Finite Group Databasebeta

← C31×#20601 #21477360

Subgroups — C31×#20601 #21477360

Order 7936.

4038 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C31C211yesmaximal, characteristic
2C2×C31×#417C214yesmaximal, complemented
3C2×C31×#407C211yesmaximal, characteristic, complemented
4C2×C31×#380C211yesmaximal, characteristic, complemented
5C31×#1788C214yesmaximal, complemented
6C31×#1787C212yesmaximal, complemented
7C31×#1786C212yesmaximal, complemented
8C22×#41×C31—216no—
9C22×#41×C31—24no—
10C22×#41×C31—21no—
11C22×#41×C31—21no—
12C22×#41×C31C418yescomplemented
13C22×#41×C31C2212yes—
14C22×#41×C31C2212yes—
15C23×Q8×C31C2211yescharacteristic
16C23×D4×C31—21no—
17C23×D4×C31C2211yescharacteristic
18C24×C4×C31—21no—
19C24×C4×C31C2211yescharacteristic
20C22×M16×C31C2212yescomplemented
21C22×C4⋊C4×C31C2212yescomplemented
22C22×C22⋊C4×C31—21no—
23C31×#417—216no—
24C2×C31×#103C2218yescomplemented
25C2×C31×#102C2218yescomplemented
26C31×#407—24no—
27C2×C31×C22⋊C8C2212yes—
28C2×C31×C22⋊C8C2212yescomplemented
29C31×#380—21no—
30C31×#380—21no—
31C31×#380C2212yes—
32C31×#380C2212yescomplemented
33C2×C31×#95—21no—
34C25×C31—21no—
35C25×C31D411yescharacteristic
36C2×#41×C31—216no—
37C2×#41×C31—264no—
38C2×#41×C31—216no—
39C2×#41×C31—264no—
40C2×#41×C31—28no—
41C2×#41×C31—28no—
42C2×#41×C31—28no—
43C2×#41×C31—28no—
44C2×#41×C31—24no—
45C2×#41×C31—24no—
46C2×#41×C31—28no—
47C2×#41×C31C2×C418yes—
48C2×#41×C31C2×C418yescomplemented
49C2×#41×C31D4116yes—
50C22×Q8×C31—28no—
51C22×Q8×C31—22no—
52C22×Q8×C31—21no—
53C22×Q8×C31C2×C414yescomplemented
54C22×Q8×C31C2312yes—
55C22×D4×C31—216no—
56C22×D4×C31—28no—
57C22×D4×C31—22no—
58C22×D4×C31—22no—
59C22×D4×C31—28no—
60C22×D4×C31—21no—
61C22×D4×C31—22no—
62C22×D4×C31C2×C414yescomplemented
63C22×D4×C31C2312yes—
64C23×C4×C31—216no—
65C23×C4×C31—21no—
66C23×C4×C31—28no—
67C23×C4×C31—22no—
68C23×C4×C31—22no—
69C23×C4×C31—28no—
70C23×C4×C31—22no—
71C23×C4×C31C2×C414yes—
72C23×C4×C31D411yescharacteristic
73C23×C4×C31D411yescharacteristic
74C23×C4×C31D411yescharacteristic
75C23×C4×C31D414yes—
76C23×C4×C31C2311yescharacteristic
77C23×C4×C31C2311yescharacteristic
78C23×C4×C31C2311yescharacteristic
79C2×M16×C31—24no—
80C2×M16×C31—28no—
81C22×C8×C31C2314yes—
82C2×C4⋊C4×C31—28no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31C2314yes—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—44no—
88C31×#103—216no—
89C31×#102—216no—
90C31×C22⋊C8—24no—
91C31×C22⋊C8—24no—
92C31×#95—22no—
93C31×#95—22no—
94C24×C31—28no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—42no—
101C24×C31—44no—
102C24×C31—42no—
103C24×C31—42no—
104C24×C31C22⋊C414yes—
105C24×C31C2×D411yescharacteristic
106C24×C31C2×D411yescharacteristic
107C24×C31C2×D411yescharacteristic
108#41×C31—264no—
109#41×C31—232no—
110#41×C31—264no—
111#41×C31—216no—
112#41×C31—216no—
113#41×C31—216no—
114#41×C31—232no—
115#41×C31C22⋊C4132yescomplemented
116C2×Q8×C31—216no—
117C2×Q8×C31—28no—
118C2×Q8×C31—216no—
119C2×Q8×C31—24no—
120C2×Q8×C31—22no—
121C2×Q8×C31—24no—
122C2×Q8×C31C22×C414yes—
123C2×Q8×C31C2×D418yes—
124C2×D4×C31—264no—
125C2×D4×C31—216no—
126C2×D4×C31—28no—
127C2×D4×C31—216no—
128C2×D4×C31—28no—
129C2×D4×C31—28no—
130C2×D4×C31—216no—
131C2×D4×C31—216no—
132C2×D4×C31—24no—
133C2×D4×C31—24no—
134C2×D4×C31—22no—
135C2×D4×C31C22×C414yes—
