Finite Group Databasebeta

← C31×#18432 #21475191

Subgroups — C31×#18432 #21475191

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic, complemented
2C31×#3217C211yesmaximal, characteristic
3C31×#3208C211yesmaximal, characteristic, complemented
4C31×#2641C211yesmaximal, characteristic, complemented
5C31×#1599C214yesmaximal, complemented
6C31×#1597C214yesmaximal, complemented
7C31×#1506C211yesmaximal, characteristic, complemented
8C31×#1500C211yesmaximal, characteristic, complemented
9C31×#1470C211yesmaximal, characteristic, complemented
10C31×#564—21no—
11C31×#563—21no—
12C31×#557—21no—
13C31×#554—21no—
14C31×#552—21no—
15C31×#552—21no—
16C31×#552—22no—
17C31×#544—21no—
18C31×#544C2212yes—
19C31×#541—21no—
20C31×#541—22no—
21C31×#539C2212yes—
22C31×#533C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C2×C31×#125C2211yescharacteristic
26C31×#496—21no—
27C31×#492—21no—
28C31×#490—21no—
29C31×#486—21no—
30C31×#460—22no—
31C31×#460—21no—
32C31×#459—21no—
33C31×#450C2214yes—
34C31×#450C2212yes—
35C31×#447C2212yes—
36C31×#432C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#394—22no—
39C31×#394C2214yes—
40C31×#390—21no—
41C31×#390—21no—
42C31×#376C2211yescharacteristic
43C31×#328C2214yes—
44C31×#328C2212yes—
45C31×#327C2212yes—
46C31×#324C2214yes—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31C2312yes—
50C2×SD16×C31—24no—
51C2×SD16×C31—22no—
52C2×D8×C31—22no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8—22no—
56C31×C4⋊Q8D411yescharacteristic
57C31×C4⋊Q8D411yescharacteristic
58C31×#127—21no—
59C31×#127—21no—
60C31×#127—21no—
61C31×#127—21no—
62C31×#126—22no—
63C31×#126—22no—
64C31×#126—22no—
65C31×#126—22no—
66C31×#126—21no—
67C31×#125—22no—
68C31×#125—22no—
69C31×#125—22no—
70C31×#125—22no—
71C31×#125—21no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—21no—
77C31×#125—21no—
78C31×#125—21no—
79C31×#125—21no—
80C31×#125—21no—
81C31×#125D411yescharacteristic
82C31×#124D411yescharacteristic
83C31×#123—22no—
84C31×#123—21no—
85C31×#123—22no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8C2311yescharacteristic
94C31×C22⋊Q8C2312yes—
95C31×#121C2311yescharacteristic
96C4×Q8×C31—21no—
97C4×Q8×C31—22no—
98C4×Q8×C31—21no—
99C4×Q8×C31—21no—
100C4×Q8×C31D411yescharacteristic
101C4×Q8×C31D412yes—
102C4×D4×C31—21no—
103C4×D4×C31—22no—
104C4×D4×C31D411yescharacteristic
105C31×#117—22no—
106C31×#117—21no—
107C31×#117—21no—
108C31×#117—22no—
109C31×#117—21no—
110C2×C4⋊C4×C31—21no—
111C2×C4⋊C4×C31D411yescharacteristic
112C2×C4⋊C4×C31D412yes—
113C2×C4⋊C4×C31D411yescharacteristic
114C2×C4⋊C4×C31C2312yes—
115C2×C22⋊C4×C31—24no—
116C2×C22⋊C4×C31—24no—
117C2×C42×C31—21no—
118C2×C42×C31C2311yescharacteristic
119C31×C4⋊C8—21no—
120C31×C4⋊C8—21no—
121C31×#102—22no—
122C31×#102—22no—
123C31×#102—24no—
124C31×C22⋊C8C2312yes—
125C31×C22⋊C8C2312yes—
126C31×#97—21no—
127C4×C8×C31—21no—
128C31×#95—21no—
129C31×#95C2312yes—
130C24×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—21no—
133C2×Q8×C31C2×D412yes—
134C2×Q8×C31C2×D411yescharacteristic
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—22no—
138C2×D4×C31—44no—
139C2×D4×C31—41no—
140C2×D4×C31—41no—
141C2×D4×C31—41no—
142C2×D4×C31—41no—
143C2×D4×C31—82no—
144C2×D4×C31—82no—
145C2×D4×C31C2×D411yescharacteristic
146C22×C4×C31—22no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2411yescharacteristic
154SD16×C31—82no—
155SD16×C31—84no—
156D8×C31—82no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—21no—
