Finite Group Databasebeta

← C31×#24835 #21481594

Subgroups — C31×#24835 #21481594

Order 7936.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#474C211yesmaximal, characteristic, complemented
2C22×C31×C4⋊C8C211yesmaximal, characteristic, complemented
3C31×#2857C211yesmaximal, characteristic, complemented
4C4×C31×#121C211yesmaximal, characteristic
5C31×#2086C212yesmaximal, complemented
6C31×#2067C212yesmaximal, complemented
7C31×#1988C211yesmaximal, characteristic, complemented
8C31×#1947C211yesmaximal, characteristic, complemented
9C31×#1893C212yesmaximal
10C31×#1808C212yesmaximal, complemented
11C31×#1802C211yesmaximal, characteristic, complemented
12C23×C8×C31C2212yes—
13C2×C31×C4⋊Q8C2211yescharacteristic
14C2×C31×C22⋊Q8—21no—
15C2×C31×#121C2211yescharacteristic
16C2×C4×D4×C31—21no—
17C2×C4×D4×C31C2211yescharacteristic
18C22×C42×C31C2211yescharacteristic
19C31×#474—24no—
20C2×C31×#107C2212yes—
21C2×C31×#107C2212yes—
22C2×C31×C4⋊C8—24no—
23C2×C31×C4⋊C8C2211yescharacteristic
24C2×C31×C4⋊C8C2212yes—
25C2×C31×C4⋊C8C2211yescharacteristic
26C2×C31×#102C2212yes—
27C2×C31×#102C2212yes—
28C31×#391—21no—
29C31×#390C2211yescharacteristic
30C31×#389—21no—
31C31×#389C2212yes—
32C31×#389C2211yescharacteristic
33C31×#388—21no—
34C31×#386—21no—
35C31×#384C2211yescharacteristic
36C31×#380—21no—
37C31×#380C2211yescharacteristic
38C31×#380C2212yes—
39C4×C4⋊C4×C31C2211yescharacteristic
40C4×C22⋊C4×C31—21no—
41C31×#340C2212yes—
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C22×Q8×C31—21no—
46C22×D4×C31—21no—
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2311yescharacteristic
52C22×C8×C31—24no—
53C22×C8×C31—24no—
54C22×C8×C31—42no—
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
59C31×C22⋊Q8—44no—
60C31×#121—22no—
61C31×#121Q814yescomplemented
62C4×D4×C31—22no—
63C4×D4×C31—22no—
64C4×D4×C31—44no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31C2312yes—
71C2×C4⋊C4×C31C2312yes—
72C2×C4⋊C4×C31C2311yescharacteristic
73C2×C4⋊C4×C31C2311yescharacteristic
74C2×C4⋊C4×C31C2311yescharacteristic
75C2×C22⋊C4×C31—21no—
76C2×C22⋊C4×C31—21no—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—21no—
81C2×C42×C31—22no—
82C2×C42×C31—22no—
83C2×C42×C31—21no—
84C2×C42×C31D411yescharacteristic
85C2×C42×C31D411yescharacteristic
86C2×C42×C31C2311yescharacteristic
87C2×C42×C31C2311yescharacteristic
88C31×#107—24no—
89C31×#107—42no—
90C31×C4⋊C8—24no—
91C31×C4⋊C8—22no—
92C31×C4⋊C8D414yescomplemented
93C31×#102—24no—
94C31×#102—42no—
95C31×#95—21no—
96C31×#95—21no—
97C31×#95—21no—
98C31×#95—22no—
99C31×#95—22no—
100C24×C31—21no—
101C24×C31C2×D411yescharacteristic
102C2×Q8×C31—42no—
103C2×Q8×C31—42no—
104C2×Q8×C31—42no—
105C2×D4×C31—24no—
106C2×D4×C31—21no—
107C2×D4×C31—21no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31C2×Q812yes—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—41no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31D814yes—
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×Q812yes—
138C22×C4×C31#4112yes—
139C22×C4×C31C2411yescharacteristic
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C2×C8×C31—22no—
145C2×C8×C31—48no—
146C2×C8×C31C2×D414yes—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—42no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—44no—
161C4⋊C4×C31C2×Q812yes—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—42no—
164C22⋊C4×C31—42no—
165C22⋊C4×C31—44no—
166C22⋊C4×C31—42no—
167C22⋊C4×C31—42no—
168C42×C31—22no—
169C42×C31—22no—
170C42×C31—21no—
171C42×C31—21no—
172C42×C31—42no—
173C42×C31C2×D412yes—
174C42×C31#4112yes—
175#24835C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—21no—
