Finite Group Databasebeta

← C31×#18357 #21475116

Subgroups — C31×#18357 #21475116

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3491C211yesmaximal, characteristic
2C31×#3218C212yesmaximal, complemented
3C31×#2628C211yesmaximal, characteristic, complemented
4C31×#1603C214yesmaximal, complemented
5C31×#1602C212yesmaximal, complemented
6C31×#1601C212yesmaximal
7C31×#1512C212yesmaximal, complemented
8C31×#1464C211yesmaximal, characteristic, complemented
9C31×#552—22no—
10C31×#552—22no—
11C31×#551—22no—
12C31×#551—22no—
13C31×#543C2211yescharacteristic
14C31×#539—21no—
15C31×#539C2212yes—
16C31×#534C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic, complemented
18C31×#529—21no—
19C31×#529C2212yes—
20C2×C31×#123—21no—
21C31×#518—21no—
22C2×C31×#117C2211yescharacteristic
23C31×#497—22no—
24C31×#491—22no—
25C31×#462—22no—
26C31×#461—22no—
27C31×#451C2214yescomplemented
28C31×#448C2214yes—
29C31×#432C2212yescomplemented
30C31×#400—21no—
31C31×#399—21no—
32C31×#395C2212yescomplemented
33C31×#393C2212yes—
34C31×#389—21no—
35C31×#382C2211yescharacteristic, complemented
36C31×#380—21no—
37C31×#328C2214yescomplemented
38C31×#327C2214yes—
39C31×#324C2212yescomplemented
40C31×#323C2212yes—
41C22×Q8×C31C2312yes—
42C22×D4×C31—21no—
43C23×C4×C31D411yescharacteristic
44C2×Q16×C31—24no—
45C2×SD16×C31—24no—
46C31×C4⋊Q8—22no—
47C31×C4⋊Q8—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#125—21no—
53C31×#125—22no—
54C31×#125—22no—
55C31×#125D412yes—
56C31×#124—21no—
57C31×#124D412yes—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8C2312yes—
71C31×#121—21no—
72C31×#121C2312yes—
73C31×#120—22no—
74C4×Q8×C31—21no—
75C4×Q8×C31D412yes—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C4×D4×C31—21no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31D412yes—
82C31×#117—22no—
83C31×#117—22no—
84C31×#117—22no—
85C31×#117—22no—
86C31×#117—22no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—22no—
93C2×C4⋊C4×C31D411yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—22no—
100C2×C22⋊C4×C31D411yescharacteristic
101C2×C22⋊C4×C31C2311yescharacteristic
102C2×C42×C31D411yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C31×C4⋊C8—22no—
105C31×#103—24no—
106C31×#103—24no—
107C31×C22⋊C8C2314yes—
108C31×#97—22no—
109C31×#95—22no—
110C31×#95—21no—
111C31×#95—22no—
112C31×#95C2312yes—
113C24×C31—21no—
114C24×C31C2×D411yescharacteristic
115C2×Q8×C31—21no—
116C2×Q8×C31—24no—
117C2×Q8×C31—82no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—21no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31C2×D412yes—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—41no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2411yescharacteristic
142Q16×C31—84no—
143SD16×C31—84no—
144C2×C8×C31—24no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31—41no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31—82no—
162C4⋊C4×C31C2×D412yes—
163C4⋊C4×C31C2×D412yes—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—82no—
177C42×C31—22no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—22no—
181C42×C31C2×D412yes—
182C42×C31C2×D412yes—
183#18357C3111yesmaximal, characteristic, direct product factor, Sylow
184C23×C31—21no—
185C23×C31—21no—
186C23×C31—22no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—81no—
191C23×C31—81no—
192C23×C31#13712yes—
193C23×C31C22×D411yescharacteristic
194Q8×C31—42no—
195Q8×C31—44no—
196Q8×C31—44no—
197Q8×C31—81no—
198D4×C31—42no—
199D4×C31—44no—
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—81no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—82no—
227C2×C4×C31—81no—
228C2×C4×C31—82no—
229C2×C4×C31—81no—
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#12012yes—
233C2×C4×C31C22×D412yes—
234C8×C31—44no—
235#3491C2×C3111yescharacteristic
236#3218C2×C3112yescomplemented
237#2628C2×C3111yescharacteristic, complemented
238#1603C2×C3114yescomplemented
239#1602C2×C3112yescomplemented
240#1601C2×C3112yes—
241#1512C2×C3112yescomplemented
242#1464C2×C3111yescharacteristic, complemented
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—22no—
246C22×C31—42no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—81no—
253C22×C31—162no—
254C22×C31C2×#12011yescharacteristic
255C22×C31C2×#13711yescharacteristic
256C22×C31#57511yescharacteristic
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—41no—
262C4×C31—42no—
263C4×C31—41no—
264C4×C31—42no—
265C4×C31—41no—
266C4×C31—42no—
