
171C22⋊C4×C31—28no—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—42no—
174C22⋊C4×C31—44no—
175C22⋊C4×C31—82no—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31C2×D412yes—
179C42×C31C2×D412yes—
180#24326C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31C22×D411yescharacteristic
196Q8×C31—28no—
197Q8×C31—416no—
198Q8×C31—48no—
199D4×C31—22no—
200D4×C31—22no—
201D4×C31—22no—
202D4×C31—22no—
203D4×C31—22no—
204D4×C31—41no—
205D4×C31—41no—
206D4×C31C22⋊Q814yescomplemented
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—24no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—416no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—48no—
222C2×C4×C31—41no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—84no—
229C2×C4×C31#12012yes—
230C2×C4×C31#12012yes—
231C2×C4×C31C2×D814yes—
232C2×C4×C31C2×D812yes—
233C2×C4×C31AGL(1,ℤ/8)12yes—
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×Q811yescharacteristic
237C2×C4×C31C2×#4111yescharacteristic
238C2×C4×C312−1+411yescharacteristic
239C2×C4×C312−1+411yescharacteristic
240C8×C31—22no—
241C8×C31—41no—
242C8×C31—42no—
243C2×D4×Q8C2×C3111yescharacteristic, complemented
244C2×#474C2×C3111yescharacteristic, complemented
245C2×C4×D8C2×C3111yescharacteristic
246#2538C2×C3111yescharacteristic, complemented
247#2057C2×C3112yescomplemented
248#2023C2×C3112yescomplemented
249#1990C2×C3111yescharacteristic, complemented
250#1984C2×C3111yescharacteristic, complemented
251#1879C2×C3112yes—
252#1862C2×C3112yescomplemented
253#1802C2×C3111yescharacteristic, complemented
254C22×C31—24no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31—81no—
268C22×C31C2×#12011yescharacteristic
269C22×C31#54811yescharacteristic
270C22×C31D4×Q811yescharacteristic
271C22×C31C22×D811yescharacteristic
272C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C31#57711yescharacteristic
274C22×C31#57811yescharacteristic
275C4×C31—24no—
276C4×C31—22no—
277C4×C31—44no—
278C4×C31—42no—
279C4×C31—42no—
280C4×C31—82no—
281C4×C31—81no—
282C4×C31#44714yes—
283C4×C31C2×C22⋊Q811yescharacteristic
284C4×C31#53611yescharacteristic
285C4×C31#54811yescharacteristic
286C4×C31D4×Q811yescharacteristic
287C23×Q8—21no—
288C22×D8C22×C3111yescharacteristic
289D4×Q8—24no—
290D4×Q8—24no—
291C2×C4⋊Q8C22×C3112yes—
292C2×C4⋊Q8C22×C3111yescharacteristic
293C2×C22⋊Q8—21no—
294C2×C22⋊Q8—21no—
295C2×C22⋊Q8—22no—
296C2×C4×Q8C22×C3111yescharacteristic
297C2×C4×D4C22×C3112yes—
298C2×C4×D4C22×C3111yescharacteristic
299C2×C4×D4C22×C3111yescharacteristic
300#474—24no—
301C4×D8—24no—
302C2×#107C22×C3111yescharacteristic
303C2×#107C22×C3112yes—
304C2×C4⋊C8C22×C3111yescharacteristic
305C2×#102C22×C3112yes—
306C2×#102C22×C3111yescharacteristic
307C2×#102C22×C3112yes—
308C2×#102C22×C3111yescharacteristic
309C2×C4×C8C22×C3111yescharacteristic
310#396—22no—
311#390C22×C3112yes—
312#389C22×C3112yes—
313#384C22×C3111yescharacteristic
314#380—21no—
315#340C22×C3112yes—
316#340C22×C3112yes—
317#340C22×C3112yes—
318#336C22×C3112yes—
319#336C22×C3112yes—
320C2×C31—44no—
321C2×C31—82no—
322C2×C31#240911yescharacteristic
323C2×C31C2×#44711yescharacteristic
324C2×C31#301311yescharacteristic
325C2×C31#335811yescharacteristic
326C2×C31#336711yescharacteristic
327C2×C31Q8×D811yescharacteristic
328C2×C31#338411yescharacteristic
329C22×Q8—28no—
330C22×Q8—22no—
331C22×Q8—21no—
332C22×Q8—44no—
333C22×Q8—44no—
334C22×Q8D4×C3111yescharacteristic
335C22×D4C23×C3111yescharacteristic
336C22×D4C23×C3111yescharacteristic
337C23×C4—22no—
338C23×C4—21no—
339C23×C4—21no—
340C2×D8—22no—
341C2×D8—22no—
342C2×D8Q8×C3114yescomplemented
343C22×C8C23×C3111yescharacteristic
344C22×C8C23×C3112yes—
345C22×C8C23×C3111yescharacteristic
346C4⋊Q8—24no—
347C4⋊Q8—24no—
348C4⋊Q8—22no—
349C4⋊Q8D4×C3114yes—
350C22⋊Q8—28no—
351C22⋊Q8—216no—
352C22⋊Q8—44no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×D4—24no—
