Finite Group Databasebeta

← C2×C31×#3092 #21517266

Subgroups — C2×C31×#3092 #21517266

Order 7936.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#525C211yesmaximal, characteristic
2C23×C4⋊C4×C31C211yesmaximal, characteristic, complemented
3C31×#3092C2116yesmaximal, direct product factor
4C2×C31×#484C212yesmaximal, complemented
5C2×C31×#481C212yesmaximal, complemented
6C2×C31×#483C212yesmaximal, complemented
7C2×C31×#480C212yesmaximal, complemented
8C2×C31×#428C212yesmaximal, complemented
9C2×C31×#417C212yesmaximal, complemented
10C2×C31×#407C211yesmaximal, characteristic, complemented
11C22×#41×C31C2211yescharacteristic
12C24×C4×C31—21no—
13C24×C4×C31C2211yescharacteristic
14C22×M16×C31C2212yes—
15C31×#525—24no—
16C31×#525C2218yes—
17C2×C31×#123—21no—
18C2×C31×C22⋊Q8C2211yescharacteristic
19C2×C31×C22⋊Q8C2211yescharacteristic
20C2×C31×#121C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C2×C31×#120—21no—
23C2×C4×D4×C31—21no—
24C2×C4×D4×C31—21no—
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31—24no—
27C22×C4⋊C4×C31—21no—
28C22×C4⋊C4×C31—21no—
29C22×C4⋊C4×C31—21no—
30C22×C4⋊C4×C31C2212yes—
31C22×C4⋊C4×C31C2212yes—
32C22×C4⋊C4×C31C2212yes—
33C22×C4⋊C4×C31C2218yescomplemented
34C31×#484C22116yescomplemented
35C31×#483C22116yescomplemented
36C31×#481C22116yescomplemented
37C31×#480C22116yescomplemented
38C31×#428C22116yescomplemented
39C2×C31×#107C2214yescomplemented
40C2×C31×#106C2214yescomplemented
41C31×#417C22116yescomplemented
42C2×C31×#103C2214yescomplemented
43C2×C31×#102C2214yescomplemented
44C31×#407C2218yescomplemented
45C2×C31×C22⋊C8C2212yes—
46C2×C31×C22⋊C8C2212yes—
47C25×C31D411yescharacteristic
48C2×#41×C31—22no—
49C2×#41×C31—28no—
50C2×#41×C31C2314yes—
51C22×Q8×C31C2311yescharacteristic
52C22×D4×C31—21no—
53C22×D4×C31—21no—
54C22×D4×C31—21no—
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—28no—
57C23×C4×C31—22no—
58C23×C4×C31—22no—
59C23×C4×C31—22no—
60C23×C4×C31—28no—
61C23×C4×C31—21no—
62C23×C4×C31—48no—
63C23×C4×C31D411yescharacteristic
64C23×C4×C31D411yescharacteristic
65C23×C4×C31D411yescharacteristic
66C23×C4×C31D414yes—
67C23×C4×C31C2311yescharacteristic
68C23×C4×C31C2311yescharacteristic
69C23×C4×C31C2314yes—
70C23×C4×C31C2311yescharacteristic
71C2×M16×C31—28no—
72C2×M16×C31C2318yes—
73C22×C8×C31C2314yes—
74C31×#123—28no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8C2314yes—
78C31×C22⋊Q8C2314yes—
79C31×#121—22no—
80C31×#121—22no—
81C31×#121C2314yes—
82C31×#121C2314yes—
83C31×#120—28no—
84C4×D4×C31—28no—
85C4×D4×C31—28no—
86C31×#117—22no—
87C31×#117C2314yes—
88C2×C4⋊C4×C31—24no—
89C2×C4⋊C4×C31—24no—
90C2×C4⋊C4×C31—24no—
91C2×C4⋊C4×C31—22no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31—22no—
94C2×C4⋊C4×C31—28no—
95C2×C4⋊C4×C31—28no—
96C2×C4⋊C4×C31—28no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—21no—
99C2×C4⋊C4×C31C2314yes—
100C2×C4⋊C4×C31C2318yes—
101C2×C4⋊C4×C31C2311yescharacteristic
102C2×C4⋊C4×C31C2311yescharacteristic
103C2×C4⋊C4×C31C2318yes—
104C2×C4⋊C4×C31C2318yes—
105C2×C22⋊C4×C31—21no—
106C2×C22⋊C4×C31—21no—
107C2×C22⋊C4×C31—21no—
108C2×C22⋊C4×C31—21no—
109C2×C22⋊C4×C31—21no—
110C2×C42×C31—21no—
111C31×#107C23116yescomplemented
112C31×#106C23116yescomplemented
113C31×#103C23116yescomplemented
114C31×#102C23116yescomplemented
115C31×C22⋊C8C2318yes—
116C31×C22⋊C8C2318yes—
117C24×C31—24no—
118C24×C31—24no—
119C24×C31—24no—
120C24×C31—21no—
121C24×C31C2×D414yes—
122C24×C31C2×D411yescharacteristic
