Finite Group Databasebeta

← C31×#22500 #21479259

Subgroups — C31×#22500 #21479259

Order 7936.

1370 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C2×C31×#436C211yesmaximal, characteristic, complemented
3C31×#2354C211yesmaximal, characteristic, complemented
4C31×#1990C211yesmaximal, characteristic, complemented
5C31×#1988C211yesmaximal, characteristic, complemented
6C31×#1895C212yesmaximal, complemented
7C31×#1875C212yesmaximal, complemented
8C31×#1824C212yesmaximal, complemented
9C31×#1802C211yesmaximal, characteristic, complemented
10C4×C31×#102C211yesmaximal, characteristic, complemented
11C31×#1736C212yesmaximal, complemented
12C22×#41×C31—21no—
13C22×M16×C31C2212yes—
14C31×#547—24no—
15C31×#547C418yescomplemented
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#123—22no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×#121C2212yes—
20C2×C4×D4×C31C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C2×C31×#117C2212yes—
23C22×C4⋊C4×C31C2212yes—
24C31×#436—24no—
25C2×C31×C4⋊C8C2211yescharacteristic
26C2×C31×#102C2211yescharacteristic
27C2×C31×#102C2211yescharacteristic
28C2×C31×#102C2212yes—
29C2×C31×C22⋊C8C2212yes—
30C2×C4×C8×C31C2211yescharacteristic
31C31×#390—21no—
32C31×#388—22no—
33C31×#384C2211yescharacteristic
34C31×#380C2212yes—
35C4×C4⋊C4×C31—21no—
36C4×C4⋊C4×C31C2211yescharacteristic
37C31×#340C2212yes—
38C31×#340C2211yescharacteristic
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#340C2211yescharacteristic
44C2×#41×C31—28no—
45C2×#41×C31—44no—
46C2×#41×C31—44no—
47C22×Q8×C31—21no—
48C22×D4×C31—22no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—22no—
52C23×C4×C31—21no—
53C23×C4×C31C2312yes—
54C2×M16×C31—24no—
55C2×M16×C31—24no—
56C2×M16×C31—42no—
57C22×C8×C31C2311yescharacteristic
58C22×C8×C31C2312yes—
59C22×C8×C31C2311yescharacteristic
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8C2×C414yes—
62C31×#123—216no—
63C31×C22⋊Q8—28no—
64C31×#121—24no—
65C31×#121C2×C418yes—
66C4×D4×C31—22no—
67C4×D4×C31C2×C414yes—
68C4×D4×C31C2×C414yes—
69C4×D4×C31D414yes—
70C31×#117—24no—
71C31×#117—42no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—24no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31C2×C418yes—
78C2×C4⋊C4×C31D411yescharacteristic
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31D412yes—
89C2×C42×C31—22no—
90C2×C42×C31—21no—
91C2×C42×C31C2311yescharacteristic
92C31×C4⋊C8—22no—
93C31×#102—22no—
94C31×#102—22no—
95C31×#102—42no—
96C31×C22⋊C8—24no—
97C4×C8×C31—22no—
98C31×#95—22no—
99C31×#95—21no—
100C24×C31—21no—
101C24×C31C2×D412yes—
102#41×C31—432no—
103C2×Q8×C31—24no—
104C2×Q8×C31—42no—
105C2×Q8×C31—42no—
106C2×D4×C31—28no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—21no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31C22⋊C414yes—
117C2×D4×C31C22×C412yes—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—28no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—24no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—44no—
130C22×C4×C31—44no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31C22×C414yes—
134C22×C4×C31C2×D412yes—
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D414yes—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2411yescharacteristic
139M16×C31—48no—
140C2×C8×C31—21no—
141C2×C8×C31—21no—
142C2×C8×C31—21no—
143C2×C8×C31—22no—
144C2×C8×C31—21no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—21no—
148C2×C8×C31#4112yescomplemented
149C4⋊C4×C31—28no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—81no—
160C4⋊C4×C31C22⋊C414yes—
161C4⋊C4×C31C22×C412yes—
162C4⋊C4×C31C22×C414yes—
163C4⋊C4×C31#4112yescomplemented
164C22⋊C4×C31—24no—
165C22⋊C4×C31—28no—
166C22⋊C4×C31—44no—
167C22⋊C4×C31—82no—
168C22⋊C4×C31C22⋊C418yes—
169C42×C31—44no—
170C42×C31—42no—
171C42×C31C22×C412yes—
172C42×C31C2×D412yes—
