Finite Group Databasebeta

← C31×#15510 #21472269

Subgroups — C31×#15510 #21472269

Order 7936.

1478 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#395C212yesmaximal
2C2×C31×#391C212yesmaximal
3C31×#1446C211yesmaximal, characteristic
4C31×#1446C211yesmaximal, characteristic
5C31×#1444C211yesmaximal, characteristic
6C31×#1444C212yesmaximal
7C31×#1441C212yesmaximal
8C31×#1436C212yesmaximal
9C31×C4⋊#95C212yesmaximal
10C23×Q8×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2212yes—
13C22×C4⋊C4×C31C2212yes—
14C22×C4⋊C4×C31C2212yes—
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C4⋊C4×C31C2212yes—
17C22×C42×C31C2212yes—
18C31×#395—28no—
19C31×#391—28no—
20C2×C31×#95C2212yes—
21C2×C31×#95C2212yes—
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2211yescharacteristic
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2211yescharacteristic
33C22×Q8×C31—24no—
34C22×Q8×C31—24no—
35C22×Q8×C31—24no—
36C22×Q8×C31—22no—
37C22×Q8×C31—41no—
38C22×Q8×C31—41no—
39C22×Q8×C31—41no—
40C22×Q8×C31—42no—
41C22×Q8×C31—41no—
42C22×Q8×C31Q814yescomplemented
43C23×C4×C31C2312yes—
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2312yes—
47C23×C4×C31C2311yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2312yes—
51C23×C4×C31C2312yes—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31—24no—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—24no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—42no—
59C2×C4⋊C4×C31—42no—
60C2×C4⋊C4×C31—42no—
61C2×C4⋊C4×C31—42no—
62C2×C4⋊C4×C31—42no—
63C2×C4⋊C4×C31—41no—
64C2×C4⋊C4×C31—42no—
65C2×C4⋊C4×C31Q818yescomplemented
66C2×C42×C31—24no—
67C2×C42×C31—24no—
68C2×C42×C31—42no—
69C31×#95—24no—
70C31×#95—24no—
71C31×#95—24no—
72C31×#95—24no—
73C31×#95—24no—
74C31×#95—24no—
75C31×#95—41no—
76C31×#95—42no—
77C31×#95—42no—
78C31×#95—41no—
79C31×#95—41no—
80C31×#95D418yes—
81C31×#95Q814yescomplemented
82C24×C31C2411yescharacteristic
83C2×Q8×C31—216no—
84C2×Q8×C31—44no—
85C2×Q8×C31—44no—
86C2×Q8×C31—48no—
87C2×Q8×C31—44no—
88C2×Q8×C31—44no—
89C2×Q8×C31—48no—
90C2×Q8×C31—48no—
91C2×Q8×C31—44no—
92C2×Q8×C31—44no—
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
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—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—21no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31C2×D414yes—
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D414yes—
141C22×C4×C31C2×Q812yes—
142C22×C4×C31C2×Q814yes—
143C22×C4×C31C2×Q814yes—
144C22×C4×C31C2×Q812yes—
145C22×C4×C31C2×Q814yes—
146C22×C4×C31C2×Q814yes—
147C22×C4×C31C2×Q814yes—
148C22×C4×C31#4112yes—
149C22×C4×C31#4114yes—
150C4⋊C4×C31—48no—
151C4⋊C4×C31—48no—
152C4⋊C4×C31—48no—
153C4⋊C4×C31—48no—
154C4⋊C4×C31—44no—
155C42×C31—48no—
156#15510C3111yesmaximal, characteristic, direct product factor, Sylow
157C23×C31C22×D412yes—
158C23×C31C22×Q811yescharacteristic
159C23×C31C22×Q811yescharacteristic
160C23×C31C22×Q812yes—
161C23×C31C2×#4112yes—
162C23×C31C2×#4111yescharacteristic
163C23×C31H(5,𝔽2)11yescharacteristic
