Finite Group Databasebeta

← C31×#18641 #21475400

Subgroups — C31×#18641 #21475400

Order 7936.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3499C211yesmaximal, characteristic
2C31×#3227C211yesmaximal, characteristic, complemented
3C2×C31×#487C211yesmaximal, characteristic, complemented
4C31×#3079C211yesmaximal, characteristic, complemented
5C31×#1637C214yesmaximal, complemented
6C31×#1635C214yesmaximal, complemented
7C31×#1546C211yesmaximal, characteristic, complemented
8C31×#1492C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×2−1+4—21no—
11D4×Q8×C31—21no—
12D4×Q8×C31C2212yes—
13C31×#547—22no—
14C31×#547—21no—
15C31×#547—22no—
16C31×#547C2212yes—
17C31×#536—22no—
18C31×#536—21no—
19C31×#532—21no—
20C31×#532C2211yescharacteristic
21C2×C31×C4⋊Q8—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C4×#41×C31C2211yescharacteristic
25C31×#498—21no—
26C31×#487—22no—
27C31×#487C2214yes—
28C31×#486—21no—
29C31×#481—22no—
30C31×#477C2212yes—
31C31×#476C2212yes—
32C31×#476C2214yes—
33C31×#475—21no—
34C31×#474—21no—
35C31×#431C2211yescharacteristic
36C31×#428—21no—
37C31×#423C2211yescharacteristic
38C31×#417—21no—
39C2×C31×#103—22no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#332C2214yes—
42C31×Q8⋊C8C2212yes—
43C31×D4⋊C8C2212yes—
44C31×D4⋊C8C2214yes—
45C31×2−1+4—44no—
46C31×2−1+4—44no—
47C2×#41×C31—22no—
48C2×#41×C31—24no—
49C2×#41×C31—22no—
50C2×#41×C31—21no—
51C2×#41×C31D411yescharacteristic
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31—21no—
55C22×Q8×C31—21no—
56C2×M16×C31—21no—
57C22×C8×C31—21no—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8D412yes—
67C31×C4⋊Q8D411yescharacteristic
68C31×C4⋊Q8C2312yes—
69C31×C4⋊Q8C2312yes—
70C31×#127D411yescharacteristic
71C31×#124—22no—
72C31×#124—21no—
73C31×#123—24no—
74C31×#123—24no—
75C31×#123—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—22no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121D412yes—
84C4×Q8×C31D411yescharacteristic
85C4×Q8×C31C2311yescharacteristic
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31D411yescharacteristic
92C4×D4×C31C2311yescharacteristic
93C4×D4×C31C2312yes—
94C31×#117—21no—
95C31×#117D411yescharacteristic
96C2×C4⋊C4×C31—22no—
97C2×C4⋊C4×C31—21no—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31D412yes—
102C2×C42×C31—21no—
103C2×C42×C31C2311yescharacteristic
104C31×#107—22no—
105C31×#106—22no—
106C31×#106—24no—
107C31×C4⋊C8—21no—
108C31×C4⋊C8C2312yes—
109C31×C4⋊C8C2312yes—
110C31×#103—22no—
111C31×#103—28no—
112C31×#102—22no—
113C31×#102—24no—
114C31×C22⋊C8—21no—
115C4×C8×C31—21no—
116C4×C8×C31—21no—
117C4×C8×C31C2312yes—
118#41×C31—22no—
119#41×C31—48no—
120#41×C31—416no—
121#41×C31—44no—
122#41×C31—44no—
123#41×C31—42no—
124#41×C31—42no—
125#41×C31SD1614yes—
126C2×Q8×C31—24no—
127C2×Q8×C31—22no—
128C2×Q8×C31—21no—
129C2×Q8×C31—22no—
130C2×Q8×C31—22no—
131C2×Q8×C31—48no—
132C2×Q8×C31—44no—
133C2×Q8×C31—44no—
134C2×Q8×C31—44no—
135C2×Q8×C31—41no—
136C2×Q8×C31—42no—
137C2×Q8×C31—42no—
138C2×Q8×C31C2×D411yescharacteristic
139C2×D4×C31—21no—
140C2×D4×C31—24no—
141C2×D4×C31—22no—
142C2×D4×C31—21no—
143C2×D4×C31—21no—
144C2×D4×C31C2×D412yes—
145C2×D4×C31C2×D411yescharacteristic
146C22×C4×C31—21no—
147C22×C4×C31—24no—
148C22×C4×C31—24no—
149C22×C4×C31—22no—
150C22×C4×C31—21no—
151C22×C4×C31—21no—
152C22×C4×C31—22no—
153C22×C4×C31—41no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157M16×C31—42no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—22no—
