Finite Group Databasebeta

← C31×#33570 #21490329

Subgroups — C31×#33570 #21490329

Order 7936.

1212 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3549C211yesmaximal, characteristic, complemented
2C31×#3021C212yesmaximal, complemented
3C31×#3018C211yesmaximal, characteristic, complemented
4C31×#2204C214yesmaximal, complemented
5C31×#2202C214yesmaximal, complemented
6C31×#2134C212yesmaximal, complemented
7C31×#2132C211yesmaximal, characteristic, complemented
8C31×#585C2211yescharacteristic, complemented
9C31×#575—21no—
10C2×C31×AGL(1,ℤ/8)—22no—
11C31×H(3,𝔽4)—21no—
12C31×H(3,𝔽4)C2212yescomplemented
13C31×#555—22no—
14C31×#555C2212yescomplemented
15D42×C31—22no—
16D42×C31—21no—
17C31×#535—21no—
18C31×#535—24no—
19C31×#535C2212yes—
20C31×#532—21no—
21C31×#532C2211yescharacteristic
22C31×#456C2212yescomplemented
23C31×#455—21no—
24C31×#454C2212yescomplemented
25C31×#454C2218yescomplemented
26C31×#453—21no—
27C31×#453—24no—
28C31×#421—22no—
29C2×C31×#104—21no—
30C31×#413C2212yescomplemented
31C31×#413C2211yescharacteristic, complemented
32C31×#356C2212yescomplemented
33C31×#355C2212yescomplemented
34C31×#355C2218yescomplemented
35C31×2−1+4—41no—
36C31×2−1+4C2312yes—
37C31×H(5,𝔽2)—22no—
38C31×H(5,𝔽2)—21no—
39C31×H(5,𝔽2)—41no—
40C2×#41×C31—22no—
41C2×#41×C31—21no—
42C2×#41×C31—22no—
43C2×#41×C31—42no—
44C2×#41×C31D411yescharacteristic
45C2×#41×C31C2311yescharacteristic
46C22×D4×C31—24no—
47C22×D4×C31—21no—
48C22×D4×C31D412yes—
49C31×#137—22no—
50C31×#137—41no—
51C31×AGL(1,ℤ/8)—28no—
52C31×AGL(1,ℤ/8)—22no—
53C31×AGL(1,ℤ/8)—41no—
54C31×AGL(1,ℤ/8)—44no—
55C31×#135—42no—
56C2×SD16×C31—42no—
57C2×D8×C31—42no—
58C2×M16×C31—21no—
59C2×M16×C31—22no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8C2311yescharacteristic
62C31×#127—21no—
63C31×#127—22no—
64C31×#127D411yescharacteristic
65C31×#127D414yes—
66C31×#124—24no—
67C31×#124—22no—
68C31×#124—22no—
69C31×#124—24no—
70C31×#124D411yescharacteristic
71C31×#124C2311yescharacteristic
72C31×#124C2314yes—
73C31×#121—22no—
74C31×#121—22no—
75C31×#121—24no—
76C31×#121—22no—
77C31×#121—24no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#120—22no—
81C31×#120—24no—
82C31×#120—22no—
83C31×#120—22no—
84C31×#120—24no—
85C31×#120—22no—
86C4×D4×C31—22no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C31×#117—21no—
90C31×#117—24no—
91C31×#117D412yes—
92C2×C42×C31D411yescharacteristic
93C31×#104—22no—
94C31×#104—22no—
95C31×#104—28no—
96C31×#101C2314yes—
97C31×#101C2312yes—
98C31×#100—22no—
99C31×#100—21no—
100C24×C31—22no—
101C24×C31—22no—
102#41×C31—22no—
103#41×C31—24no—
104#41×C31—24no—
105#41×C31—22no—
106#41×C31—41no—
107#41×C31—41no—
108#41×C31—42no—
109#41×C31—41no—
110#41×C31—41no—
111#41×C31—41no—
112#41×C31—42no—
113#41×C31—44no—
114#41×C31—42no—
115#41×C31—41no—
116#41×C31—84no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—21no—
120C2×Q8×C31—42no—
121C2×Q8×C31C2×D411yescharacteristic
122C2×Q8×C31C2411yescharacteristic
123C2×D4×C31—22no—
124C2×D4×C31—21no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—24no—
130C2×D4×C31—24no—
131C2×D4×C31—21no—
132C2×D4×C31—24no—
133C2×D4×C31—22no—
134C2×D4×C31—42no—
135C2×D4×C31—42no—
136C2×D4×C31—44no—
137C2×D4×C31—42no—
138C2×D4×C31—42no—
139C2×D4×C31—41no—
140C2×D4×C31—88no—
141C2×D4×C31—81no—
142C2×D4×C31—81no—
143C2×D4×C31C2×D411yescharacteristic
144C2×D4×C31C2×D414yes—
145C2×D4×C31C2×D411yescharacteristic
146C2×D4×C31C2×D412yes—
147C22×C4×C31—24no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—41no—
151C22×C4×C31—42no—
152C22×C4×C31C2×D411yescharacteristic
153C22×C4×C31C2×D412yes—
154Q16×C31—42no—
155SD16×C31—42no—
156SD16×C31—48no—
157SD16×C31—42no—
158D8×C31—48no—
159D8×C31—42no—
160M16×C31—24no—
161M16×C31—22no—
162M16×C31—42no—
163M16×C31—41no—
164C2×C8×C31—42no—
165C2×C8×C31—41no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C22⋊C4×C31—24no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—24no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—24no—
177C42×C31—22no—
178C42×C31—21no—
179C42×C31C2×D411yescharacteristic
180C42×C31C2×D411yescharacteristic
181C42×C31C2×D414yes—
182#33570C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—24no—