164C23×C31H(5,𝔽2)12yes—
165C23×C31H(5,𝔽2)11yescharacteristic
166C23×C31H(5,𝔽2)11yescharacteristic
167C23×C312−1+411yescharacteristic
168Q8×C31—416no—
169Q8×C31—416no—
170C2×C4×C31—24no—
171C2×C4×C31—24no—
172C2×C4×C31—24no—
173C2×C4×C31—24no—
174C2×C4×C31—24no—
175C2×C4×C31—24no—
176C2×C4×C31—24no—
177C2×C4×C31—24no—
178C2×C4×C31—24no—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—24no—
182C2×C4×C31—24no—
183C2×C4×C31—24no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—24no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—24no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—24no—
193C2×C4×C31—24no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
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—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31C22⋊Q818yes—
226C2×C4×C31C4⋊Q818yes—
227C2×#395C2×C3112yes—
228C2×#391C2×C3112yes—
229#1446C2×C3111yescharacteristic
230#1446C2×C3111yescharacteristic
231#1444C2×C3111yescharacteristic
232#1444C2×C3112yes—
233#1441C2×C3112yes—
234#1436C2×C3112yes—
235C4⋊#95C2×C3112yes—
236C22×C31C2×C22⋊Q812yes—
237C22×C31C2×C4⋊Q812yes—
238C22×C31#53312yes—
239C22×C31#53512yes—
240C22×C31#53811yescharacteristic
241C22×C31#54212yes—
242C22×C31#54312yes—
243C22×C31#54311yescharacteristic
244C22×C31#54411yescharacteristic
245C22×C31D4211yescharacteristic
246C22×C31#54612yes—
247C22×C31#54612yes—
248C22×C31D4×Q812yes—
249C22×C31#55111yescharacteristic
250C22×C31#55412yes—
251C22×C31#55412yes—
252C22×C31#55411yescharacteristic
253C22×C31#55711yescharacteristic
254C22×C31Q8211yescharacteristic
255C22×C31Q8212yes—
256C22×C31Q8211yescharacteristic
257C22×C31H(3,𝔽4)11yescharacteristic
258C22×C31#56211yescharacteristic
259C4×C31—28no—
260C4×C31—28no—
261C4×C31—44no—
262C4×C31—42no—
263C4×C31—44no—
264C4×C31—44no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—82no—
268C23×Q8C22×C3111yescharacteristic
269C23×Q8C22×C3111yescharacteristic
270C22×C4⋊C4C22×C3112yes—
271C22×C4⋊C4C22×C3112yes—
272C22×C4⋊C4C22×C3112yes—
273C22×C4⋊C4C22×C3111yescharacteristic
274C22×C4⋊C4C22×C3112yes—
275C22×C42C22×C3112yes—
276#395—28no—
277#391—28no—
278C2×#95C22×C3112yes—
279C2×#95C22×C3112yes—
280C2×#95C22×C3112yes—
281C2×#95C22×C3112yes—
282C2×#95C22×C3112yes—
283C2×#95C22×C3112yes—
284C2×#95C22×C3111yescharacteristic
285C2×#95C22×C3112yes—
286C2×#95C22×C3112yes—
287C2×#95C22×C3111yescharacteristic
288C2×#95C22×C3111yescharacteristic
289C2×#95C22×C3112yes—
290C2×#95C22×C3111yescharacteristic
291C2×C31#243711yescharacteristic
292C2×C31#245112yes—
293C2×C31#254912yes—
294C2×C31#255111yescharacteristic
295C2×C31#256712yes—
296C2×C31#258611yescharacteristic
297C2×C31#267011yescharacteristic
298C2×C31#269212yes—
299C2×C31#276311yescharacteristic
300C2×C31#280611yescharacteristic
301C2×C31#281111yescharacteristic
302C22×Q8—24no—
303C22×Q8—24no—
304C22×Q8—24no—
305C22×Q8—22no—
306C22×Q8—41no—
307C22×Q8—41no—
308C22×Q8—41no—
309C22×Q8—42no—
310C22×Q8—41no—
311C22×Q8Q8×C3114yescomplemented
312C23×C4C23×C3112yes—
313C23×C4C23×C3112yes—
314C23×C4C23×C3112yes—