161C2×C8×C31—42no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—41no—
173C4⋊C4×C31—41no—
174C4⋊C4×C31C2×D414yes—
175C4⋊C4×C31C2×D412yes—
176C4⋊C4×C31C2×D411yescharacteristic
177C4⋊C4×C31C2×D411yescharacteristic
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C42×C31—21no—
186C42×C31—21no—
187C42×C31—21no—
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191C42×C31C2411yescharacteristic
192#18641C3111yesmaximal, characteristic, direct product factor, Sylow
193C23×C31—21no—
194C23×C31—22no—
195C23×C31#12011yescharacteristic
196Q8×C31—21no—
197Q8×C31—48no—
198Q8×C31—44no—
199Q8×C31—42no—
200Q8×C31—42no—
201Q8×C31—42no—
202Q8×C31—44no—
203Q8×C31—82no—
204Q8×C31C2×SD1612yes—
205D4×C31—24no—
206D4×C31—21no—
207D4×C31—24no—
208D4×C31—48no—
209D4×C31—44no—
210D4×C31—42no—
211D4×C31—42no—
212D4×C31C2×SD1612yes—
213C2×C4×C31—21no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—24no—
220C2×C4×C31—22no—
221C2×C4×C31—48no—
222C2×C4×C31—48no—
223C2×C4×C31—44no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—82no—
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31C2×SD1612yes—
234C2×C4×C31AGL(1,ℤ/8)12yes—
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C8×C31—42no—
239C8×C31—42no—
240C8×C31—42no—
241#3499C2×C3111yescharacteristic
242#3227C2×C3111yescharacteristic, complemented
243C2×#487C2×C3111yescharacteristic, complemented
244#3079C2×C3111yescharacteristic, complemented
245#1637C2×C3114yescomplemented
246#1635C2×C3114yescomplemented
247#1546C2×C3111yescharacteristic, complemented
248#1492C2×C3111yescharacteristic, complemented
249#1476C2×C3111yescharacteristic, complemented
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—22no—
253C22×C31—41no—
254C22×C31—41no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31#45012yes—
259C22×C31C2×#12011yescharacteristic
260C4×C31—22no—
261C4×C31—21no—
262C4×C31—21no—
263C4×C31—44no—
264C4×C31—42no—
265C4×C31—41no—
266C4×C31—41no—
267C4×C31—82no—
268C4×C31#45012yes—
269C4×C31#45412yes—
270C4×C31C22×SD1611yescharacteristic
271C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C31#57812yes—
273C2×2−1+4—21no—
274D4×Q8—21no—
275D4×Q8C22×C3112yes—
276#547—22no—
277#547—21no—
278#547—22no—
279#547C22×C3112yes—
280#536—22no—
281#536—21no—
282#532—21no—
283#532C22×C3111yescharacteristic
284C2×C4⋊Q8—21no—
285C2×C4⋊Q8C22×C3111yescharacteristic
286C2×C4⋊Q8C22×C3111yescharacteristic
287C4×#41C22×C3111yescharacteristic
288#498—21no—
289#487—22no—
290#487C22×C3114yes—
291#486—21no—
292#481—22no—
293#477C22×C3112yes—
294#476C22×C3112yes—
295#476C22×C3114yes—
296#475—21no—
297#474—21no—
298#431C22×C3111yescharacteristic
299#428—21no—
300#423C22×C3111yescharacteristic
301#417—21no—
302C2×#103—22no—
303C2×C4×C8C22×C3111yescharacteristic
304#332C22×C3114yes—
305Q8⋊C8C22×C3112yes—
306D4⋊C8C22×C3112yes—
307D4⋊C8C22×C3114yes—
308C2×C31—22no—
309C2×C31—42no—
310C2×C31—82no—
311C2×C31C2×#45011yescharacteristic
312C2×C31#301111yescharacteristic
313C2×C31C2×#45411yescharacteristic
3142−1+4—44no—
3152−1+4—44no—
316C2×#41—22no—
317C2×#41—24no—
318C2×#41—22no—
319C2×#41—21no—
320C2×#41D4×C3111yescharacteristic
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—21no—
324C22×Q8—21no—
325C2×M16—21no—
326C22×C8—21no—
327C4⋊Q8—22no—
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8D4×C3111yescharacteristic
336C4⋊Q8D4×C3112yes—
337C4⋊Q8C23×C3112yes—
338C4⋊Q8C23×C3112yes—
339#127D4×C3111yescharacteristic
340#124—22no—
341#124—21no—
342#123—24no—
343#123—24no—
344#123—24no—