123C24×C31C2×D411yescharacteristic
124C24×C31C2×D411yescharacteristic
125#41×C31—216no—
126#41×C31—48no—
127C2×Q8×C31—21no—
128C2×Q8×C31—24no—
129C2×Q8×C31C2412yes—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—24no—
133C2×D4×C31—21no—
134C2×D4×C31—24no—
135C2×D4×C31—44no—
136C2×D4×C31—44no—
137C2×D4×C31—44no—
138C2×D4×C31C2412yes—
139C22×C4×C31—24no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31—216no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—216no—
146C22×C4×C31—28no—
147C22×C4×C31—28no—
148C22×C4×C31—28no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—48no—
155C22×C4×C31—48no—
156C22×C4×C31—432no—
157C22×C4×C31—48no—
158C22×C4×C31—44no—
159C22×C4×C31C2×D414yes—
160C22×C4×C31C2×D414yes—
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D412yes—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2×D411yescharacteristic
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D411yescharacteristic
168C22×C4×C31#4118yes—
169C22×C4×C31C2412yes—
170C22×C4×C31C2412yes—
171C22×C4×C31C2412yes—
172C22×C4×C31C2411yescharacteristic
173M16×C31—216no—
174C2×C8×C31—28no—
175C2×C8×C31C2418yes—
176C4⋊C4×C31—28no—
177C4⋊C4×C31—28no—
178C4⋊C4×C31—28no—
179C4⋊C4×C31—24no—
180C4⋊C4×C31—24no—
181C4⋊C4×C31—24no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—21no—
184C4⋊C4×C31C2418yes—
185C4⋊C4×C31C2412yes—
186C4⋊C4×C31C2412yes—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C22⋊C4×C31—24no—
192C42×C31—24no—
193C2×#3092C3111yesmaximal, characteristic, direct product factor, Sylow
194C23×C31—232no—
195C23×C31—28no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—28no—
199C23×C31—22no—
200C23×C31—22no—
201C23×C31—28no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—22no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31#12314yes—
209C23×C31AGL(1,ℤ/8)14yes—
210C23×C31#13714yes—
211C23×C31C22×D412yes—
212C23×C31C22×D412yes—
213C23×C31C22×D412yes—
214C23×C31C22×D411yescharacteristic
215Q8×C31—24no—
216Q8×C31—42no—
217D4×C31—24no—
218D4×C31—48no—
219D4×C31—48no—
220D4×C31—48no—
221D4×C31—42no—
222C2×C4×C31—28no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—28no—
226C2×C4×C31—28no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—24no—
231C2×C4×C31—22no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—28no—
235C2×C4×C31—416no—
236C2×C4×C31—416no—
237C2×C4×C31—42no—
238C2×C4×C31—416no—
239C2×C4×C31—48no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—48no—
243C2×C4×C31#12314yes—
244C2×C4×C31#12318yes—
245C2×C4×C31C22×D412yes—
246C2×C4×C31C22×D412yes—
247C2×C4×C31C22×D412yes—
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C2×#4114yes—
250C2×C4×C31C2×#4118yes—
251C2×C4×C31C2511yescharacteristic
252C8×C31—28no—
253C2×#525C2×C3111yescharacteristic
254C23×C4⋊C4C2×C3111yescharacteristic, complemented
255#3092C2×C31116yesdirect product factor
256C2×#484C2×C3112yescomplemented
257C2×#481C2×C3112yescomplemented
258C2×#483C2×C3112yescomplemented
259C2×#480C2×C3112yescomplemented
260C2×#428C2×C3112yescomplemented
261C2×#417C2×C3112yescomplemented
262C2×#407C2×C3111yescharacteristic, complemented
263C22×C31—232no—
264C22×C31—28no—
265C22×C31—28no—
266C22×C31—28no—
267C22×C31—22no—
268C22×C31—22no—
269C22×C31—22no—
270C22×C31—22no—
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31—42no—
276C22×C31—84no—
277C22×C31C2×#12314yes—
278C22×C31C2×#12312yes—
279C22×C31C2×AGL(1,ℤ/8)14yes—
280C22×C31C2×AGL(1,ℤ/8)12yes—
281C22×C31C2×#13714yes—
282C22×C31C2×#13712yes—