136C2×D4×C31C2×D418yes—
137C22×C4×C31—28no—
138C22×C4×C31—264no—
139C22×C4×C31—216no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31—22no—
143C22×C4×C31—28no—
144C22×C4×C31—28no—
145C22×C4×C31—216no—
146C22×C4×C31—24no—
147C22×C4×C31—216no—
148C22×C4×C31—24no—
149C22×C4×C31—216no—
150C22×C4×C31—24no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31—44no—
156C22×C4×C31C22⋊C414yes—
157C22×C4×C31C22⋊C414yes—
158C22×C4×C31C22⋊C414yes—
159C22×C4×C31C22×C412yes—
160C22×C4×C31C22×C412yes—
161C22×C4×C31C22×C412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D411yescharacteristic
167C22×C4×C31C2×D418yes—
168C22×C4×C31C2×D412yes—
169C22×C4×C31C2411yescharacteristic
170M16×C31—48no—
171C2×C8×C31—24no—
172C2×C8×C31—28no—
173C4⋊C4×C31—28no—
174C4⋊C4×C31—44no—
175C22⋊C4×C31—44no—
176C22⋊C4×C31—44no—
177#20601C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—22no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—216no—
184C23×C31—24no—
185C23×C31—22no—
186C23×C31—28no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—22no—
190C23×C31—28no—
191C23×C31—28no—
192C23×C31—44no—
193C23×C31—48no—
194C23×C31—48no—
195C23×C31—48no—
196C23×C31—416no—
197C23×C31—416no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31—48no—
202C23×C31C2×C22⋊C412yes—
203C23×C31C2×C22⋊C412yes—
204C23×C31C2×C22⋊C412yes—
205C23×C31#12014yes—
206C23×C31AGL(1,ℤ/8)12yes—
207C23×C31#13712yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—216no—
210Q8×C31—24no—
211Q8×C31—28no—
212Q8×C31C2×C22⋊C418yescomplemented
213D4×C31—232no—
214D4×C31—216no—
215D4×C31—232no—
216D4×C31—24no—
217D4×C31—28no—
218D4×C31C2×C22⋊C418yescomplemented
219C2×C4×C31—216no—
220C2×C4×C31—232no—
221C2×C4×C31—24no—
222C2×C4×C31—22no—
223C2×C4×C31—28no—
224C2×C4×C31—22no—
225C2×C4×C31—28no—
226C2×C4×C31—24no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—28no—
231C2×C4×C31—232no—
232C2×C4×C31—28no—
233C2×C4×C31—44no—
234C2×C4×C31—44no—
235C2×C4×C31—44no—
236C2×C4×C31—88no—
237C2×C4×C31C2×C22⋊C412yes—
238C2×C4×C31C2×C22⋊C412yes—
239C2×C4×C31C2×C22⋊C418yes—
240C2×C4×C31C2×C22⋊C412yes—
241C2×C4×C31#12018yes—
242C2×C4×C31#12014yes—
243C2×C4×C31C23×C411yescharacteristic
244C2×C4×C31C22×D414yes—
245C2×C4×C31C22×D411yescharacteristic
246C8×C31—44no—
247C23×#41C2×C3111yescharacteristic
248C2×#417C2×C3114yescomplemented
249C2×#407C2×C3111yescharacteristic, complemented
250C2×#380C2×C3111yescharacteristic, complemented
251#1788C2×C3114yescomplemented
252#1787C2×C3112yescomplemented
253#1786C2×C3112yescomplemented
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—24no—
258C22×C31—24no—
259C22×C31—24no—
260C22×C31—24no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—22no—
265C22×C31—28no—
266C22×C31—416no—
267C22×C31—432no—
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—42no—
271C22×C31—48no—
272C22×C31—42no—
273C22×C31—48no—
274C22×C31#37914yes—
275C22×C31#41714yescomplemented
276C22×C31C22×C22⋊C411yescharacteristic
277C22×C31C2×#12012yes—
278C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C31C2×#13711yescharacteristic
280C22×C31#57512yes—
281C4×C31—22no—
282C4×C31—28no—
283C4×C31—22no—
284C4×C31—84no—
285C4×C31#37914yes—
286C4×C31C22×C22⋊C412yes—
287C4×C31C2×#12012yes—
288C22×#41—216no—
289C22×#41—24no—
290C22×#41—21no—
291C22×#41—21no—
292C22×#41C4×C3118yescomplemented
