Finite Group Databasebeta

← C31×#14720 #21471479

Subgroups — C31×#14720 #21471479

Order 7936.

2494 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C211yesmaximal, characteristic
2C23×C4⋊C4×C31C211yesmaximal, characteristic
3C2×C31×#395C212yesmaximal, complemented
4C2×C31×#385C212yesmaximal
5C31×#1444C214yesmaximal, complemented
6C31×#1441C212yesmaximal, complemented
7C31×C22⋊#95C212yesmaximal
8C31×C22⋊#95C211yesmaximal, characteristic
9C23×Q8×C31C2211yescharacteristic
10C24×C4×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C24×C4×C31C2211yescharacteristic
13C2×C31×C22⋊Q8—24no—
14C2×C31×C22⋊Q8—28no—
15C22×C4⋊C4×C31—24no—
16C22×C4⋊C4×C31—28no—
17C22×C4⋊C4×C31C2212yes—
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2212yes—
20C22×C4⋊C4×C31C2214yes—
21C22×C4⋊C4×C31C2212yes—
22C22×C22⋊C4×C31C2212yes—
23C22×C22⋊C4×C31C2212yes—
24C31×#395—28no—
25C31×#385—28no—
26C2×C31×#95C2212yes—
27C2×C31×#95C2214yes—
28C2×C31×#95C2214yes—
29C2×C31×#95C2214yes—
30C2×C31×#95C2212yes—
31C25×C31C2311yescharacteristic
32C22×Q8×C31—24no—
33C22×Q8×C31—22no—
34C22×Q8×C31—24no—
35C22×Q8×C31—42no—
36C23×C4×C31—22no—
37C23×C4×C31—22no—
38C23×C4×C31—24no—
39C23×C4×C31—24no—
40C23×C4×C31—22no—
41C23×C4×C31—24no—
42C23×C4×C31—24no—
43C23×C4×C31—24no—
44C23×C4×C31—24no—
45C23×C4×C31—42no—
46C23×C4×C31—42no—
47C23×C4×C31D414yes—
48C23×C4×C31Q814yescomplemented
49C23×C4×C31C2312yes—
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2314yes—
54C23×C4×C31C2311yescharacteristic
55C23×C4×C31C2312yes—
56C31×C22⋊Q8—232no—
57C2×C4⋊C4×C31—232no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—24no—
60C2×C4⋊C4×C31—24no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—24no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—24no—
66C2×C4⋊C4×C31—24no—
67C2×C4⋊C4×C31—24no—
68C2×C4⋊C4×C31—28no—
69C2×C4⋊C4×C31—28no—
70C2×C4⋊C4×C31—44no—
71C2×C4⋊C4×C31D418yes—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31D418yes—
77C2×C22⋊C4×C31Q818yescomplemented
78C31×#95—24no—
79C31×#95—28no—
80C31×#95—28no—
81C31×#95—28no—
82C31×#95—42no—
83C24×C31—22no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31C2×D414yes—
88C24×C31C2×D412yes—
89C24×C31C2×Q814yes—
90C24×C31C2×Q812yes—
91C24×C31#4112yes—
92C24×C31C2411yescharacteristic
93C2×Q8×C31—28no—
94C2×Q8×C31—416no—
95C2×Q8×C31—44no—
96C2×Q8×C31—44no—
97C22×C4×C31—28no—
98C22×C4×C31—216no—
99C22×C4×C31—28no—
100C22×C4×C31—28no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—216no—
108C22×C4×C31—28no—
109C22×C4×C31—28no—
110C22×C4×C31—24no—
111C22×C4×C31—24no—
112C22×C4×C31—24no—
113C22×C4×C31—24no—
114C22×C4×C31—22no—
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—22no—
122C22×C4×C31—24no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—24no—
126C22×C4×C31—22no—
127C22×C4×C31—24no—
128C22×C4×C31—24no—
129C22×C4×C31—24no—
130C22×C4×C31—24no—
131C22×C4×C31—28no—
132C22×C4×C31—44no—
133C22×C4×C31—416no—
134C22×C4×C31—416no—
135C22×C4×C31—44no—
136C22×C4×C31—44no—
137C22×C4×C31C2×D414yes—
138C22×C4×C31C2×D414yes—
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D414yes—
141C22×C4×C31C2×D414yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D414yes—
144C22×C4×C31C2×Q818yes—
145C22×C4×C31C2×Q814yes—
146C22×C4×C31#4114yes—
147C4⋊C4×C31—216no—
148C4⋊C4×C31—28no—
149C4⋊C4×C31—216no—
150C4⋊C4×C31—216no—
151C4⋊C4×C31—416no—
152C22⋊C4×C31—216no—
153C22⋊C4×C31—216no—
154#14720C3111yesmaximal, characteristic, direct product factor, Sylow
155C23×C31—22no—
156C23×C31—24no—
157C23×C31—22no—
