Finite Group Databasebeta

← C31×#11002 #21467761

Subgroups — C31×#11002 #21467761

Order 7936.

1430 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C22×C31×#104C212yesmaximal, complemented
3C2×C31×#417C212yesmaximal, complemented
4C2×C31×#343C211yesmaximal, characteristic, complemented
5C31×C4⋊#95C211yesmaximal, characteristic, complemented
6C22×#41×C31C2212yes—
7C22×M16×C31C2212yes—
8C31×#537—24no—
9C31×#537—22no—
10C31×#537C414yescomplemented
11C2×C31×C22⋊Q8—22no—
12C2×C31×#121—22no—
13C2×C31×#121—22no—
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C42×C31C2211yescharacteristic
17C2×C31×#104—24no—
18C2×C31×#104—24no—
19C2×C31×#104—24no—
20C31×#417—24no—
21C31×#417—24no—
22C2×C31×#103—22no—
23C2×C31×#102—22no—
24C2×C31×#95—22no—
25C31×#343—22no—
26C2×#41×C31—22no—
27C2×#41×C31—22no—
28C2×#41×C31—22no—
29C2×#41×C31—22no—
30C2×#41×C31—22no—
31C2×#41×C31—22no—
32C2×#41×C31—22no—
33C2×#41×C31—22no—
34C2×#41×C31—22no—
35C2×#41×C31—22no—
36C2×#41×C31C2×C414yes—
37C2×#41×C31D414yes—
38C22×Q8×C31D412yes—
39C22×D4×C31—22no—
40C22×D4×C31—22no—
41C22×D4×C31D412yes—
42C23×C4×C31—21no—
43C23×C4×C31—22no—
44C23×C4×C31—21no—
45C23×C4×C31D412yes—
46C23×C4×C31C2311yescharacteristic
47C2×M16×C31—22no—
48C2×M16×C31—22no—
49C2×M16×C31—22no—
50C2×M16×C31—22no—
51C2×M16×C31—22no—
52C2×M16×C31D414yescomplemented
53C22×C8×C31—22no—
54C31×C22⋊Q8—28no—
55C31×C22⋊Q8—28no—
56C31×#121—28no—
57C31×#121—28no—
58C31×#121—28no—
59C31×#121—28no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31C2×C412yes—
67C2×C4⋊C4×C31D412yes—
68C2×C4⋊C4×C31D412yescomplemented
69C2×C22⋊C4×C31—24no—
70C2×C42×C31—22no—
71C2×C42×C31—22no—
72C2×C42×C31—21no—
73C2×C42×C31D412yescomplemented
74C31×#104—216no—
75C31×#103—28no—
76C31×#102—28no—
77C31×#95—44no—
78C24×C31—22no—
79C24×C31C2×D411yescharacteristic
80#41×C31—24no—
81#41×C31—24no—
82#41×C31—24no—
83#41×C31—24no—
84#41×C31—24no—
85#41×C31—24no—
86#41×C31—24no—
87#41×C31—24no—
88#41×C31—42no—
89#41×C31—42no—
90#41×C31—42no—
91#41×C31—42no—
92#41×C31—42no—
93#41×C31—42no—
94#41×C31—42no—
95#41×C31—42no—
96C2×Q8×C31—22no—
97C2×Q8×C31—22no—
98C2×Q8×C31—22no—
99C2×Q8×C31—22no—
100C2×Q8×C31—22no—
101C2×Q8×C31C22⋊C414yes—
102C2×D4×C31—24no—
103C2×D4×C31—24no—
104C2×D4×C31—28no—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—22no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—42no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31—42no—
114C2×D4×C31—82no—
115C2×D4×C31—82no—
116C2×D4×C31C22⋊C414yes—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—24no—
124C22×C4×C31—24no—
125C22×C4×C31—22no—
126C22×C4×C31—41no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—41no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—42no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31C22⋊C414yes—
141C22×C4×C31C22×C411yescharacteristic
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31#4112yes—
149C22×C4×C31#4111yescharacteristic
150C22×C4×C31#4111yescharacteristic
151M16×C31—24no—
152M16×C31—24no—
153M16×C31—42no—
154M16×C31—42no—
155C2×C8×C31—24no—
156C2×C8×C31—42no—
157C2×C8×C31—42no—
158C4⋊C4×C31—28no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—42no—
163C22⋊C4×C31—216no—
164C42×C31—24no—
165C42×C31—42no—
166#11002C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—21no—
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—21no—
171C23×C31—24no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31—42no—
177C23×C31—42no—
