Finite Group Databasebeta

← C31×#17387 #21474146

Subgroups — C31×#17387 #21474146

Order 7936.

984 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×#539C211yesmaximal, characteristic
2C31×#2973C211yesmaximal, characteristic, complemented
3C31×#2972C211yesmaximal, characteristic, complemented
4C31×#1473C212yesmaximal, complemented
5C31×#1472C212yesmaximal, complemented
6C31×#1470C212yesmaximal, complemented
7C31×#1468C212yesmaximal, complemented
8C31×#1459C212yesmaximal, complemented
9C2×C31×#324C212yesmaximal, complemented
10C31×#539—24no—
11C31×#539—24no—
12C2×C31×#126C411yescharacteristic
13C2×C31×#126C2211yescharacteristic
14C2×C31×#125C411yescharacteristic
15C2×C31×#125C2211yescharacteristic
16C2×C31×#123C411yescharacteristic
17C2×C31×#123C411yescharacteristic
18C2×C31×#123C411yescharacteristic
19C2×C31×#123C2211yescharacteristic
20C2×C31×C22⋊Q8C411yescharacteristic
21C2×C31×C22⋊Q8C411yescharacteristic
22C2×C31×C22⋊Q8C2211yescharacteristic
23C2×C31×#121C411yescharacteristic
24C2×C4×D4×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#435—24no—
28C8×D4×C31—24no—
29C31×#432C2212yes—
30C31×#431C2212yes—
31C31×#407C2212yes—
32C2×C31×C22⋊C8C2212yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×C22⋊C8C2212yes—
35C31×#324C2214yes—
36C31×#324C2214yes—
37C31×#324C2214yes—
38C31×#323C2214yes—
39C22×Q8×C31C2×C411yescharacteristic
40C22×D4×C31C2×C411yescharacteristic
41C22×D4×C31C2×C411yescharacteristic
42C22×D4×C31C2×C411yescharacteristic
43C23×C4×C31C2×C411yescharacteristic
44C23×C4×C31C2×C411yescharacteristic
45C23×C4×C31C2×C411yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C2×M16×C31—22no—
49C22×C8×C31—22no—
50C22×C8×C31—22no—
51C22×C8×C31—22no—
52C31×#126—24no—
53C31×#126—22no—
54C31×#126—22no—
55C31×#125—24no—
56C31×#125—22no—
57C31×#125—22no—
58C31×#123—24no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—24no—
62C31×#123—24no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8—24no—
67C31×#121—24no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31D414yes—
71C4×D4×C31D414yes—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31C2×C411yescharacteristic
75C2×C4⋊C4×C31C2×C411yescharacteristic
76C2×C4⋊C4×C31C2×C411yescharacteristic
77C2×C4⋊C4×C31C2×C411yescharacteristic
78C2×C4⋊C4×C31C2×C411yescharacteristic
79C2×C4⋊C4×C31C2×C411yescharacteristic
80C2×C4⋊C4×C31C2×C411yescharacteristic
81C2×C4⋊C4×C31C2×C411yescharacteristic
82C2×C4⋊C4×C31C2×C411yescharacteristic
83C2×C4⋊C4×C31C2×C411yescharacteristic
84C2×C4⋊C4×C31C2×C411yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C22⋊C4×C31C2×C411yescharacteristic
89C2×C22⋊C4×C31C2×C411yescharacteristic
90C2×C22⋊C4×C31C2×C411yescharacteristic
91C2×C22⋊C4×C31C2×C411yescharacteristic
92C2×C22⋊C4×C31C2×C411yescharacteristic
93C2×C22⋊C4×C31C2×C411yescharacteristic
94C2×C22⋊C4×C31C2×C411yescharacteristic
95C2×C22⋊C4×C31C2×C411yescharacteristic
96C2×C22⋊C4×C31C2×C411yescharacteristic
97C2×C22⋊C4×C31C2311yescharacteristic
98C2×C42×C31C2×C411yescharacteristic
99C2×C42×C31C2311yescharacteristic
100C31×C4⋊C8—22no—
101C31×C4⋊C8—22no—
102C31×C22⋊C8—22no—
103C31×C22⋊C8—22no—
104C31×C22⋊C8—22no—
105C31×C22⋊C8—22no—
106C31×C22⋊C8C2314yes—
107C31×C22⋊C8C2314yes—
108C31×#97—22no—
109C4×C8×C31—22no—
110C24×C31C22×C411yescharacteristic
111C24×C31C22×C411yescharacteristic
112C24×C31C22×C411yescharacteristic
113C2×Q8×C31—22no—
114C2×Q8×C31—81no—
115C2×D4×C31—24no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—81no—
120C2×D4×C31C22⋊C414yes—
121C22×C4×C31—22no—
122C22×C4×C31—24no—
123C22×C4×C31—22no—
124C22×C4×C31—22no—
125C22×C4×C31—81no—
126C22×C4×C31—81no—
127C22×C4×C31—81no—
128C22×C4×C31—81no—
129C22×C4×C31C22×C411yescharacteristic
130C22×C4×C31C22×C411yescharacteristic
131C22×C4×C31C22×C411yescharacteristic
132C22×C4×C31C22×C411yescharacteristic
133C22×C4×C31C22×C411yescharacteristic
134C22×C4×C31C22×C411yescharacteristic
135C22×C4×C31C22×C411yescharacteristic