183C4⋊C4×C31—22no—
184C4⋊C4×C31C2×D411yescharacteristic
185C4⋊C4×C31C2×D411yescharacteristic
186C4⋊C4×C31C2×D412yes—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—21no—
192C22⋊C4×C31—21no—
193C22⋊C4×C31—21no—
194C22⋊C4×C31—21no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—84no—
197C22⋊C4×C31—84no—
198C42×C31—22no—
199C42×C31—22no—
200C42×C31—22no—
201C42×C31—22no—
202C42×C31—21no—
203C42×C31—42no—
204C42×C31C2×D411yescharacteristic
205C42×C31C2×D411yescharacteristic
206C42×C31C2×D411yescharacteristic
207C42×C31C2×D411yescharacteristic
208#18432C3111yesmaximal, characteristic, direct product factor, Sylow
209C23×C31—21no—
210C23×C31—44no—
211C23×C31—44no—
212C23×C31—42no—
213C23×C31—44no—
214C23×C31C22×D411yescharacteristic
215Q8×C31—41no—
216Q8×C31—42no—
217Q8×C31—42no—
218Q8×C31—82no—
219D4×C31—41no—
220D4×C31—42no—
221D4×C31—42no—
222D4×C31—44no—
223D4×C31—42no—
224D4×C31—82no—
225D4×C31—82no—
226D4×C31—82no—
227D4×C31—82no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—21no—
232C2×C4×C31—21no—
233C2×C4×C31—22no—
234C2×C4×C31—21no—
235C2×C4×C31—22no—
236C2×C4×C31—22no—
237C2×C4×C31—24no—
238C2×C4×C31—41no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—82no—
245C2×C4×C31#12011yescharacteristic
246C2×C4×C31#12011yescharacteristic
247C2×C4×C31#12011yescharacteristic
248C2×C4×C31#12011yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C2×C4×C31C22×D411yescharacteristic
251C8×C31—42no—
252C8×C31—42no—
253#3563C2×C3111yescharacteristic, complemented
254#3217C2×C3111yescharacteristic
255#3208C2×C3111yescharacteristic, complemented
256#2641C2×C3111yescharacteristic, complemented
257#1599C2×C3114yescomplemented
258#1597C2×C3114yescomplemented
259#1506C2×C3111yescharacteristic, complemented
260#1500C2×C3111yescharacteristic, complemented
261#1470C2×C3111yescharacteristic, complemented
262C22×C31—21no—
263C22×C31—21no—
264C22×C31—41no—
265C22×C31—41no—
266C22×C31—41no—
267C22×C31—88no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31C2×#12011yescharacteristic
272C22×C31#57712yes—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—42no—
279C4×C31—41no—
280C4×C31—41no—
281C4×C31—42no—
282C4×C31—42no—
283C4×C31—42no—
284C4×C31—41no—
285C4×C31—82no—
286C4×C31#45412yes—
287#564—21no—
288#563—21no—
289#557—21no—
290#554—21no—
291#552—21no—
292#552—21no—
293#552—22no—
294#544—21no—
295#544C22×C3112yes—
296#541—21no—
297#541—22no—
298#539C22×C3112yes—
299#533C22×C3111yescharacteristic
300C2×#127C22×C3111yescharacteristic
301#529C22×C3111yescharacteristic
302C2×#125C22×C3111yescharacteristic
303#496—21no—
304#492—21no—
305#490—21no—
306#486—21no—
307#460—22no—
308#460—21no—
309#459—21no—
310#450C22×C3114yes—
311#450C22×C3112yes—
312#447C22×C3112yes—
313#432C22×C3111yescharacteristic
314#431C22×C3111yescharacteristic
315#394—22no—
316#394C22×C3114yes—
317#390—21no—
318#390—21no—
319#376C22×C3111yescharacteristic
320#328C22×C3114yes—
321#328C22×C3112yes—
322#327C22×C3112yes—
323#324C22×C3114yes—
324C2×C31—22no—
325C2×C31—81no—
326C2×C31—162no—
327C2×C31#301011yescharacteristic
328C2×C31C2×#45411yescharacteristic
329C2×C31#302711yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4C23×C3112yes—
333C2×SD16—24no—
334C2×SD16—22no—
335C2×D8—22no—
336C4⋊Q8—21no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8D4×C3111yescharacteristic
340C4⋊Q8D4×C3111yescharacteristic
341#127—21no—
342#127—21no—
343#127—21no—
344#127—21no—
345#126—22no—
346#126—22no—
