Finite Group Databasebeta

← C31×#18370 #21475129

Subgroups — C31×#18370 #21475129

Order 7936.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3506C211yesmaximal, characteristic, complemented
2C31×#3215C211yesmaximal, characteristic, complemented
3C31×#3213C211yesmaximal, characteristic, complemented
4C31×#2631C211yesmaximal, characteristic, complemented
5C31×#1606C214yesmaximal, complemented
6C31×#1600C214yesmaximal, complemented
7C31×#1515C211yesmaximal, characteristic, complemented
8C31×#1514C211yesmaximal, characteristic, complemented
9C31×#1466C211yesmaximal, characteristic, complemented
10C31×#559—22no—
11C31×#556—21no—
12C31×#555—21no—
13C31×#553—22no—
14C31×#550—21no—
15C31×#547—21no—
16C31×#538C2214yes—
17C31×#535—21no—
18C31×#533C2211yescharacteristic
19C31×#532C2211yescharacteristic
20C2×C31×#124C2211yescharacteristic
21C31×#525—22no—
22C31×#520—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C31×#497—21no—
25C31×#496—21no—
26C31×#489—21no—
27C31×#489—21no—
28C31×#464—22no—
29C31×#463—22no—
30C31×#451C2212yes—
31C31×#450C2212yes—
32C31×#448C2212yes—
33C31×#447C2212yes—
34C31×#432C2211yescharacteristic
35C31×#432C2211yescharacteristic
36C31×#397—21no—
37C31×#393C2212yes—
38C31×#392—21no—
39C31×#392C2212yes—
40C31×#386—22no—
41C31×#383C2211yescharacteristic
42C31×#328C2214yes—
43C31×#327C2214yes—
44C31×#323C2214yes—
45C2×#41×C31—22no—
46C2×#41×C31—21no—
47C22×Q8×C31C2311yescharacteristic
48C22×D4×C31—22no—
49C22×D4×C31D411yescharacteristic
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31D412yes—
52C2×Q16×C31—22no—
53C2×SD16×C31—22no—
54C2×SD16×C31—22no—
55C2×D8×C31—22no—
56C31×C4⋊Q8D411yescharacteristic, complemented
57C31×C4⋊Q8D411yescharacteristic, complemented
58C31×C4⋊Q8D411yescharacteristic, complemented
59C31×#127—22no—
60C31×#127D411yescharacteristic, complemented
61C31×#126—24no—
62C31×#125—21no—
63C31×#124—22no—
64C31×#124—24no—
65C31×#124—21no—
66C31×#124—21no—
67C31×#124—21no—
68C31×#124—21no—
69C31×#124—21no—
70C31×#123—24no—
71C31×#123—24no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8C2312yes—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—24no—
79C31×#121—21no—
80C31×#121C2312yes—
81C31×#120—24no—
82C4×Q8×C31—22no—
83C4×Q8×C31—21no—
84C4×Q8×C31D412yescomplemented
85C4×D4×C31—22no—
86C4×D4×C31—24no—
87C4×D4×C31—24no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31D412yescomplemented
94C31×#117—22no—
95C31×#117—22no—
96C31×#117—21no—
97C2×C4⋊C4×C31—22no—
98C2×C4⋊C4×C31D411yescharacteristic
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—22no—
102C2×C22⋊C4×C31—22no—
103C2×C22⋊C4×C31C2312yes—
104C2×C42×C31C2311yescharacteristic
105C31×C4⋊C8—21no—
106C31×C4⋊C8—21no—
107C31×#103—22no—
108C31×#103—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×C22⋊C8C2312yes—
112C31×C22⋊C8C2312yes—
113C31×#97—21no—
114C31×#97—21no—
115C31×#95—21no—
116C31×#95—22no—
117C31×#95—21no—
118C31×#95C2312yes—
119C24×C31—21no—
120C24×C31C2×D412yes—
121#41×C31—44no—
122#41×C31—84no—
123C2×Q8×C31—21no—
124C2×Q8×C31—21no—
125C2×Q8×C31—21no—
126C2×Q8×C31—21no—
127C2×Q8×C31—81no—
128C2×Q8×C31C2×D412yes—
129C2×D4×C31—22no—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—21no—
137C2×D4×C31—21no—
138C2×D4×C31—42no—
139C2×D4×C31—44no—
140C2×D4×C31—81no—
141C2×D4×C31—82no—
142C2×D4×C31—82no—
143C2×D4×C31C2×D412yes—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—21no—
150C22×C4×C31—82no—
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D411yescharacteristic
