Finite Group Databasebeta

← C31×#9923 #21466682

Subgroups — C31×#9923 #21466682

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#450C211yesmaximal, characteristic
2C31×#2627C211yesmaximal, characteristic, complemented
3C2×C31×#394C211yesmaximal, characteristic
4C31×#1786C211yesmaximal, characteristic
5C2×C31×#328C211yesmaximal, characteristic, complemented
6C2×C31×#324C211yesmaximal, characteristic, complemented
7C31×#1294C211yesmaximal, characteristic, complemented
8C23×D4×C31C2211yescharacteristic
9C22×SD16×C31—21no—
10C2×C31×C22⋊Q8C2211yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C22×C22⋊C4×C31—21no—
13C22×C22⋊C4×C31—21no—
14C31×#450—22no—
15C31×#450—22no—
16C2×C31×#102—21no—
17C2×C31×#102—22no—
18C2×C31×C22⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8C2211yescharacteristic
20C31×#400—21no—
21C31×#398—22no—
22C31×#395—21no—
23C31×#394—22no—
24C31×#394—22no—
25C31×#385—21no—
26C31×#381—21no—
27C31×#380C2211yescharacteristic
28C2×C31×#95C2211yescharacteristic
29C31×#328—22no—
30C31×#324—22no—
31C25×C31—21no—
32C22×Q8×C31D411yescharacteristic
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—
40C23×C4×C31—21no—
41C23×C4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×SD16×C31—24no—
45C2×SD16×C31—44no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C4⋊C4×C31D412yes—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—21no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—44no—
68C2×C22⋊C4×C31—42no—
69C31×#102—24no—
70C31×#102—44no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8D412yes—
75C31×#95—22no—
76C31×#95—22no—
77C31×#95—22no—
78C31×#95—21no—
79C31×#95—21no—
80C31×#95D412yes—
81C24×C31—24no—
82C24×C31—22no—
83C24×C31—42no—
84C24×C31—42no—
85C24×C31—42no—
86C24×C31—41no—
87C24×C31—41no—
88C24×C31—41no—
89C24×C31—41no—
90C24×C31—41no—
91C24×C31—41no—
92C24×C31C2×D411yescharacteristic
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—42no—
96C2×D4×C31—24no—
97C2×D4×C31—24no—
98C2×D4×C31—22no—
99C2×D4×C31—22no—
100C2×D4×C31—42no—
101C2×D4×C31—42no—
102C2×D4×C31—44no—
103C2×D4×C31—44no—
104C2×D4×C31—44no—
105C2×D4×C31—44no—
106C2×D4×C31—41no—
107C2×D4×C31—41no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—21no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—42no—
122C22×C4×C31—42no—
123C22×C4×C31—42no—
124C22×C4×C31—42no—
125C22×C4×C31—41no—
126C22×C4×C31—41no—
127C22×C4×C31SD1612yescomplemented
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31#4111yescharacteristic
134SD16×C31—48no—
135C2×C8×C31—22no—
136C2×C8×C31—42no—
137C2×C8×C31—42no—
138C2×C8×C31—41no—
139C4⋊C4×C31—21no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—42no—
142C4⋊C4×C31—41no—
143C4⋊C4×C31—41no—
144C4⋊C4×C31—41no—
145C4⋊C4×C31—41no—
146C4⋊C4×C31—41no—
147C4⋊C4×C31—41no—
148C4⋊C4×C31—82no—
149C22⋊C4×C31—48no—
150C22⋊C4×C31—48no—
151C22⋊C4×C31—41no—
152C22⋊C4×C31—41no—
153C22⋊C4×C31—41no—
154C22⋊C4×C31—41no—
155#9923C3111yesmaximal, characteristic, direct product factor, Sylow
156C23×C31—22no—
157C23×C31—22no—
158C23×C31—24no—
159C23×C31—42no—
160C23×C31—44no—
161C23×C31—44no—
162C23×C31—44no—
163C23×C31—44no—
164C23×C31—44no—
165C23×C31—42no—
166C23×C31—42no—
167C23×C31—42no—
168C23×C31—42no—
169C23×C31—42no—
170C23×C31—44no—
171C23×C31—82no—
172C23×C31—82no—
173C23×C31—82no—
174C23×C31—82no—
175C23×C31—82no—
176C23×C31—81no—
177C23×C31—81no—
178C23×C31—82no—
179C23×C31—81no—
180C23×C31—81no—
181C23×C31#12011yescharacteristic
182C23×C31#12011yescharacteristic
183C23×C31#12111yescharacteristic
184C23×C31C2×SD1611yescharacteristic
185C23×C31#13511yescharacteristic
186C23×C31AGL(1,ℤ/8)11yescharacteristic
187C23×C31#13711yescharacteristic
188Q8×C31—42no—
189Q8×C31—81no—
190D4×C31—24no—
191D4×C31—24no—
192D4×C31—42no—
193D4×C31—44no—
194D4×C31—42no—
195D4×C31—44no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—22no—
201C2×C4×C31—41no—
202C2×C4×C31—41no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—82no—
223C2×C4×C31—82no—
224C2×C4×C31—81no—
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12111yescharacteristic
227C2×C4×C31#12111yescharacteristic
228C2×C4×C31#12711yescharacteristic
229C8×C31—42no—
230C8×C31—81no—
231C2×#450C2×C3111yescharacteristic
232#2627C2×C3111yescharacteristic, complemented
233C2×#394C2×C3111yescharacteristic
234#1786C2×C3111yescharacteristic
235C2×#328C2×C3111yescharacteristic, complemented
236C2×#324C2×C3111yescharacteristic, complemented
237#1294C2×C3111yescharacteristic, complemented
238C22×C31—22no—
239C22×C31—22no—
240C22×C31—22no—
241C22×C31—22no—
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—44no—