178C23×C31C2×C22⋊C411yescharacteristic
179C23×C31#12011yescharacteristic
180C23×C31#12011yescharacteristic
181C23×C31#12111yescharacteristic
182C23×C31#12111yescharacteristic
183C23×C31#12411yescharacteristic
184C23×C31#12711yescharacteristic
185Q8×C31—24no—
186Q8×C31—24no—
187Q8×C31—42no—
188Q8×C31—42no—
189D4×C31—24no—
190D4×C31—24no—
191D4×C31—44no—
192D4×C31—44no—
193D4×C31—44no—
194D4×C31—44no—
195D4×C31—42no—
196D4×C31—42no—
197D4×C31—88no—
198C2×C4×C31—24no—
199C2×C4×C31—24no—
200C2×C4×C31—24no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—44no—
212C2×C4×C31—44no—
213C2×C4×C31—44no—
214C2×C4×C31—44no—
215C2×C4×C31—44no—
216C2×C4×C31—44no—
217C2×C4×C31—42no—
218C2×C4×C31—82no—
219C2×C4×C31—82no—
220C2×C4×C31—81no—
221C2×C4×C31—81no—
222C2×C4×C31C2×C22⋊C412yes—
223C2×C4×C31C4×D412yes—
224C2×C4×C31C4×D411yescharacteristic
225C2×C4×C31C4×D411yescharacteristic
226C2×C4×C31#12012yes—
227C2×C4×C31#12012yes—
228C2×C4×C31#12012yes—
229C2×C4×C31#12112yes—
230C2×C4×C31#12112yes—
231C2×C4×C31#12112yes—
232C2×C4×C31#12112yes—
233C2×C4×C31#12112yes—
234C2×C4×C31#12312yes—
235C2×C4×C31#12312yes—
236C2×C4×C31#12411yescharacteristic
237C2×C4×C31#12711yescharacteristic
238C8×C31—44no—
239C2×#537C2×C3111yescharacteristic
240C22×#104C2×C3112yescomplemented
241C2×#417C2×C3112yescomplemented
242C2×#343C2×C3111yescharacteristic, complemented
243C4⋊#95C2×C3111yescharacteristic, complemented
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—21no—
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—44no—
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—82no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31#37911yescharacteristic
264C22×C31#39011yescharacteristic
265C22×C31#39011yescharacteristic
266C22×C31#39211yescharacteristic
267C22×C31#39211yescharacteristic
268C22×C31#39411yescharacteristic
269C22×C31#39411yescharacteristic
270C22×C31#45311yescharacteristic
271C22×C31#45411yescharacteristic
272C22×C31#45511yescharacteristic
273C22×C31#45611yescharacteristic
274C4×C31—44no—
275C4×C31—44no—
276C4×C31—44no—
277C4×C31—82no—
278C4×C31#38612yes—
279C4×C31#38612yes—
280C4×C31#39212yes—
281C4×C31#39412yes—
282C22×#41C22×C3112yes—
283C22×M16C22×C3112yes—
284#537—24no—
285#537—22no—
286#537C4×C3114yescomplemented
287C2×C22⋊Q8—22no—
288C2×#121—22no—
289C2×#121—22no—
290C22×C4⋊C4C22×C3111yescharacteristic
291C22×C4⋊C4C22×C3111yescharacteristic
292C22×C42C22×C3111yescharacteristic
293C2×#104—24no—
294C2×#104—24no—
295C2×#104—24no—
296#417—24no—
297#417—24no—
298C2×#103—22no—
299C2×#102—22no—
300C2×#95—22no—
301#343—22no—
302C2×C31—21no—
303C2×C31—21no—
304C2×C31—21no—
305C2×C31—21no—
306C2×C31—84no—
307C2×C31#143811yescharacteristic
308C2×C31#179211yescharacteristic
309C2×C31#179311yescharacteristic
310C2×C31#200611yescharacteristic
311C2×C31#200711yescharacteristic
312C2×C31#203911yescharacteristic
313C2×C31#204011yescharacteristic
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41C2×C4×C3114yes—
325C2×#41D4×C3114yes—
326C22×Q8D4×C3112yes—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4D4×C3112yes—
330C23×C4—21no—
331C23×C4—22no—
332C23×C4—21no—
333C23×C4D4×C3112yes—
334C23×C4C23×C3111yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16—22no—
340C2×M16D4×C3114yescomplemented
341C22×C8—22no—
342C22⋊Q8—28no—
343C22⋊Q8—28no—
344#121—28no—
345#121—28no—
346#121—28no—
347#121—28no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C2×C4×C3112yes—
355C2×C4⋊C4D4×C3112yes—
356C2×C4⋊C4D4×C3112yescomplemented