356C4×D4—24no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4D4×C3114yes—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4D4×C3111yescharacteristic
367C2×C4⋊C4D4×C3112yes—
368C2×C4⋊C4C23×C3111yescharacteristic
369C2×C4⋊C4C23×C3112yes—
370C2×C4⋊C4C23×C3112yes—
371C2×C4⋊C4C23×C3111yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C42C23×C3112yes—
378C2×C42C23×C3111yescharacteristic
379#107—22no—
380#107—24no—
381C4⋊C8—22no—
382#102—22no—
383#102—24no—
384#102—22no—
385#102—24no—
386C4×C8—22no—
387#95—22no—
388C31#2432611yescharacteristic, direct product factor, Sylow
389C24—21no—
390C24—21no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—432no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8D8×C3114yescomplemented
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×D4×C3112yes—
410C2×D4C2×Q8×C3112yes—
411C2×D4C2×Q8×C3112yescomplemented
412C2×D4#41×C3112yes—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4C2×D4×C3112yes—
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3112yes—
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C24×C3111yescharacteristic
431D8—28no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8C2×Q8×C3112yescomplemented
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—81no—
454C4⋊C4C2×D4×C3114yes—
455C4⋊C4C2×D4×C3112yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C42—22no—
462C42—22no—
463C42C2×D4×C3112yes—
464C42C2×D4×C3112yes—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C22×D4×C3111yescharacteristic
480Q8—28no—
481Q8—416no—
482Q8—48no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4C31×C22⋊Q814yescomplemented
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4C31×#12012yes—
514C2×C4C31×#12012yes—
515C2×C4C2×D8×C3114yes—
516C2×C4C2×D8×C3112yes—
517C2×C4C31×AGL(1,ℤ/8)12yes—
518C2×C4C22×D4×C3111yescharacteristic
519C2×C4C22×D4×C3111yescharacteristic
520C2×C4C22×Q8×C3111yescharacteristic
521C2×C4C2×#41×C3111yescharacteristic
522C2×C4C31×2−1+411yescharacteristic
523C2×C4C31×2−1+411yescharacteristic
524C8—22no—
525C8—41no—
526C8—42no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C2×C31×#12011yescharacteristic
542C22C31×#54811yescharacteristic
543C22D4×Q8×C3111yescharacteristic
544C22C22×D8×C3111yescharacteristic
545C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
546C22C31×#57711yescharacteristic
547C22C31×#57811yescharacteristic
548C4—24no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—81no—
555C4C31×#44714yes—
556C4C2×C31×C22⋊Q811yescharacteristic
557C4C31×#53611yescharacteristic
558C4C31×#54811yescharacteristic
559C4D4×Q8×C3111yescharacteristic
560C2—44no—
561C2—82no—
562C2C31×#240911yescharacteristic
563C2C2×C31×#44711yescharacteristic
564C2C31×#301311yescharacteristic
565C2C31×#335811yescharacteristic
566C2C31×#336711yescharacteristic
567C2Q8×D8×C3111yescharacteristic
568C2C31×#338411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D8×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×D4×Q8×C31C2semidirect
2. C2×C31×#474C2semidirect
4. C31×#2538C2semidirect
5. C31×#2057C2semidirect
6. C31×#2023C2semidirect
7. C31×#1990C2semidirect
8. C31×#1984C2semidirect
10. C31×#1862C2semidirect
11. C31×#1802C2semidirect
58. C2×D8×C31Q8semidirect
115. C2×Q8×C31D8semidirect
126. C2×D4×C31C2×Q8semidirect
155. C2×C8×C31C2×Q8semidirect
180. #24326C31direct (N × H)
206. D4×C31C22⋊Q8semidirect
243. C2×D4×Q8C2×C31semidirect
244. C2×#474C2×C31semidirect
246. #2538C2×C31semidirect
247. #2057C2×C31semidirect
248. #2023C2×C31semidirect
249. #1990C2×C31semidirect
250. #1984C2×C31semidirect
252. #1862C2×C31semidirect
253. #1802C2×C31semidirect
342. C2×D8Q8×C31semidirect
388. C31#24326direct (N × H)
400. C2×Q8D8×C31semidirect
411. C2×D4C2×Q8×C31semidirect
440. C2×C8C2×Q8×C31semidirect
490. D4C31×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q8×C312
C2×C31×#4742
C2×C4×D8×C312
C31×#25382
C31×#20572
C31×#20232
C31×#19902
C31×#19842
C31×#18792
C31×#18622
C31×#18022
#2432631