247C22×C31—44no—
248C22×C31—44no—
249C22×C31—82no—
250C22×C31—82no—
251C22×C31—84no—
252C22×C31—84no—
253C22×C31—84no—
254C22×C31—84no—
255C22×C31—82no—
256C22×C31—82no—
257C22×C31#39211yescharacteristic
258C22×C31#44811yescharacteristic
259C22×C31#44911yescharacteristic
260C22×C31#45011yescharacteristic
261C22×C31#45211yescharacteristic
262C22×C31#45412yes—
263C22×C31B(4,𝔽2)12yes—
264C22×C31#46511yescharacteristic
265C22×C31#46811yescharacteristic
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—21no—
269C4×C31—21no—
270C4×C31—42no—
271C4×C31—82no—
272C4×C31—82no—
273C4×C31—81no—
274C4×C31—81no—
275C4×C31—81no—
276C23×D4C22×C3111yescharacteristic
277C22×SD16—21no—
278C2×C22⋊Q8C22×C3111yescharacteristic
279C22×C4⋊C4C22×C3111yescharacteristic
280C22×C22⋊C4—21no—
281C22×C22⋊C4—21no—
282#450—22no—
283#450—22no—
284C2×#102—21no—
285C2×#102—22no—
286C2×C22⋊C8C22×C3111yescharacteristic
287C2×C22⋊C8C22×C3111yescharacteristic
288#400—21no—
289#398—22no—
290#395—21no—
291#394—22no—
292#394—22no—
293#385—21no—
294#381—21no—
295#380C22×C3111yescharacteristic
296C2×#95C22×C3111yescharacteristic
297#328—22no—
298#324—22no—
299C2×C31—21no—
300C2×C31—21no—
301C2×C31—21no—
302C2×C31—21no—
303C2×C31—84no—
304C2×C31#159712yes—
305C2×C31#159912yes—
306C2×C31#200011yescharacteristic
307C2×C31#200311yescharacteristic
308C2×C31#202111yescharacteristic
309C25—21no—
310C22×Q8D4×C3111yescharacteristic
311C22×D4—21no—
312C22×D4—21no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—44no—
317C22×D4D4×C3112yes—
318C23×C4—21no—
319C23×C4D4×C3111yescharacteristic
320C23×C4D4×C3111yescharacteristic
321C23×C4C23×C3111yescharacteristic
322C2×SD16—24no—
323C2×SD16—44no—
324C22×C8—21no—
325C22×C8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4D4×C3111yescharacteristic
339C2×C4⋊C4D4×C3112yes—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—42no—
347#102—24no—
348#102—44no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8D4×C3112yes—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95D4×C3112yes—
359C31#992311yescharacteristic, direct product factor, Sylow
360C24—24no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24C2×D4×C3111yescharacteristic
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—42no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—41no—
386C2×D4—41no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4SD16×C3112yescomplemented
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C2×D4×C3111yescharacteristic
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4#41×C3111yescharacteristic
413SD16—48no—
414C2×C8—22no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—41no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—82no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—81no—
456C23—82no—
457C23—81no—
458C23—81no—
459C23C31×#12011yescharacteristic
460C23C31×#12011yescharacteristic
461C23C31×#12111yescharacteristic
462C23C2×SD16×C3111yescharacteristic
463C23C31×#13511yescharacteristic
464C23C31×AGL(1,ℤ/8)11yescharacteristic
465C23C31×#13711yescharacteristic
466Q8—42no—
467Q8—81no—
468D4—24no—
469D4—24no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—44no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4C31×#12011yescharacteristic
504C2×C4C31×#12111yescharacteristic
505C2×C4C31×#12111yescharacteristic
506C2×C4C31×#12711yescharacteristic
507C8—42no—
508C8—81no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—82no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22C31×#39211yescharacteristic
529C22C31×#44811yescharacteristic
530C22C31×#44911yescharacteristic
531C22C31×#45011yescharacteristic
532C22C31×#45211yescharacteristic
533C22C31×#45412yes—
534C22C31×B(4,𝔽2)12yes—
535C22C31×#46511yescharacteristic
536C22C31×#46811yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—81no—
545C4—81no—
546C4—81no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—84no—
552C2C31×#159712yes—
553C2C31×#159912yes—
554C2C31×#200011yescharacteristic
555C2C31×#200311yescharacteristic
556C2C31×#202111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC23×D4×C31C22
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×#2627C2semidirect
5. C2×C31×#328C2semidirect
6. C2×C31×#324C2semidirect
7. C31×#1294C2semidirect
127. C22×C4×C31SD16semidirect
155. #9923C31direct (N × H)
232. #2627C2×C31semidirect
235. C2×#328C2×C31semidirect
236. C2×#324C2×C31semidirect
237. #1294C2×C31semidirect
359. C31#9923direct (N × H)
406. C22×C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4502
C31×#26272
C2×C31×#3942
C31×#17862
C2×C31×#3282
C2×C31×#3242
C31×#12942
#992331