154C22×C4×C31C2411yescharacteristic
155Q16×C31—82no—
156SD16×C31—82no—
157SD16×C31—82no—
158D8×C31—82no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C22⋊C4×C31—24no—
173C22⋊C4×C31—24no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—41no—
181C22⋊C4×C31—41no—
182C22⋊C4×C31—41no—
183C22⋊C4×C31—41no—
184C22⋊C4×C31—82no—
185C22⋊C4×C31—82no—
186C42×C31—24no—
187C42×C31—22no—
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191C42×C31C2×D411yescharacteristic
192#18370C3111yesmaximal, characteristic, direct product factor, Sylow
193C23×C31—22no—
194C23×C31—22no—
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—41no—
200C23×C31—41no—
201C23×C31—42no—
202C23×C31—41no—
203C23×C31—42no—
204C23×C31C22×D411yescharacteristic
205Q8×C31—42no—
206Q8×C31—42no—
207Q8×C31—42no—
208Q8×C31—42no—
209Q8×C31—42no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—42no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—44no—
218D4×C31—84no—
219D4×C31—84no—
220D4×C31—88no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—44no—
229C2×C4×C31—44no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31—82no—
237C2×C4×C31—84no—
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C22×D411yescharacteristic
244C8×C31—42no—
245C8×C31—42no—
246#3506C2×C3111yescharacteristic, complemented
247#3215C2×C3111yescharacteristic, complemented
248#3213C2×C3111yescharacteristic, complemented
249#2631C2×C3111yescharacteristic, complemented
250#1606C2×C3114yescomplemented
251#1600C2×C3114yescomplemented
252#1515C2×C3111yescharacteristic, complemented
253#1514C2×C3111yescharacteristic, complemented
254#1466C2×C3111yescharacteristic, complemented
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—44no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31—42no—
264C22×C31—82no—
265C22×C31—82no—
266C22×C31—82no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31—81no—
270C22×C31C2×#12011yescharacteristic
271C22×C31#57512yes—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—21no—
278C4×C31—42no—
279C4×C31—42no—
280C4×C31—42no—
281C4×C31—42no—
282C4×C31—82no—
283C4×C31—81no—
284#559—22no—
285#556—21no—
286#555—21no—
287#553—22no—
288#550—21no—
289#547—21no—
290#538C22×C3114yes—
291#535—21no—
292#533C22×C3111yescharacteristic
293#532C22×C3111yescharacteristic
294C2×#124C22×C3111yescharacteristic
295#525—22no—
296#520—21no—
297C2×C4×D4C22×C3111yescharacteristic
298#497—21no—
299#496—21no—
300#489—21no—
301#489—21no—
302#464—22no—
303#463—22no—
304#451C22×C3112yes—
305#450C22×C3112yes—
306#448C22×C3112yes—
307#447C22×C3112yes—
308#432C22×C3111yescharacteristic
309#432C22×C3111yescharacteristic
310#397—21no—
311#393C22×C3112yes—
312#392—21no—
313#392C22×C3112yes—
314#386—22no—
315#383C22×C3111yescharacteristic
316#328C22×C3114yes—
317#327C22×C3114yes—
318#323C22×C3114yes—
319C2×C31—22no—
320C2×C31—82no—
321C2×C31—82no—
322C2×C31—161no—
323C2×C31#300511yescharacteristic
324C2×C31#301911yescharacteristic
325C2×C31#302711yescharacteristic
326C2×#41—22no—
327C2×#41—21no—
328C22×Q8C23×C3111yescharacteristic
329C22×D4—22no—
330C22×D4D4×C3111yescharacteristic
331C22×D4C23×C3111yescharacteristic
332C23×C4D4×C3112yes—
333C2×Q16—22no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×D8—22no—
337C4⋊Q8D4×C3111yescharacteristic, complemented
338C4⋊Q8D4×C3111yescharacteristic, complemented
339C4⋊Q8D4×C3111yescharacteristic, complemented
340#127—22no—
341#127D4×C3111yescharacteristic, complemented
342#126—24no—
343#125—21no—