173#22500C3111yesmaximal, characteristic, direct product factor, Sylow
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—22no—
180C23×C31—22no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31C2×C22⋊C414yes—
188C23×C31C22×D411yescharacteristic
189Q8×C31—48no—
190D4×C31—22no—
191D4×C31—22no—
192D4×C31—28no—
193D4×C31—416no—
194D4×C31—41no—
195D4×C31—41no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—22no—
207C2×C4×C31—416no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—48no—
215C2×C4×C31—42no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31C2×C22⋊C412yes—
222C2×C4×C31C2×C22⋊C414yes—
223C2×C4×C31C2×C22⋊C412yes—
224C2×C4×C31AGL(1,ℤ/8)12yes—
225C2×C4×C31AGL(1,ℤ/8)12yes—
226C2×C4×C31C23×C411yescharacteristic
227C2×C4×C31C22×D411yescharacteristic
228C2×C4×C31C2×#4111yescharacteristic
229C2×C4×C31C2×#4111yescharacteristic
230C2×C4×C31H(5,𝔽2)11yescharacteristic
231C2×C4×C312−1+411yescharacteristic
232C8×C31—22no—
233C8×C31—42no—
234C8×C31—41no—
235C2×#547C2×C3111yescharacteristic
236C2×#436C2×C3111yescharacteristic, complemented
237#2354C2×C3111yescharacteristic, complemented
238#1990C2×C3111yescharacteristic, complemented
239#1988C2×C3111yescharacteristic, complemented
240#1895C2×C3112yescomplemented
241#1875C2×C3112yescomplemented
242#1824C2×C3112yescomplemented
243#1802C2×C3111yescharacteristic, complemented
244C4×#102C2×C3111yescharacteristic, complemented
245#1736C2×C3112yescomplemented
246C22×C31—24no—
247C22×C31—24no—
248C22×C31—24no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—81no—
257C22×C31—81no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31C22×C22⋊C411yescharacteristic
261C22×C31#54611yescharacteristic
262C22×C31#54711yescharacteristic
263C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C31#57711yescharacteristic
266C22×C31#57811yescharacteristic
267C4×C31—24no—
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—44no—
272C4×C31—82no—
273C4×C31—81no—
274C4×C31#41814yes—
275C4×C31C4×#4111yescharacteristic
276C4×C31#52011yescharacteristic
277C4×C31#54611yescharacteristic
278C4×C31#54711yescharacteristic
279C22×#41—21no—
280C22×M16C22×C3112yes—
281#547—24no—
282#547C4×C3118yescomplemented
283C2×C4⋊Q8C22×C3111yescharacteristic
284C2×#123—22no—
285C2×C22⋊Q8—21no—
286C2×#121C22×C3112yes—
287C2×C4×D4C22×C3111yescharacteristic
288C2×C4×D4C22×C3111yescharacteristic
289C2×#117C22×C3112yes—
290C22×C4⋊C4C22×C3112yes—
291#436—24no—
292C2×C4⋊C8C22×C3111yescharacteristic
293C2×#102C22×C3111yescharacteristic
294C2×#102C22×C3111yescharacteristic
295C2×#102C22×C3112yes—
296C2×C22⋊C8C22×C3112yes—
297C2×C4×C8C22×C3111yescharacteristic
298#390—21no—
299#388—22no—
300#384C22×C3111yescharacteristic
301#380C22×C3112yes—
302C4×C4⋊C4—21no—
303C4×C4⋊C4C22×C3111yescharacteristic
304#340C22×C3112yes—
305#340C22×C3111yescharacteristic
306#340C22×C3112yes—
307#340C22×C3112yes—
308#340C22×C3112yes—
309#340C22×C3112yes—
310#340C22×C3111yescharacteristic
311C2×C31—44no—
312C2×C31—82no—
313C2×C31#233911yescharacteristic
314C2×C31C2×#41811yescharacteristic
315C2×C31#289811yescharacteristic
316C2×C31#333011yescharacteristic
317C2×C31#333211yescharacteristic
318C2×C31#335311yescharacteristic
319C2×C31#335411yescharacteristic
320C2×#41—28no—
321C2×#41—44no—
322C2×#41—44no—
323C22×Q8—21no—
324C22×D4—22no—
325C22×D4D4×C3111yescharacteristic
326C22×D4C23×C3111yescharacteristic
327C23×C4—22no—
328C23×C4—21no—
329C23×C4C23×C3112yes—
330C2×M16—24no—
331C2×M16—24no—
332C2×M16—42no—
333C22×C8C23×C3112yes—
334C22×C8C23×C3111yescharacteristic
335C22×C8C23×C3111yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8C2×C4×C3114yes—
338#123—216no—
339C22⋊Q8—28no—
340#121—24no—
341#121C2×C4×C3118yes—
342C4×D4—22no—
343C4×D4C2×C4×C3114yes—
344C4×D4C2×C4×C3114yes—
345C4×D4D4×C3114yes—
346#117—24no—
347#117—42no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C2×C4×C3118yes—