267C4×C31—82no—
268C4×C31—82no—
269#552—22no—
270#552—22no—
271#551—22no—
272#551—22no—
273#543C22×C3111yescharacteristic
274#539—21no—
275#539C22×C3112yes—
276#534C22×C3111yescharacteristic
277C2×C4⋊Q8C22×C3111yescharacteristic, complemented
278#529—21no—
279#529C22×C3112yes—
280C2×#123—21no—
281#518—21no—
282C2×#117C22×C3111yescharacteristic
283#497—22no—
284#491—22no—
285#462—22no—
286#461—22no—
287#451C22×C3114yescomplemented
288#448C22×C3114yes—
289#432C22×C3112yescomplemented
290#400—21no—
291#399—21no—
292#395C22×C3112yescomplemented
293#393C22×C3112yes—
294#389—21no—
295#382C22×C3111yescharacteristic, complemented
296#380—21no—
297#328C22×C3114yescomplemented
298#327C22×C3114yes—
299#324C22×C3112yescomplemented
300#323C22×C3112yes—
301C2×C31—21no—
302C2×C31—21no—
303C2×C31—42no—
304C2×C31—82no—
305C2×C31#300711yescharacteristic
306C2×C31#302211yescharacteristic
307C2×C31#302711yescharacteristic
308C22×Q8C23×C3112yes—
309C22×D4—21no—
310C23×C4D4×C3111yescharacteristic
311C2×Q16—24no—
312C2×SD16—24no—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315#126—22no—
316#126—22no—
317#126—22no—
318#126—22no—
319#125—21no—
320#125—22no—
321#125—22no—
322#125D4×C3112yes—
323#124—21no—
324#124D4×C3112yes—
325#123—22no—
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8C23×C3112yes—
338#121—21no—
339#121C23×C3112yes—
340#120—22no—
341C4×Q8—21no—
342C4×Q8D4×C3112yes—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—21no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4D4×C3112yes—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4D4×C3111yescharacteristic
361C2×C4⋊C4C23×C3111yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4D4×C3111yescharacteristic
368C2×C22⋊C4C23×C3111yescharacteristic
369C2×C42D4×C3111yescharacteristic
370C2×C42C23×C3111yescharacteristic
371C4⋊C8—22no—
372#103—24no—
373#103—24no—
374C22⋊C8C23×C3114yes—
375#97—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95C23×C3112yes—
380C31#1835711yescharacteristic, direct product factor, Sylow
381C24—21no—
382C24C2×D4×C3111yescharacteristic
383C2×Q8—21no—
384C2×Q8—24no—
385C2×Q8—82no—
386C2×Q8C2×D4×C3112yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4C2×D4×C3112yes—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3111yescharacteristic
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4C24×C3111yescharacteristic
410Q16—84no—
411SD16—84no—
412C2×C8—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C4⋊C4C2×D4×C3112yes—
431C4⋊C4C2×D4×C3112yes—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—82no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42C2×D4×C3112yes—
450C42C2×D4×C3112yes—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—81no—
458C23—81no—
459C23C31×#13712yes—
460C23C22×D4×C3111yescharacteristic
461Q8—42no—
462Q8—44no—
463Q8—44no—
464Q8—81no—
465D4—42no—
466D4—44no—
467D4—42no—
468D4—42no—
469D4—81no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4C31×#12011yescharacteristic
498C2×C4C31×#12011yescharacteristic
499C2×C4C31×#12012yes—
500C2×C4C22×D4×C3112yes—
501C8—44no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—81no—
512C22—162no—
513C22C2×C31×#12011yescharacteristic
514C22C2×C31×#13711yescharacteristic
515C22C31×#57511yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—82no—
527C4—82no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—82no—
532C2C31×#300711yescharacteristic
533C2C31×#302211yescharacteristic
534C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#534C22
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
2. C31×#3218C2semidirect
3. C31×#2628C2semidirect
4. C31×#1603C2semidirect
5. C31×#1602C2semidirect
7. C31×#1512C2semidirect
8. C31×#1464C2semidirect
17. C2×C31×C4⋊Q8C22semidirect
27. C31×#451C22semidirect
29. C31×#432C22semidirect
32. C31×#395C22semidirect
35. C31×#382C22semidirect
37. C31×#328C22semidirect
39. C31×#324C22semidirect
183. #18357C31direct (N × H)
236. #3218C2×C31semidirect
237. #2628C2×C31semidirect
238. #1603C2×C31semidirect
239. #1602C2×C31semidirect
241. #1512C2×C31semidirect
242. #1464C2×C31semidirect
277. C2×C4⋊Q8C22×C31semidirect
287. #451C22×C31semidirect
289. #432C22×C31semidirect
292. #395C22×C31semidirect
295. #382C22×C31semidirect
297. #328C22×C31semidirect
299. #324C22×C31semidirect
380. C31#18357direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#32182
C31×#26282
C31×#16032
C31×#16022
C31×#16012
C31×#15122
C31×#14642
#1835731