344#124—22no—
345#124—24no—
346#124—21no—
347#124—21no—
348#124—21no—
349#124—21no—
350#124—21no—
351#123—24no—
352#123—24no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8C23×C3112yes—
356#121—22no—
357#121—22no—
358#121—22no—
359#121—24no—
360#121—21no—
361#121C23×C3112yes—
362#120—24no—
363C4×Q8—22no—
364C4×Q8—21no—
365C4×Q8D4×C3112yescomplemented
366C4×D4—22no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4D4×C3112yescomplemented
375#117—22no—
376#117—22no—
377#117—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4D4×C3111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4C23×C3112yes—
385C2×C42C23×C3111yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—21no—
388#103—22no—
389#103—22no—
390#102—22no—
391#102—22no—
392C22⋊C8C23×C3112yes—
393C22⋊C8C23×C3112yes—
394#97—21no—
395#97—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95C23×C3112yes—
400C31#1837011yescharacteristic, direct product factor, Sylow
401C24—21no—
402C24C2×D4×C3112yes—
403#41—44no—
404#41—84no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—81no—
410C2×Q8C2×D4×C3112yes—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C3112yes—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—82no—
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3112yes—
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C24×C3111yescharacteristic
437Q16—82no—
438SD16—82no—
439SD16—82no—
440D8—82no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4C2×D4×C3112yes—
453C4⋊C4C2×D4×C3112yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C42—24no—
469C42—22no—
470C42C2×D4×C3111yescharacteristic
471C42C2×D4×C3111yescharacteristic
472C42C2×D4×C3111yescharacteristic
473C42C2×D4×C3111yescharacteristic
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23C22×D4×C3111yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—44no—
499D4—84no—
500D4—84no—
501D4—88no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4C31×#12011yescharacteristic
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12011yescharacteristic
523C2×C4C22×D4×C3111yescharacteristic
524C2×C4C22×D4×C3111yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×C31×#12011yescharacteristic
543C22C31×#57512yes—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—81no—
556C2—22no—
557C2—82no—
558C2—82no—
559C2—161no—
560C2C31×#300511yescharacteristic
561C2C31×#301911yescharacteristic
562C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
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×#3506C2semidirect
2. C31×#3215C2semidirect
3. C31×#3213C2semidirect
4. C31×#2631C2semidirect
5. C31×#1606C2semidirect
6. C31×#1600C2semidirect
7. C31×#1515C2semidirect
8. C31×#1514C2semidirect
9. C31×#1466C2semidirect
56. C31×C4⋊Q8D4semidirect
57. C31×C4⋊Q8D4semidirect
58. C31×C4⋊Q8D4semidirect
60. C31×#127D4semidirect
84. C4×Q8×C31D4semidirect
93. C4×D4×C31D4semidirect
192. #18370C31direct (N × H)
246. #3506C2×C31semidirect
247. #3215C2×C31semidirect
248. #3213C2×C31semidirect
249. #2631C2×C31semidirect
250. #1606C2×C31semidirect
251. #1600C2×C31semidirect
252. #1515C2×C31semidirect
253. #1514C2×C31semidirect
254. #1466C2×C31semidirect
337. C4⋊Q8D4×C31semidirect
338. C4⋊Q8D4×C31semidirect
339. C4⋊Q8D4×C31semidirect
341. #127D4×C31semidirect
365. C4×Q8D4×C31semidirect
374. C4×D4D4×C31semidirect
400. C31#18370direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35062
C31×#32152
C31×#32132
C31×#26312
C31×#16062
C31×#16002
C31×#15152
C31×#15142
C31×#14662
#1837031