136C22×C4×C31C22×C411yescharacteristic
137C22×C4×C31C22×C411yescharacteristic
138C22×C4×C31C22×C411yescharacteristic
139C22×C4×C31C22×C411yescharacteristic
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2411yescharacteristic
145M16×C31—82no—
146C2×C8×C31—24no—
147C2×C8×C31—24no—
148C2×C8×C31—82no—
149C2×C8×C31—82no—
150C2×C8×C31—82no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31C22⋊C414yes—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31C22⋊C414yes—
178C22⋊C4×C31C22⋊C414yes—
179C42×C31—22no—
180C42×C31C2×D412yes—
181C42×C31C2×D412yes—
182#17387C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—21no—
185C23×C31—22no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31—41no—
192C23×C31C2×C22⋊C412yes—
193C23×C31C2×C22⋊C412yes—
194C23×C31#13114yes—
195C23×C31C23×C411yescharacteristic
196Q8×C31—82no—
197D4×C31—28no—
198D4×C31—44no—
199D4×C31—82no—
200C2×C4×C31—22no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—44no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—82no—
225C2×C4×C31—82no—
226C2×C4×C31C2×C22⋊C412yes—
227C2×C4×C31C2×C22⋊C412yes—
228C2×C4×C31C2×C22⋊C412yes—
229C2×C4×C31C2×C22⋊C412yes—
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C22×D411yescharacteristic
232C8×C31—44no—
233C8×C31—44no—
234C2×#539C2×C3111yescharacteristic
235#2973C2×C3111yescharacteristic, complemented
236#2972C2×C3111yescharacteristic, complemented
237#1473C2×C3112yescomplemented
238#1472C2×C3112yescomplemented
239#1470C2×C3112yescomplemented
240#1468C2×C3112yescomplemented
241#1459C2×C3112yescomplemented
242C2×#324C2×C3112yescomplemented
243C22×C31—21no—
244C22×C31—22no—
245C22×C31—21no—
246C22×C31—22no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—22no—
250C22×C31—44no—
251C22×C31—41no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31—82no—
255C22×C31—82no—
256C22×C31#40814yes—
257C22×C31C22×C22⋊C411yescharacteristic
258C22×C31C2×#13112yes—
259C4×C31—22no—
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—22no—
263C4×C31—22no—
264C4×C31—22no—
265C4×C31—41no—
266C4×C31—41no—
267C4×C31—41no—
268C4×C31—41no—
269C4×C31—41no—
270C4×C31—41no—
271C4×C31—41no—
272#539—24no—
273#539—24no—
274C2×#126C4×C3111yescharacteristic
275C2×#126C22×C3111yescharacteristic
276C2×#125C4×C3111yescharacteristic
277C2×#125C22×C3111yescharacteristic
278C2×#123C4×C3111yescharacteristic
279C2×#123C4×C3111yescharacteristic
280C2×#123C4×C3111yescharacteristic
281C2×#123C22×C3111yescharacteristic
282C2×C22⋊Q8C4×C3111yescharacteristic
283C2×C22⋊Q8C4×C3111yescharacteristic
284C2×C22⋊Q8C22×C3111yescharacteristic
285C2×#121C4×C3111yescharacteristic
286C2×C4×D4C22×C3111yescharacteristic
287C2×C4×D4C22×C3111yescharacteristic
288C22×C4⋊C4C22×C3111yescharacteristic
289#435—24no—
290C8×D4—24no—
291#432C22×C3112yes—
292#431C22×C3112yes—
293#407C22×C3112yes—
294C2×C22⋊C8C22×C3112yes—
295C2×C22⋊C8C22×C3112yes—
296C2×C22⋊C8C22×C3112yes—
297#324C22×C3114yes—
298#324C22×C3114yes—
299#324C22×C3114yes—
300#323C22×C3114yes—
301C2×C31—22no—
302C2×C31—24no—
303C2×C31—42no—
304C2×C31—81no—
305C2×C31C2×#40811yescharacteristic
306C2×C31#288311yescharacteristic
307C2×C31#288911yescharacteristic
308C22×Q8C2×C4×C3111yescharacteristic
309C22×D4C2×C4×C3111yescharacteristic
310C22×D4C2×C4×C3111yescharacteristic
311C22×D4C2×C4×C3111yescharacteristic
312C23×C4C2×C4×C3111yescharacteristic
313C23×C4C2×C4×C3111yescharacteristic
314C23×C4C2×C4×C3111yescharacteristic
315C23×C4C23×C3111yescharacteristic
316C23×C4C23×C3111yescharacteristic
317C2×M16—22no—
318C22×C8—22no—
319C22×C8—22no—
320C22×C8—22no—
321#126—24no—
322#126—22no—
323#126—22no—
324#125—24no—
325#125—22no—
326#125—22no—
327#123—24no—
328#123—22no—
329#123—22no—
330#123—24no—
331#123—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336#121—24no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4D4×C3114yes—
340C4×D4D4×C3114yes—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2×C4×C3111yescharacteristic