158C23×C31—24no—
159C23×C31—22no—
160C23×C31—22no—
161C23×C31—24no—
162C23×C31—24no—
163C23×C31—28no—
164C23×C31—28no—
165C23×C31—28no—
166C23×C31—28no—
167C23×C31#12014yes—
168C23×C31C22⋊Q818yes—
169C23×C31C22⋊Q814yes—
170C23×C31C4⋊Q814yes—
171C23×C31C4⋊Q818yes—
172C23×C31C22×D411yescharacteristic
173C23×C31C22×D412yes—
174C23×C31C22×D412yes—
175C23×C31C22×Q811yescharacteristic
176C23×C31C22×Q812yes—
177C23×C31C2×#4111yescharacteristic
178C23×C31C2×#4112yes—
179C23×C31H(5,𝔽2)12yes—
180C23×C312−1+412yes—
181Q8×C31—416no—
182C2×C4×C31—232no—
183C2×C4×C31—24no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—24no—
189C2×C4×C31—24no—
190C2×C4×C31—22no—
191C2×C4×C31—28no—
192C2×C4×C31—24no—
193C2×C4×C31—24no—
194C2×C4×C31—22no—
195C2×C4×C31—24no—
196C2×C4×C31—22no—
197C2×C4×C31—28no—
198C2×C4×C31—22no—
199C2×C4×C31—28no—
200C2×C4×C31—28no—
201C2×C4×C31—28no—
202C2×C4×C31—28no—
203C2×C4×C31—28no—
204C2×C4×C31—416no—
205C2×C4×C31—416no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—44no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—44no—
214C2×C4×C31—44no—
215C2×C4×C31—44no—
216C2×C4×C31—44no—
217C2×C4×C31—44no—
218C2×C4×C31—44no—
219C2×C4×C31—44no—
220C2×C4×C31#12014yes—
221C2×C4×C31C22⋊Q818yescomplemented
222C22×C22⋊Q8C2×C3111yescharacteristic
223C23×C4⋊C4C2×C3111yescharacteristic
224C2×#395C2×C3112yescomplemented
225C2×#385C2×C3112yes—
226#1444C2×C3114yescomplemented
227#1441C2×C3112yescomplemented
228C22⋊#95C2×C3112yes—
229C22⋊#95C2×C3111yescharacteristic
230C22×C31—24no—
231C22×C31—24no—
232C22×C31—24no—
233C22×C31—24no—
234C22×C31—24no—
235C22×C31—24no—
236C22×C31—216no—
237C22×C31—216no—
238C22×C31#39518yescomplemented
239C22×C31C2×#12011yescharacteristic
240C22×C31C2×#12011yescharacteristic
241C22×C31C2×C22⋊Q811yescharacteristic
242C22×C31C2×C22⋊Q812yes—
243C22×C31C2×C22⋊Q812yes—
244C22×C31#52511yescharacteristic
245C22×C31#52512yes—
246C22×C31C2×C4⋊Q811yescharacteristic
247C22×C31C2×C4⋊Q812yes—
248C22×C31#53411yescharacteristic
249C22×C31#53611yescharacteristic
250C22×C31#53912yes—
251C22×C31#54412yes—
252C22×C31D4214yes—
253C22×C31#54714yes—
254C22×C31D4×Q814yes—
255C22×C31#55414yes—
256C4×C31—28no—
257C4×C31—24no—
258C4×C31—42no—
259C4×C31—44no—
260C4×C31—44no—
261C4×C31—44no—
262C4×C31—48no—
263C23×Q8C22×C3111yescharacteristic
264C24×C4C22×C3111yescharacteristic
265C24×C4C22×C3111yescharacteristic
266C24×C4C22×C3111yescharacteristic
267C2×C22⋊Q8—24no—
268C2×C22⋊Q8—28no—
269C22×C4⋊C4—24no—
270C22×C4⋊C4—28no—
271C22×C4⋊C4C22×C3112yes—
272C22×C4⋊C4C22×C3111yescharacteristic
273C22×C4⋊C4C22×C3112yes—
274C22×C4⋊C4C22×C3114yes—
275C22×C4⋊C4C22×C3112yes—
276C22×C22⋊C4C22×C3112yes—
277C22×C22⋊C4C22×C3112yes—
278#395—28no—
279#385—28no—
280C2×#95C22×C3112yes—
281C2×#95C22×C3114yes—
282C2×#95C22×C3114yes—
283C2×#95C22×C3114yes—
284C2×#95C22×C3112yes—
285C2×C31—28no—
286C2×C31C2×#39511yescharacteristic
287C2×C31#240311yescharacteristic
288C2×C31#240611yescharacteristic
289C2×C31#243411yescharacteristic
290C2×C31#244014yes—
291C2×C31#244914yes—
292C2×C31#258311yescharacteristic
293C2×C31#262711yescharacteristic
294C2×C31#284811yescharacteristic
295C25C23×C3111yescharacteristic
296C22×Q8—24no—
297C22×Q8—22no—
298C22×Q8—24no—
299C22×Q8—42no—
300C23×C4—22no—
301C23×C4—22no—
302C23×C4—24no—
303C23×C4—24no—
304C23×C4—22no—
305C23×C4—24no—
306C23×C4—24no—
307C23×C4—24no—
308C23×C4—24no—
309C23×C4—42no—
310C23×C4—42no—
311C23×C4D4×C3114yes—
312C23×C4Q8×C3114yescomplemented
313C23×C4C23×C3112yes—
314C23×C4C23×C3112yes—