283C22×C31C23×D411yescharacteristic
284C4×C31—22no—
285C4×C31—22no—
286C4×C31—42no—
287C4×C31—48no—
288C4×C31—42no—
289C4×C31—42no—
290C4×C31C2×#12314yes—
291C4×C31C2×#12312yes—
292C4×C31C22×#4112yes—
293C22×#41C22×C3111yescharacteristic
294C24×C4—21no—
295C24×C4C22×C3111yescharacteristic
296C22×M16C22×C3112yes—
297#525—24no—
298#525C22×C3118yes—
299C2×#123—21no—
300C2×C22⋊Q8C22×C3111yescharacteristic
301C2×C22⋊Q8C22×C3111yescharacteristic
302C2×#121C22×C3111yescharacteristic
303C2×#121C22×C3111yescharacteristic
304C2×#120—21no—
305C2×C4×D4—21no—
306C2×C4×D4—21no—
307C2×#117C22×C3111yescharacteristic
308C22×C4⋊C4—24no—
309C22×C4⋊C4—21no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4—21no—
312C22×C4⋊C4C22×C3112yes—
313C22×C4⋊C4C22×C3112yes—
314C22×C4⋊C4C22×C3112yes—
315C22×C4⋊C4C22×C3118yescomplemented
316#484C22×C31116yescomplemented
317#483C22×C31116yescomplemented
318#481C22×C31116yescomplemented
319#480C22×C31116yescomplemented
320#428C22×C31116yescomplemented
321C2×#107C22×C3114yescomplemented
322C2×#106C22×C3114yescomplemented
323#417C22×C31116yescomplemented
324C2×#103C22×C3114yescomplemented
325C2×#102C22×C3114yescomplemented
326#407C22×C3118yescomplemented
327C2×C22⋊C8C22×C3112yes—
328C2×C22⋊C8C22×C3112yes—
329C2×C31—24no—
330C2×C31—24no—
331C2×C31—24no—
332C2×C31—82no—
333C2×C31#309214yesdirect product factor
334C2×C31C22×#12311yescharacteristic
335C2×C31C22×AGL(1,ℤ/8)11yescharacteristic
336C2×C31C22×#13711yescharacteristic
337C25D4×C3111yescharacteristic
338C2×#41—22no—
339C2×#41—28no—
340C2×#41C23×C3114yes—
341C22×Q8C23×C3111yescharacteristic
342C22×D4—21no—
343C22×D4—21no—
344C22×D4—21no—
345C22×D4C23×C3111yescharacteristic
346C23×C4—28no—
347C23×C4—22no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—28no—
351C23×C4—21no—
352C23×C4—48no—
353C23×C4D4×C3111yescharacteristic
354C23×C4D4×C3111yescharacteristic
355C23×C4D4×C3111yescharacteristic
356C23×C4D4×C3114yes—
357C23×C4C23×C3111yescharacteristic
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3114yes—
360C23×C4C23×C3111yescharacteristic
361C2×M16—28no—
362C2×M16C23×C3118yes—
363C22×C8C23×C3114yes—
364#123—28no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8C23×C3114yes—
368C22⋊Q8C23×C3114yes—
369#121—22no—
370#121—22no—
371#121C23×C3114yes—
372#121C23×C3114yes—
373#120—28no—
374C4×D4—28no—
375C4×D4—28no—
376#117—22no—
377#117C23×C3114yes—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C23×C3114yes—
390C2×C4⋊C4C23×C3118yes—
391C2×C4⋊C4C23×C3118yes—
392C2×C4⋊C4C23×C3118yes—
393C2×C4⋊C4C23×C3111yescharacteristic
394C2×C4⋊C4C23×C3111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—21no—
401#107C23×C31116yescomplemented
402#106C23×C31116yescomplemented
403#103C23×C31116yescomplemented
404#102C23×C31116yescomplemented
405C22⋊C8C23×C3118yes—
406C22⋊C8C23×C3118yes—
407C31C2×#309211yescharacteristic, direct product factor, Sylow
408C24—24no—
409C24—24no—
410C24—24no—
411C24—21no—
412C24C2×D4×C3114yes—
413C24C2×D4×C3111yescharacteristic
414C24C2×D4×C3111yescharacteristic
415C24C2×D4×C3111yescharacteristic
416#41—216no—
417#41—48no—
418C2×Q8—21no—
419C2×Q8—24no—
420C2×Q8C24×C3112yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4C24×C3112yes—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—432no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3114yes—
453C22×C4C2×D4×C3114yes—
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×D4×C3114yes—
457C22×C4C2×D4×C3111yescharacteristic
458C22×C4C2×D4×C3112yes—