344C2×C4⋊C4C2×C4×C3111yescharacteristic
345C2×C4⋊C4C2×C4×C3111yescharacteristic
346C2×C4⋊C4C2×C4×C3111yescharacteristic
347C2×C4⋊C4C2×C4×C3111yescharacteristic
348C2×C4⋊C4C2×C4×C3111yescharacteristic
349C2×C4⋊C4C2×C4×C3111yescharacteristic
350C2×C4⋊C4C2×C4×C3111yescharacteristic
351C2×C4⋊C4C2×C4×C3111yescharacteristic
352C2×C4⋊C4C2×C4×C3111yescharacteristic
353C2×C4⋊C4C2×C4×C3111yescharacteristic
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C22⋊C4C2×C4×C3111yescharacteristic
358C2×C22⋊C4C2×C4×C3111yescharacteristic
359C2×C22⋊C4C2×C4×C3111yescharacteristic
360C2×C22⋊C4C2×C4×C3111yescharacteristic
361C2×C22⋊C4C2×C4×C3111yescharacteristic
362C2×C22⋊C4C2×C4×C3111yescharacteristic
363C2×C22⋊C4C2×C4×C3111yescharacteristic
364C2×C22⋊C4C2×C4×C3111yescharacteristic
365C2×C22⋊C4C2×C4×C3111yescharacteristic
366C2×C22⋊C4C23×C3111yescharacteristic
367C2×C42C2×C4×C3111yescharacteristic
368C2×C42C23×C3111yescharacteristic
369C4⋊C8—22no—
370C4⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375C22⋊C8C23×C3114yes—
376C22⋊C8C23×C3114yes—
377#97—22no—
378C4×C8—22no—
379C31#1738711yescharacteristic, direct product factor, Sylow
380C24C22×C4×C3111yescharacteristic
381C24C22×C4×C3111yescharacteristic
382C24C22×C4×C3111yescharacteristic
383C2×Q8—22no—
384C2×Q8—81no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—81no—
390C2×D4C22⋊C4×C3114yes—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4C22×C4×C3111yescharacteristic
400C22×C4C22×C4×C3111yescharacteristic
401C22×C4C22×C4×C3111yescharacteristic
402C22×C4C22×C4×C3111yescharacteristic
403C22×C4C22×C4×C3111yescharacteristic
404C22×C4C22×C4×C3111yescharacteristic
405C22×C4C22×C4×C3111yescharacteristic
406C22×C4C22×C4×C3111yescharacteristic
407C22×C4C22×C4×C3111yescharacteristic
408C22×C4C22×C4×C3111yescharacteristic
409C22×C4C22×C4×C3111yescharacteristic
410C22×C4C2×D4×C3112yes—
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3112yes—
414C22×C4C24×C3111yescharacteristic
415M16—82no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—82no—
419C2×C8—82no—
420C2×C8—82no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4C22⋊C4×C3114yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4C22⋊C4×C3114yes—
448C22⋊C4C22⋊C4×C3114yes—
449C42—22no—
450C42C2×D4×C3112yes—
451C42C2×D4×C3112yes—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23C2×C22⋊C4×C3112yes—
462C23C2×C22⋊C4×C3112yes—
463C23C31×#13114yes—
464C23C23×C4×C3111yescharacteristic
465Q8—82no—
466D4—28no—
467D4—44no—
468D4—82no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4C2×C22⋊C4×C3112yes—
496C2×C4C2×C22⋊C4×C3112yes—
497C2×C4C2×C22⋊C4×C3112yes—
498C2×C4C2×C22⋊C4×C3112yes—
499C2×C4C22×D4×C3111yescharacteristic
500C2×C4C22×D4×C3111yescharacteristic
501C8—44no—
502C8—44no—
503C22—21no—
504C22—22no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—82no—
515C22—82no—
516C22C31×#40814yes—
517C22C22×C22⋊C4×C3111yescharacteristic
518C22C2×C31×#13112yes—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C2—22no—
533C2—24no—
534C2—42no—
535C2—81no—
536C2C2×C31×#40811yescharacteristic
537C2C31×#288311yescharacteristic
538C2C31×#288911yescharacteristic
subgroupG/H
center, socleC22×C31C22×C22⋊C4
commutatorC23C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2973C2semidirect
3. C31×#2972C2semidirect
4. C31×#1473C2semidirect
5. C31×#1472C2semidirect
6. C31×#1470C2semidirect
7. C31×#1468C2semidirect
8. C31×#1459C2semidirect
9. C2×C31×#324C2semidirect
182. #17387C31direct (N × H)
235. #2973C2×C31semidirect
236. #2972C2×C31semidirect
237. #1473C2×C31semidirect
238. #1472C2×C31semidirect
239. #1470C2×C31semidirect
240. #1468C2×C31semidirect
241. #1459C2×C31semidirect
242. C2×#324C2×C31semidirect
379. C31#17387direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C31×#29732
C31×#29722
C31×#14732
C31×#14722
C31×#14702
C31×#14682
C31×#14592
C2×C31×#3242
#1738731