347#126—22no—
348#126—22no—
349#126—21no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125—21no—
355#125—22no—
356#125—22no—
357#125—22no—
358#125—22no—
359#125—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125D4×C3111yescharacteristic
365#124D4×C3111yescharacteristic
366#123—22no—
367#123—21no—
368#123—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8C23×C3111yescharacteristic
377C22⋊Q8C23×C3112yes—
378#121C23×C3111yescharacteristic
379C4×Q8—21no—
380C4×Q8—22no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8D4×C3111yescharacteristic
384C4×Q8D4×C3112yes—
385C4×D4—21no—
386C4×D4—22no—
387C4×D4D4×C3111yescharacteristic
388#117—22no—
389#117—21no—
390#117—21no—
391#117—22no—
392#117—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C3111yescharacteristic
395C2×C4⋊C4D4×C3112yes—
396C2×C4⋊C4D4×C3111yescharacteristic
397C2×C4⋊C4C23×C3112yes—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C42—21no—
401C2×C42C23×C3111yescharacteristic
402C4⋊C8—21no—
403C4⋊C8—21no—
404#102—22no—
405#102—22no—
406#102—24no—
407C22⋊C8C23×C3112yes—
408C22⋊C8C23×C3112yes—
409#97—21no—
410C4×C8—21no—
411#95—21no—
412#95C23×C3112yes—
413C31#1843211yescharacteristic, direct product factor, Sylow
414C24—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8C2×D4×C3112yes—
418C2×Q8C2×D4×C3111yescharacteristic
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4C2×D4×C3111yescharacteristic
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3112yes—
437C22×C4C24×C3111yescharacteristic
438SD16—82no—
439SD16—84no—
440D8—82no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4C2×D4×C3111yescharacteristic
469C4⋊C4C2×D4×C3111yescharacteristic
470C4⋊C4C2×D4×C3112yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—42no—
488C42C2×D4×C3111yescharacteristic
489C42C2×D4×C3111yescharacteristic
490C42C2×D4×C3111yescharacteristic
491C42C2×D4×C3111yescharacteristic
492C23—21no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23C22×D4×C3111yescharacteristic
498Q8—41no—
499Q8—42no—
500Q8—42no—
501Q8—82no—
502D4—41no—
503D4—42no—
504D4—42no—
505D4—44no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4C31×#12011yescharacteristic
529C2×C4C31×#12011yescharacteristic
530C2×C4C31×#12011yescharacteristic
531C2×C4C31×#12011yescharacteristic
532C2×C4C22×D4×C3111yescharacteristic
533C2×C4C22×D4×C3111yescharacteristic
534C8—42no—
535C8—42no—
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—88no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×C31×#12011yescharacteristic
546C22C31×#57712yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—82no—
560C4C31×#45412yes—
561C2—22no—
562C2—81no—
563C2—162no—
564C2C31×#301011yescharacteristic
565C2C2×C31×#45411yescharacteristic
566C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3217C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3563C2semidirect
3. C31×#3208C2semidirect
4. C31×#2641C2semidirect
5. C31×#1599C2semidirect
6. C31×#1597C2semidirect
7. C31×#1506C2semidirect
8. C31×#1500C2semidirect
9. C31×#1470C2semidirect
208. #18432C31direct (N × H)
253. #3563C2×C31semidirect
255. #3208C2×C31semidirect
256. #2641C2×C31semidirect
257. #1599C2×C31semidirect
258. #1597C2×C31semidirect
259. #1506C2×C31semidirect
260. #1500C2×C31semidirect
261. #1470C2×C31semidirect
413. C31#18432direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#32172
C31×#32082
C31×#26412
C31×#15992
C31×#15972
C31×#15062
C31×#15002
C31×#14702
#1843231