315C23×C4C23×C3111yescharacteristic
316C23×C4C23×C3112yes—
317C23×C4C23×C3111yescharacteristic
318C23×C4C23×C3111yescharacteristic
319C23×C4C23×C3112yes—
320C23×C4C23×C3112yes—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4Q8×C3118yescomplemented
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—42no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—41no—
345#95—42no—
346#95—42no—
347#95—41no—
348#95—41no—
349#95D4×C3118yes—
350#95Q8×C3114yescomplemented
351C31#1551011yescharacteristic, direct product factor, Sylow
352C24C24×C3111yescharacteristic
353C2×Q8—216no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4C2×D4×C3114yes—
409C22×C4C2×D4×C3114yes—
410C22×C4C2×D4×C3114yes—
411C22×C4C2×Q8×C3112yes—
412C22×C4C2×Q8×C3114yes—
413C22×C4C2×Q8×C3114yes—
414C22×C4C2×Q8×C3112yes—
415C22×C4C2×Q8×C3114yes—
416C22×C4C2×Q8×C3114yes—
417C22×C4C2×Q8×C3114yes—
418C22×C4#41×C3112yes—
419C22×C4#41×C3114yes—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C42—48no—
426C23C22×D4×C3112yes—
427C23C22×Q8×C3111yescharacteristic
428C23C22×Q8×C3111yescharacteristic
429C23C22×Q8×C3112yes—
430C23C2×#41×C3112yes—
431C23C2×#41×C3111yescharacteristic
432C23C31×H(5,𝔽2)11yescharacteristic
433C23C31×H(5,𝔽2)12yes—
434C23C31×H(5,𝔽2)11yescharacteristic
435C23C31×H(5,𝔽2)11yescharacteristic
436C23C31×2−1+411yescharacteristic
437Q8—416no—
438Q8—416no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4C31×C22⋊Q818yes—
495C2×C4C31×C4⋊Q818yes—
496C22C2×C31×C22⋊Q812yes—
497C22C2×C31×C4⋊Q812yes—
498C22C31×#53312yes—
499C22C31×#53512yes—
500C22C31×#53811yescharacteristic
501C22C31×#54212yes—
502C22C31×#54312yes—
503C22C31×#54311yescharacteristic
504C22C31×#54411yescharacteristic
505C22D42×C3111yescharacteristic
506C22C31×#54612yes—
507C22C31×#54612yes—
508C22D4×Q8×C3112yes—
509C22C31×#55111yescharacteristic
510C22C31×#55412yes—
511C22C31×#55412yes—
512C22C31×#55411yescharacteristic
513C22C31×#55711yescharacteristic
514C22Q82×C3111yescharacteristic
515C22Q82×C3112yes—
516C22Q82×C3111yescharacteristic
517C22C31×H(3,𝔽4)11yescharacteristic
518C22C31×#56211yescharacteristic
519C4—28no—
520C4—28no—
521C4—44no—
522C4—42no—
523C4—44no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—82no—
528C2C31×#243711yescharacteristic
529C2C31×#245112yes—
530C2C31×#254912yes—
531C2C31×#255111yescharacteristic
532C2C31×#256712yes—
533C2C31×#258611yescharacteristic
534C2C31×#267011yescharacteristic
535C2C31×#269212yes—
536C2C31×#276311yescharacteristic
537C2C31×#280611yescharacteristic
538C2C31×#281111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C31C24
commutator, Frattini, cosocleC24C24×C31
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
42. C22×Q8×C31Q8semidirect
65. C2×C4⋊C4×C31Q8semidirect
81. C31×#95Q8semidirect
156. #15510C31direct (N × H)
311. C22×Q8Q8×C31semidirect
334. C2×C4⋊C4Q8×C31semidirect
350. #95Q8×C31semidirect
351. C31#15510direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3952
C2×C31×#3912
C31×#14462
C31×#14442
C31×#14412
C31×#14362
C31×C4⋊#952
#1551031