354C2×C4⋊C4D4×C3111yescharacteristic
355C2×C4⋊C4C23×C3112yes—
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C4⋊C4C23×C3112yes—
359C2×C4⋊C4C23×C3111yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4D4×C3112yes—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42C23×C3111yescharacteristic
368C4⋊C8—22no—
369#102—22no—
370#102—22no—
371#102—42no—
372C22⋊C8—24no—
373C4×C8—22no—
374#95—22no—
375#95—21no—
376C31#2250011yescharacteristic, direct product factor, Sylow
377C24—21no—
378C24C2×D4×C3112yes—
379#41—432no—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4C22⋊C4×C3114yes—
394C2×D4C22×C4×C3112yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C22×C4×C3114yes—
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3114yes—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C24×C3111yescharacteristic
416M16—48no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8#41×C3112yescomplemented
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4C22⋊C4×C3114yes—
438C4⋊C4C22×C4×C3112yes—
439C4⋊C4C22×C4×C3114yes—
440C4⋊C4#41×C3112yescomplemented
441C22⋊C4—24no—
442C22⋊C4—28no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4C22⋊C4×C3118yes—
446C42—44no—
447C42—42no—
448C42C22×C4×C3112yes—
449C42C2×D4×C3112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C2×C22⋊C4×C3114yes—
464C23C22×D4×C3111yescharacteristic
465Q8—48no—
466D4—28no—
467D4—22no—
468D4—22no—
469D4—416no—
470D4—41no—
471D4—41no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—416no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4C2×C22⋊C4×C3112yes—
498C2×C4C2×C22⋊C4×C3114yes—
499C2×C4C2×C22⋊C4×C3112yes—
500C2×C4C31×AGL(1,ℤ/8)12yes—
501C2×C4C31×AGL(1,ℤ/8)12yes—
502C2×C4C23×C4×C3111yescharacteristic
503C2×C4C22×D4×C3111yescharacteristic
504C2×C4C2×#41×C3111yescharacteristic
505C2×C4C2×#41×C3111yescharacteristic
506C2×C4C31×H(5,𝔽2)11yescharacteristic
507C2×C4C31×2−1+411yescharacteristic
508C8—22no—
509C8—42no—
510C8—41no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22C22×C22⋊C4×C3111yescharacteristic
526C22C31×#54611yescharacteristic
527C22C31×#54711yescharacteristic
528C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
529C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
530C22C31×#57711yescharacteristic
531C22C31×#57811yescharacteristic
532C4—24no—
533C4—22no—
534C4—42no—
535C4—44no—
536C4—42no—
537C4—82no—
538C4—81no—
539C4C31×#41814yes—
540C4C4×#41×C3111yescharacteristic
541C4C31×#52011yescharacteristic
542C4C31×#54611yescharacteristic
543C4C31×#54711yescharacteristic
544C2—44no—
545C2—82no—
546C2C31×#233911yescharacteristic
547C2C2×C31×#41811yescharacteristic
548C2C31×#289811yescharacteristic
549C2C31×#333011yescharacteristic
550C2C31×#333211yescharacteristic
551C2C31×#335311yescharacteristic
552C2C31×#335411yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#547C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#436C2semidirect
3. C31×#2354C2semidirect
4. C31×#1990C2semidirect
5. C31×#1988C2semidirect
6. C31×#1895C2semidirect
7. C31×#1875C2semidirect
8. C31×#1824C2semidirect
9. C31×#1802C2semidirect
10. C4×C31×#102C2semidirect
11. C31×#1736C2semidirect
15. C31×#547C4semidirect
148. C2×C8×C31#41semidirect
163. C4⋊C4×C31#41semidirect
173. #22500C31direct (N × H)
236. C2×#436C2×C31semidirect
237. #2354C2×C31semidirect
238. #1990C2×C31semidirect
239. #1988C2×C31semidirect
240. #1895C2×C31semidirect
241. #1875C2×C31semidirect
242. #1824C2×C31semidirect
243. #1802C2×C31semidirect
244. C4×#102C2×C31semidirect
245. #1736C2×C31semidirect
282. #547C4×C31semidirect
376. C31#22500direct (N × H)
425. C2×C8#41×C31semidirect
440. C4⋊C4#41×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C2×C31×#4362
C31×#23542
C31×#19902
C31×#19882
C31×#18952
C31×#18752
C31×#18242
C31×#18022
C4×C31×#1022
C31×#17362
#2250031