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—22no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31C22⋊Q812yes—
188C23×C31C2×D812yes—
189C23×C31#13712yes—
190C23×C31C22×D411yescharacteristic
191Q8×C31—84no—
192D4×C31—22no—
193D4×C31—22no—
194D4×C31—48no—
195D4×C31—41no—
196D4×C31—41no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—28no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—44no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—42no—
225C2×C4×C31—41no—
226C2×C4×C31—42no—
227C2×C4×C31—84no—
228C2×C4×C31#12112yes—
229C2×C4×C31#12112yes—
230C2×C4×C31C22⋊Q812yes—
231C2×C4×C31C2×D812yes—
232C2×C4×C31C2×D812yes—
233C2×C4×C31#13512yes—
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C22×Q811yescharacteristic
236C2×C4×C31C2×#4111yescharacteristic
237C2×C4×C31C2×#4111yescharacteristic
238C2×C4×C31H(5,𝔽2)11yescharacteristic
239C2×C4×C312−1+411yescharacteristic
240C8×C31—24no—
241C8×C31—42no—
242C2×#474C2×C3111yescharacteristic, complemented
243C22×C4⋊C8C2×C3111yescharacteristic, complemented
244#2857C2×C3111yescharacteristic, complemented
245C4×#121C2×C3111yescharacteristic
246#2086C2×C3112yescomplemented
247#2067C2×C3112yescomplemented
248#1988C2×C3111yescharacteristic, complemented
249#1947C2×C3111yescharacteristic, complemented
250#1893C2×C3112yes—
251#1808C2×C3112yescomplemented
252#1802C2×C3111yescharacteristic, complemented
253C22×C31—22no—
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—22no—
257C22×C31—22no—
258C22×C31—22no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31#47414yescomplemented
267C22×C31C2×#12111yescharacteristic
268C22×C31C2×C22⋊Q811yescharacteristic
269C22×C31#53711yescharacteristic
270C22×C31C22×D811yescharacteristic
271C22×C31C2×#13511yescharacteristic
272C22×C31C2×#13711yescharacteristic
273C22×C31#57511yescharacteristic
274C4×C31—22no—
275C4×C31—22no—
276C4×C31—22no—
277C4×C31—41no—
278C4×C31—42no—
279C4×C31—81no—
280C4×C31—82no—
281C4×C31—81no—
282C4×C31—82no—
283C4×C31#46614yes—
284C4×C31#54611yescharacteristic
285C4×C31D4×Q811yescharacteristic
286C4×C31#55211yescharacteristic
287C4×C31#55411yescharacteristic
288C23×C8C22×C3112yes—
289C2×C4⋊Q8C22×C3111yescharacteristic
290C2×C22⋊Q8—21no—
291C2×#121C22×C3111yescharacteristic
292C2×C4×D4—21no—
293C2×C4×D4C22×C3111yescharacteristic
294C22×C42C22×C3111yescharacteristic
295#474—24no—
296C2×#107C22×C3112yes—
297C2×#107C22×C3112yes—
298C2×C4⋊C8—24no—
299C2×C4⋊C8C22×C3111yescharacteristic
300C2×C4⋊C8C22×C3112yes—
301C2×C4⋊C8C22×C3111yescharacteristic
302C2×#102C22×C3112yes—
303C2×#102C22×C3112yes—
304#391—21no—
305#390C22×C3111yescharacteristic
306#389—21no—
307#389C22×C3112yes—
308#389C22×C3111yescharacteristic
309#388—21no—
310#386—21no—
311#384C22×C3111yescharacteristic
312#380—21no—
313#380C22×C3111yescharacteristic
314#380C22×C3112yes—
315C4×C4⋊C4C22×C3111yescharacteristic
316C4×C22⋊C4—21no—
317#340C22×C3112yes—
318#340C22×C3112yes—
319#340C22×C3112yes—
320#340C22×C3112yes—
321C2×C31—24no—
322C2×C31—82no—
323C2×C31#241711yescharacteristic
324C2×C31C2×#46611yescharacteristic
325C2×C31#305611yescharacteristic
326C2×C31C2×#47411yescharacteristic
327C2×C31#307611yescharacteristic
328C2×C31#324511yescharacteristic
329C2×C31#324811yescharacteristic
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4C23×C3111yescharacteristic
333C23×C4—21no—
334C23×C4D4×C3111yescharacteristic
335C23×C4D4×C3111yescharacteristic
336C23×C4C23×C3111yescharacteristic
337C22×C8—24no—
338C22×C8—24no—
339C22×C8—42no—
340C22×C8C23×C3112yes—
341C22×C8C23×C3112yes—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C22⋊Q8—44no—
345#121—22no—
346#121Q8×C3114yescomplemented