293C22×#41C22×C3112yes—
294C22×#41C22×C3112yes—
295C23×Q8C22×C3111yescharacteristic
296C23×D4—21no—
297C23×D4C22×C3111yescharacteristic
298C24×C4—21no—
299C24×C4C22×C3111yescharacteristic
300C22×M16C22×C3112yescomplemented
301C22×C4⋊C4C22×C3112yescomplemented
302C22×C22⋊C4—21no—
303#417—216no—
304C2×#103C22×C3118yescomplemented
305C2×#102C22×C3118yescomplemented
306#407—24no—
307C2×C22⋊C8C22×C3112yes—
308C2×C22⋊C8C22×C3112yescomplemented
309#380—21no—
310#380—21no—
311#380C22×C3112yes—
312#380C22×C3112yescomplemented
313C2×#95—21no—
314C2×C31—24no—
315C2×C31—24no—
316C2×C31—42no—
317C2×C31—48no—
318C2×C31C2×#37911yescharacteristic
319C2×C31C2×#41711yescharacteristic
320C2×C31#289611yescharacteristic
321C2×C31#300612yes—
322C2×C31#300712yes—
323C25—21no—
324C25D4×C3111yescharacteristic
325C2×#41—216no—
326C2×#41—264no—
327C2×#41—216no—
328C2×#41—264no—
329C2×#41—28no—
330C2×#41—28no—
331C2×#41—28no—
332C2×#41—28no—
333C2×#41—24no—
334C2×#41—24no—
335C2×#41—28no—
336C2×#41C2×C4×C3118yes—
337C2×#41C2×C4×C3118yescomplemented
338C2×#41D4×C31116yes—
339C22×Q8—28no—
340C22×Q8—22no—
341C22×Q8—21no—
342C22×Q8C2×C4×C3114yescomplemented
343C22×Q8C23×C3112yes—
344C22×D4—216no—
345C22×D4—28no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—28no—
349C22×D4—21no—
350C22×D4—22no—
351C22×D4C2×C4×C3114yescomplemented
352C22×D4C23×C3112yes—
353C23×C4—216no—
354C23×C4—21no—
355C23×C4—28no—
356C23×C4—22no—
357C23×C4—22no—
358C23×C4—28no—
359C23×C4—22no—
360C23×C4C2×C4×C3114yes—
361C23×C4D4×C3111yescharacteristic
362C23×C4D4×C3111yescharacteristic
363C23×C4D4×C3111yescharacteristic
364C23×C4D4×C3114yes—
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C23×C4C23×C3111yescharacteristic
368C2×M16—24no—
369C2×M16—28no—
370C22×C8C23×C3114yes—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4C23×C3114yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—44no—
377#103—216no—
378#102—216no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381#95—22no—
382#95—22no—
383C31#2060111yescharacteristic, direct product factor, Sylow
384C24—28no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—44no—
392C24—42no—
393C24—42no—
394C24C22⋊C4×C3114yes—
395C24C2×D4×C3111yescharacteristic
396C24C2×D4×C3111yescharacteristic
397C24C2×D4×C3111yescharacteristic
398#41—264no—
399#41—232no—
400#41—264no—
401#41—216no—
402#41—216no—
403#41—216no—
404#41—232no—
405#41C22⋊C4×C31132yescomplemented
406C2×Q8—216no—
407C2×Q8—28no—
408C2×Q8—216no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8C22×C4×C3114yes—
413C2×Q8C2×D4×C3118yes—
414C2×D4—264no—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—216no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4C22×C4×C3114yes—
426C2×D4C2×D4×C3118yes—
427C22×C4—28no—
428C22×C4—264no—
429C22×C4—216no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—216no—
436C22×C4—24no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4C22⋊C4×C3114yes—
447C22×C4C22⋊C4×C3114yes—
448C22×C4C22⋊C4×C3114yes—
449C22×C4C22×C4×C3112yes—
450C22×C4C22×C4×C3112yes—
451C22×C4C22×C4×C3112yes—
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3112yes—
455C22×C4C2×D4×C3112yes—
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C2×D4×C3118yes—
458C22×C4C2×D4×C3112yes—
459C22×C4C24×C3111yescharacteristic
460M16—48no—
461C2×C8—24no—