315C23×C4C23×C3112yes—
316C23×C4C23×C3111yescharacteristic
317C23×C4C23×C3114yes—
318C23×C4C23×C3111yescharacteristic
319C23×C4C23×C3112yes—
320C22⋊Q8—232no—
321C2×C4⋊C4—232no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4D4×C3118yes—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4D4×C3118yes—
341C2×C22⋊C4Q8×C3118yescomplemented
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—42no—
347C31#1472011yescharacteristic, direct product factor, Sylow
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C2×D4×C3114yes—
353C24C2×D4×C3112yes—
354C24C2×Q8×C3114yes—
355C24C2×Q8×C3112yes—
356C24#41×C3112yes—
357C24C24×C3111yescharacteristic
358C2×Q8—28no—
359C2×Q8—416no—
360C2×Q8—44no—
361C2×Q8—44no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—216no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—44no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4C2×D4×C3114yes—
403C22×C4C2×D4×C3114yes—
404C22×C4C2×D4×C3112yes—
405C22×C4C2×D4×C3114yes—
406C22×C4C2×D4×C3114yes—
407C22×C4C2×D4×C3112yes—
408C22×C4C2×D4×C3114yes—
409C22×C4C2×Q8×C3118yes—
410C22×C4C2×Q8×C3114yes—
411C22×C4#41×C3114yes—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—416no—
417C22⋊C4—216no—
418C22⋊C4—216no—
419C23—22no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—24no—
427C23—28no—
428C23—28no—
429C23—28no—
430C23—28no—
431C23C31×#12014yes—
432C23C31×C22⋊Q818yes—
433C23C31×C22⋊Q814yes—
434C23C31×C4⋊Q814yes—
435C23C31×C4⋊Q818yes—
436C23C22×D4×C3111yescharacteristic
437C23C22×D4×C3112yes—
438C23C22×D4×C3112yes—
439C23C22×Q8×C3111yescharacteristic
440C23C22×Q8×C3112yes—
441C23C2×#41×C3111yescharacteristic
442C23C2×#41×C3112yes—
443C23C31×H(5,𝔽2)12yes—
444C23C31×2−1+412yes—
445Q8—416no—
446C2×C4—232no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—416no—
469C2×C4—416no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4C31×#12014yes—
485C2×C4C31×C22⋊Q818yescomplemented
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—216no—
493C22—216no—
494C22C31×#39518yescomplemented
495C22C2×C31×#12011yescharacteristic
496C22C2×C31×#12011yescharacteristic
497C22C2×C31×C22⋊Q811yescharacteristic
498C22C2×C31×C22⋊Q812yes—
499C22C2×C31×C22⋊Q812yes—
500C22C31×#52511yescharacteristic
501C22C31×#52512yes—
502C22C2×C31×C4⋊Q811yescharacteristic
503C22C2×C31×C4⋊Q812yes—
504C22C31×#53411yescharacteristic
505C22C31×#53611yescharacteristic
506C22C31×#53912yes—
507C22C31×#54412yes—
508C22D42×C3114yes—
509C22C31×#54714yes—
510C22D4×Q8×C3114yes—
511C22C31×#55414yes—
512C4—28no—
513C4—24no—
514C4—42no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—48no—
519C2—28no—
520C2C2×C31×#39511yescharacteristic
521C2C31×#240311yescharacteristic
522C2C31×#240611yescharacteristic
523C2C31×#243411yescharacteristic
524C2C31×#244014yes—
525C2C31×#244914yes—
526C2C31×#258311yescharacteristic
527C2C31×#262711yescharacteristic
528C2C31×#284811yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#395C2semidirect
5. C31×#1444C2semidirect
6. C31×#1441C2semidirect
48. C23×C4×C31Q8semidirect
77. C2×C22⋊C4×C31Q8semidirect
154. #14720C31direct (N × H)
221. C2×C4×C31C22⋊Q8semidirect
224. C2×#395C2×C31semidirect
226. #1444C2×C31semidirect
227. #1441C2×C31semidirect
238. C22×C31#395semidirect
312. C23×C4Q8×C31semidirect
341. C2×C22⋊C4Q8×C31semidirect
347. C31#14720direct (N × H)
485. C2×C4C31×C22⋊Q8semidirect
494. C22C31×#395semidirect

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C23×C4⋊C4×C312
C2×C31×#3952
C2×C31×#3852
C31×#14442
C31×#14412
C31×C22⋊#952
#1472031