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—44no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C4⋊C4C23×C3112yes—
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C4⋊C4C23×C3112yes—
359C2×C4⋊C4C23×C3111yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42D4×C3111yescharacteristic
370C2×C42D4×C3111yescharacteristic
371C2×C42C23×C3111yescharacteristic
372C2×C42C23×C3111yescharacteristic
373#107—24no—
374#107—42no—
375C4⋊C8—24no—
376C4⋊C8—22no—
377C4⋊C8D4×C3114yescomplemented
378#102—24no—
379#102—42no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385C31#2483511yescharacteristic, direct product factor, Sylow
386C24—21no—
387C24C2×D4×C3111yescharacteristic
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4C2×Q8×C3112yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4D8×C3114yes—
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C2×Q8×C3112yes—
424C22×C4#41×C3112yes—
425C22×C4C24×C3111yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—48no—
432C2×C8C2×D4×C3114yes—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4C2×Q8×C3112yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C42—22no—
455C42—22no—
456C42—21no—
457C42—21no—
458C42—42no—
459C42C2×D4×C3112yes—
460C42#41×C3112yes—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23C31×C22⋊Q812yes—
473C23C2×D8×C3112yes—
474C23C31×#13712yes—
475C23C22×D4×C3111yescharacteristic
476Q8—84no—
477D4—22no—
478D4—22no—
479D4—48no—
480D4—41no—
481D4—41no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4C31×#12112yes—
514C2×C4C31×#12112yes—
515C2×C4C31×C22⋊Q812yes—
516C2×C4C2×D8×C3112yes—
517C2×C4C2×D8×C3112yes—
518C2×C4C31×#13512yes—
519C2×C4C22×D4×C3111yescharacteristic
520C2×C4C22×Q8×C3111yescharacteristic
521C2×C4C2×#41×C3111yescharacteristic
522C2×C4C2×#41×C3111yescharacteristic
523C2×C4C31×H(5,𝔽2)11yescharacteristic
524C2×C4C31×2−1+411yescharacteristic
525C8—24no—
526C8—42no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C31×#47414yescomplemented
541C22C2×C31×#12111yescharacteristic
542C22C2×C31×C22⋊Q811yescharacteristic
543C22C31×#53711yescharacteristic
544C22C22×D8×C3111yescharacteristic
545C22C2×C31×#13511yescharacteristic
546C22C2×C31×#13711yescharacteristic
547C22C31×#57511yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—42no—
553C4—81no—
554C4—82no—
555C4—81no—
556C4—82no—
557C4C31×#46614yes—
558C4C31×#54611yescharacteristic
559C4D4×Q8×C3111yescharacteristic
560C4C31×#55211yescharacteristic
561C4C31×#55411yescharacteristic
562C2—24no—
563C2—82no—
564C2C31×#241711yescharacteristic
565C2C2×C31×#46611yescharacteristic
566C2C31×#305611yescharacteristic
567C2C2×C31×#47411yescharacteristic
568C2C31×#307611yescharacteristic
569C2C31×#324511yescharacteristic
570C2C31×#324811yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#474C2semidirect
2. C22×C31×C4⋊C8C2semidirect
3. C31×#2857C2semidirect
5. C31×#2086C2semidirect
6. C31×#2067C2semidirect
7. C31×#1988C2semidirect
8. C31×#1947C2semidirect
10. C31×#1808C2semidirect
11. C31×#1802C2semidirect
61. C31×#121Q8semidirect
92. C31×C4⋊C8D4semidirect
175. #24835C31direct (N × H)
242. C2×#474C2×C31semidirect
243. C22×C4⋊C8C2×C31semidirect
244. #2857C2×C31semidirect
246. #2086C2×C31semidirect
247. #2067C2×C31semidirect
248. #1988C2×C31semidirect
249. #1947C2×C31semidirect
251. #1808C2×C31semidirect
252. #1802C2×C31semidirect
266. C22×C31#474semidirect
346. #121Q8×C31semidirect
377. C4⋊C8D4×C31semidirect
385. C31#24835direct (N × H)
540. C22C31×#474semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4742
C22×C31×C4⋊C82
C31×#28572
C4×C31×#1212
C31×#20862
C31×#20672
C31×#19882
C31×#19472
C31×#18932
C31×#18082
C31×#18022
#2483531