462C2×C8—28no—
463C4⋊C4—28no—
464C4⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—216no—
473C23—28no—
474C23—24no—
475C23—22no—
476C23—28no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—28no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—416no—
486C23—416no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23C2×C22⋊C4×C3112yes—
492C23C2×C22⋊C4×C3112yes—
493C23C2×C22⋊C4×C3112yes—
494C23C31×#12014yes—
495C23C31×AGL(1,ℤ/8)12yes—
496C23C31×#13712yes—
497C23C22×D4×C3111yescharacteristic
498Q8—216no—
499Q8—24no—
500Q8—28no—
501Q8C2×C22⋊C4×C3118yescomplemented
502D4—232no—
503D4—216no—
504D4—232no—
505D4—24no—
506D4—28no—
507D4C2×C22⋊C4×C3118yescomplemented
508C2×C4—216no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—232no—
521C2×C4—28no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—88no—
526C2×C4C2×C22⋊C4×C3112yes—
527C2×C4C2×C22⋊C4×C3112yes—
528C2×C4C2×C22⋊C4×C3118yes—
529C2×C4C2×C22⋊C4×C3112yes—
530C2×C4C31×#12018yes—
531C2×C4C31×#12014yes—
532C2×C4C23×C4×C3111yescharacteristic
533C2×C4C22×D4×C3114yes—
534C2×C4C22×D4×C3111yescharacteristic
535C8—44no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—28no—
548C22—416no—
549C22—432no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—48no—
554C22—42no—
555C22—48no—
556C22C31×#37914yes—
557C22C31×#41714yescomplemented
558C22C22×C22⋊C4×C3111yescharacteristic
559C22C2×C31×#12012yes—
560C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
561C22C2×C31×#13711yescharacteristic
562C22C31×#57512yes—
563C4—22no—
564C4—28no—
565C4—22no—
566C4—84no—
567C4C31×#37914yes—
568C4C22×C22⋊C4×C3112yes—
569C4C2×C31×#12012yes—
570C2—24no—
571C2—24no—
572C2—42no—
573C2—48no—
574C2C2×C31×#37911yescharacteristic
575C2C2×C31×#41711yescharacteristic
576C2C31×#289611yescharacteristic
577C2C31×#300612yes—
578C2C31×#300712yes—
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC23×#41×C31C2
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. C2×C31×#417C2semidirect
3. C2×C31×#407C2semidirect
4. C2×C31×#380C2semidirect
5. C31×#1788C2semidirect
6. C31×#1787C2semidirect
7. C31×#1786C2semidirect
12. C22×#41×C31C4semidirect
20. C22×M16×C31C22semidirect
21. C22×C4⋊C4×C31C22semidirect
24. C2×C31×#103C22semidirect
25. C2×C31×#102C22semidirect
28. C2×C31×C22⋊C8C22semidirect
32. C31×#380C22semidirect
48. C2×#41×C31C2×C4semidirect
53. C22×Q8×C31C2×C4semidirect
62. C22×D4×C31C2×C4semidirect
115. #41×C31C22⋊C4semidirect
177. #20601C31direct (N × H)
212. Q8×C31C2×C22⋊C4semidirect
218. D4×C31C2×C22⋊C4semidirect
248. C2×#417C2×C31semidirect
249. C2×#407C2×C31semidirect
250. C2×#380C2×C31semidirect
251. #1788C2×C31semidirect
252. #1787C2×C31semidirect
253. #1786C2×C31semidirect
275. C22×C31#417semidirect
292. C22×#41C4×C31semidirect
300. C22×M16C22×C31semidirect
301. C22×C4⋊C4C22×C31semidirect
304. C2×#103C22×C31semidirect
305. C2×#102C22×C31semidirect
308. C2×C22⋊C8C22×C31semidirect
312. #380C22×C31semidirect
337. C2×#41C2×C4×C31semidirect
342. C22×Q8C2×C4×C31semidirect
351. C22×D4C2×C4×C31semidirect
383. C31#20601direct (N × H)
405. #41C22⋊C4×C31semidirect
501. Q8C2×C22⋊C4×C31semidirect
507. D4C2×C22⋊C4×C31semidirect
557. C22C31×#417semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C312
C2×C31×#4172
C2×C31×#4072
C2×C31×#3802
C31×#17882
C31×#17872
C31×#17862
#2060131