357C2×C22⋊C4—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42D4×C3112yescomplemented
362#104—216no—
363#103—28no—
364#102—28no—
365#95—44no—
366C31#1100211yescharacteristic, direct product factor, Sylow
367C24—22no—
368C24C2×D4×C3111yescharacteristic
369#41—24no—
370#41—24no—
371#41—24no—
372#41—24no—
373#41—24no—
374#41—24no—
375#41—24no—
376#41—24no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41—42no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8C22⋊C4×C3114yes—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4C22⋊C4×C3114yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4C22⋊C4×C3114yes—
430C22×C4C22×C4×C3111yescharacteristic
431C22×C4C2×D4×C3112yes—
432C22×C4C2×D4×C3112yes—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4#41×C3112yes—
438C22×C4#41×C3111yescharacteristic
439C22×C4#41×C3111yescharacteristic
440M16—24no—
441M16—24no—
442M16—42no—
443M16—42no—
444C2×C8—24no—
445C2×C8—42no—
446C2×C8—42no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C22⋊C4—216no—
453C42—24no—
454C42—42no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23C2×C22⋊C4×C3111yescharacteristic
467C23C31×#12011yescharacteristic
468C23C31×#12011yescharacteristic
469C23C31×#12111yescharacteristic
470C23C31×#12111yescharacteristic
471C23C31×#12411yescharacteristic
472C23C31×#12711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477D4—24no—
478D4—24no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—42no—
484D4—42no—
485D4—88no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4C2×C22⋊C4×C3112yes—
511C2×C4C4×D4×C3112yes—
512C2×C4C4×D4×C3111yescharacteristic
513C2×C4C4×D4×C3111yescharacteristic
514C2×C4C31×#12012yes—
515C2×C4C31×#12012yes—
516C2×C4C31×#12012yes—
517C2×C4C31×#12112yes—
518C2×C4C31×#12112yes—
519C2×C4C31×#12112yes—
520C2×C4C31×#12112yes—
521C2×C4C31×#12112yes—
522C2×C4C31×#12312yes—
523C2×C4C31×#12312yes—
524C2×C4C31×#12411yescharacteristic
525C2×C4C31×#12711yescharacteristic
526C8—44no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C31×#37911yescharacteristic
547C22C31×#39011yescharacteristic
548C22C31×#39011yescharacteristic
549C22C31×#39211yescharacteristic
550C22C31×#39211yescharacteristic
551C22C31×#39411yescharacteristic
552C22C31×#39411yescharacteristic
553C22C31×#45311yescharacteristic
554C22C31×#45411yescharacteristic
555C22C31×#45511yescharacteristic
556C22C31×#45611yescharacteristic
557C4—44no—
558C4—44no—
559C4—44no—
560C4—82no—
561C4C31×#38612yes—
562C4C31×#38612yes—
563C4C31×#39212yes—
564C4C31×#39412yes—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—84no—
570C2C31×#143811yescharacteristic
571C2C31×#179211yescharacteristic
572C2C31×#179311yescharacteristic
573C2C31×#200611yescharacteristic
574C2C31×#200711yescharacteristic
575C2C31×#203911yescharacteristic
576C2C31×#204011yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#104C2semidirect
3. C2×C31×#417C2semidirect
4. C2×C31×#343C2semidirect
5. C31×C4⋊#95C2semidirect
10. C31×#537C4semidirect
52. C2×M16×C31D4semidirect
68. C2×C4⋊C4×C31D4semidirect
73. C2×C42×C31D4semidirect
166. #11002C31direct (N × H)
240. C22×#104C2×C31semidirect
241. C2×#417C2×C31semidirect
242. C2×#343C2×C31semidirect
243. C4⋊#95C2×C31semidirect
286. #537C4×C31semidirect
340. C2×M16D4×C31semidirect
356. C2×C4⋊C4D4×C31semidirect
361. C2×C42D4×C31semidirect
366. C31#11002direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C22×C31×#1042
C2×C31×#4172
C2×C31×#3432
C31×C4⋊#952
#1100231