459C22×C4#41×C3118yes—
460C22×C4C24×C3112yes—
461C22×C4C24×C3112yes—
462C22×C4C24×C3111yescharacteristic
463C22×C4C24×C3112yes—
464M16—216no—
465C2×C8—28no—
466C2×C8C24×C3118yes—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4C24×C3118yes—
476C4⋊C4C24×C3112yes—
477C4⋊C4C24×C3112yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C42—24no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23C31×#12314yes—
499C23C31×AGL(1,ℤ/8)14yes—
500C23C31×#13714yes—
501C23C22×D4×C3112yes—
502C23C22×D4×C3112yes—
503C23C22×D4×C3112yes—
504C23C22×D4×C3111yescharacteristic
505Q8—24no—
506Q8—42no—
507D4—24no—
508D4—48no—
509D4—48no—
510D4—48no—
511D4—42no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—416no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4C31×#12314yes—
534C2×C4C31×#12318yes—
535C2×C4C22×D4×C3112yes—
536C2×C4C22×D4×C3112yes—
537C2×C4C22×D4×C3112yes—
538C2×C4C22×D4×C3111yescharacteristic
539C2×C4C2×#41×C3114yes—
540C2×C4C2×#41×C3118yes—
541C2×C4C25×C3111yescharacteristic
542C8—28no—
543C22—232no—
544C22—28no—
545C22—28no—
546C22—28no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22C2×C31×#12314yes—
558C22C2×C31×#12312yes—
559C22C2×C31×AGL(1,ℤ/8)14yes—
560C22C2×C31×AGL(1,ℤ/8)12yes—
561C22C2×C31×#13714yes—
562C22C2×C31×#13712yes—
563C22C23×D4×C3111yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—48no—
568C4—42no—
569C4—42no—
570C4C2×C31×#12314yes—
571C4C2×C31×#12312yes—
572C4C22×#41×C3112yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—82no—
577C2C31×#309214yesdirect product factor
578C2C22×C31×#12311yescharacteristic
579C2C22×C31×AGL(1,ℤ/8)11yescharacteristic
580C2C22×C31×#13711yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC2×C31×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C4⋊C4×C31C2semidirect
3. C31×#3092C2direct (N × H)
4. C2×C31×#484C2semidirect
5. C2×C31×#481C2semidirect
6. C2×C31×#483C2semidirect
7. C2×C31×#480C2semidirect
8. C2×C31×#428C2semidirect
9. C2×C31×#417C2semidirect
10. C2×C31×#407C2semidirect
33. C22×C4⋊C4×C31C22semidirect
34. C31×#484C22semidirect
35. C31×#483C22semidirect
36. C31×#481C22semidirect
37. C31×#480C22semidirect
38. C31×#428C22semidirect
39. C2×C31×#107C22semidirect
40. C2×C31×#106C22semidirect
41. C31×#417C22semidirect
42. C2×C31×#103C22semidirect
43. C2×C31×#102C22semidirect
44. C31×#407C22semidirect
111. C31×#107C23semidirect
112. C31×#106C23semidirect
113. C31×#103C23semidirect
114. C31×#102C23semidirect
193. C2×#3092C31direct (N × H)
254. C23×C4⋊C4C2×C31semidirect
255. #3092C2×C31direct (N × H)
256. C2×#484C2×C31semidirect
257. C2×#481C2×C31semidirect
258. C2×#483C2×C31semidirect
259. C2×#480C2×C31semidirect
260. C2×#428C2×C31semidirect
261. C2×#417C2×C31semidirect
262. C2×#407C2×C31semidirect
315. C22×C4⋊C4C22×C31semidirect
316. #484C22×C31semidirect
317. #483C22×C31semidirect
318. #481C22×C31semidirect
319. #480C22×C31semidirect
320. #428C22×C31semidirect
321. C2×#107C22×C31semidirect
322. C2×#106C22×C31semidirect
323. #417C22×C31semidirect
324. C2×#103C22×C31semidirect
325. C2×#102C22×C31semidirect
326. #407C22×C31semidirect
333. C2×C31#3092direct (N × H)
401. #107C23×C31semidirect
402. #106C23×C31semidirect
403. #103C23×C31semidirect
404. #102C23×C31semidirect
407. C31C2×#3092direct (N × H)
577. C2C31×#3092direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5252
C23×C4⋊C4×C312
C31×#30922
C2×C31×#4842
C2×C31×#4812
C2×C31×#4832
C2×C31×#4802
C2×C31×#4282
C2×C31×#4172
C2×C31×#4072
C2×#309231