345C22⋊Q8—24no—
346C22⋊Q8—22no—
347C22⋊Q8—24no—
348C22⋊Q8—22no—
349#121—22no—
350#121—22no—
351#121—21no—
352#121D4×C3112yes—
353C4×Q8D4×C3111yescharacteristic
354C4×Q8C23×C3111yescharacteristic
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4D4×C3111yescharacteristic
361C4×D4C23×C3111yescharacteristic
362C4×D4C23×C3112yes—
363#117—21no—
364#117D4×C3111yescharacteristic
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4D4×C3112yes—
371C2×C42—21no—
372C2×C42C23×C3111yescharacteristic
373#107—22no—
374#106—22no—
375#106—24no—
376C4⋊C8—21no—
377C4⋊C8C23×C3112yes—
378C4⋊C8C23×C3112yes—
379#103—22no—
380#103—28no—
381#102—22no—
382#102—24no—
383C22⋊C8—21no—
384C4×C8—21no—
385C4×C8—21no—
386C4×C8C23×C3112yes—
387C31#1864111yescharacteristic, direct product factor, Sylow
388#41—22no—
389#41—48no—
390#41—416no—
391#41—44no—
392#41—44no—
393#41—42no—
394#41—42no—
395#41SD16×C3114yes—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—41no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×D4×C3111yescharacteristic
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×D4×C3112yes—
415C2×D4C2×D4×C3111yescharacteristic
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3111yescharacteristic
426C22×C4C2×D4×C3111yescharacteristic
427M16—42no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—42no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×D4×C3114yes—
445C4⋊C4C2×D4×C3112yes—
446C4⋊C4C2×D4×C3111yescharacteristic
447C4⋊C4C2×D4×C3111yescharacteristic
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42C2×D4×C3111yescharacteristic
459C42C2×D4×C3111yescharacteristic
460C42C2×D4×C3111yescharacteristic
461C42C24×C3111yescharacteristic
462C23—21no—
463C23—22no—
464C23C31×#12011yescharacteristic
465Q8—21no—
466Q8—48no—
467Q8—44no—
468Q8—42no—
469Q8—44no—
470Q8—42no—
471Q8—42no—
472Q8—82no—
473Q8C2×SD16×C3112yes—
474D4—24no—
475D4—21no—
476D4—24no—
477D4—48no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4C2×SD16×C3112yes—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#12011yescharacteristic
502C2×C4C2×SD16×C3112yes—
503C2×C4C31×AGL(1,ℤ/8)12yes—
504C2×C4C22×D4×C3111yescharacteristic
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C22×D4×C3111yescharacteristic
507C8—42no—
508C8—42no—
509C8—42no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22C31×#45012yes—
519C22C2×C31×#12011yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—44no—
524C4—42no—
525C4—41no—
526C4—41no—
527C4—82no—
528C4C31×#45012yes—
529C4C31×#45412yes—
530C4C22×SD16×C3111yescharacteristic
531C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
532C4C31×#57812yes—
533C2—22no—
534C2—42no—
535C2—82no—
536C2C2×C31×#45011yescharacteristic
537C2C31×#301111yescharacteristic
538C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3499C2
LCSG ▷ C42 ▷ 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×#3227C2semidirect
3. C2×C31×#487C2semidirect
4. C31×#3079C2semidirect
5. C31×#1637C2semidirect
6. C31×#1635C2semidirect
7. C31×#1546C2semidirect
8. C31×#1492C2semidirect
9. C31×#1476C2semidirect
192. #18641C31direct (N × H)
242. #3227C2×C31semidirect
243. C2×#487C2×C31semidirect
244. #3079C2×C31semidirect
245. #1637C2×C31semidirect
246. #1635C2×C31semidirect
247. #1546C2×C31semidirect
248. #1492C2×C31semidirect
249. #1476C2×C31semidirect
387. C31#18641direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34992
C31×#32272
C2×C31×#4872
C31×#30792
C31×#16372
C31×#16352
C31×#15462
C31×#14922
C31×#14762
#1864131