184C23×C31—21no—
185C23×C31—41no—
186C23×C31—42no—
187C23×C31—82no—
188C23×C31—84no—
189C23×C31—84no—
190C23×C31—82no—
191C23×C31#12012yes—
192C23×C31#12011yescharacteristic
193Q8×C31—24no—
194Q8×C31—22no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—41no—
198Q8×C31—82no—
199D4×C31—24no—
200D4×C31—22no—
201D4×C31—42no—
202D4×C31—41no—
203D4×C31—44no—
204D4×C31—41no—
205D4×C31—44no—
206D4×C31—44no—
207D4×C31—42no—
208D4×C31—44no—
209D4×C31—42no—
210D4×C31—41no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—81no—
214D4×C31—84no—
215D4×C31—84no—
216D4×C31—82no—
217D4×C31—84no—
218D4×C31—82no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—24no—
227C2×C4×C31—48no—
228C2×C4×C31—42no—
229C2×C4×C31—44no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—42no—
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C22×D412yes—
239C8×C31—44no—
240C8×C31—42no—
241#3549C2×C3111yescharacteristic, complemented
242#3021C2×C3112yescomplemented
243#3018C2×C3111yescharacteristic, complemented
244#2204C2×C3114yescomplemented
245#2202C2×C3114yescomplemented
246#2134C2×C3112yescomplemented
247#2132C2×C3111yescharacteristic, complemented
248C22×C31—21no—
249C22×C31—22no—
250C22×C31—41no—
251C22×C31—42no—
252C22×C31—44no—
253C22×C31—44no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—41no—
257C22×C31—44no—
258C22×C31—161no—
259C22×C31—162no—
260C22×C31—164no—
261C22×C31—164no—
262C22×C31C2×#12011yescharacteristic
263C4×C31—22no—
264C4×C31—22no—
265C4×C31—21no—
266C4×C31—44no—
267C4×C31—41no—
268C4×C31—41no—
269C4×C31—42no—
270C4×C31B(4,𝔽2)11yescharacteristic
271C4×C31B(4,𝔽2)11yescharacteristic
272#585C22×C3111yescharacteristic, complemented
273#575—21no—
274C2×AGL(1,ℤ/8)—22no—
275H(3,𝔽4)—21no—
276H(3,𝔽4)C22×C3112yescomplemented
277#555—22no—
278#555C22×C3112yescomplemented
279D42—22no—
280D42—21no—
281#535—21no—
282#535—24no—
283#535C22×C3112yes—
284#532—21no—
285#532C22×C3111yescharacteristic
286#456C22×C3112yescomplemented
287#455—21no—
288#454C22×C3112yescomplemented
289#454C22×C3118yescomplemented
290#453—21no—
291#453—24no—
292#421—22no—
293C2×#104—21no—
294#413C22×C3112yescomplemented
295#413C22×C3111yescharacteristic, complemented
296#356C22×C3112yescomplemented
297#355C22×C3112yescomplemented
298#355C22×C3118yescomplemented
299C2×C31—21no—
300C2×C31—41no—
301C2×C31—42no—
302C2×C31—84no—
303C2×C31—81no—
304C2×C31—82no—
305C2×C31C2×B(4,𝔽2)11yescharacteristic
3062−1+4—41no—
3072−1+4C23×C3112yes—
308H(5,𝔽2)—22no—
309H(5,𝔽2)—21no—
310H(5,𝔽2)—41no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—21no—
314C2×#41—42no—
315C2×#41D4×C3111yescharacteristic
316C2×#41C23×C3111yescharacteristic
317C22×D4—24no—
318C22×D4—21no—
319C22×D4D4×C3112yes—
320#137—22no—
321#137—41no—
322AGL(1,ℤ/8)—28no—
323AGL(1,ℤ/8)—22no—
324AGL(1,ℤ/8)—41no—
325AGL(1,ℤ/8)—44no—
326#135—42no—
327C2×SD16—42no—
328C2×D8—42no—
329C2×M16—22no—
330C2×M16—21no—
331C4⋊Q8—21no—
332C4⋊Q8C23×C3111yescharacteristic
333#127—21no—
334#127—22no—
335#127D4×C3111yescharacteristic
336#127D4×C3114yes—
337#124—24no—
338#124—22no—
339#124—22no—
340#124—24no—
341#124D4×C3111yescharacteristic
342#124C23×C3111yescharacteristic
343#124C23×C3114yes—
344#121—22no—
345#121—22no—
346#121—24no—
347#121—22no—
348#121—24no—
349#121—22no—
350#121—22no—
351#120—22no—
352#120—24no—
353#120—22no—
354#120—22no—
355#120—24no—
356#120—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360#117—21no—
361#117—24no—
362#117D4×C3112yes—
363C2×C42D4×C3111yescharacteristic
364#104—22no—
365#104—22no—
366#104—28no—
367#101C23×C3112yes—
368#101C23×C3114yes—
369#100—22no—
370#100—21no—
371C31#3357011yescharacteristic, direct product factor, Sylow
372C24—22no—
373C24—22no—
374#41—22no—
375#41—24no—
376#41—24no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—42no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—42no—
385#41—44no—
386#41—42no—
387#41—41no—
388#41—84no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8C2×D4×C3111yescharacteristic
394C2×Q8C24×C3111yescharacteristic
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—88no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4C2×D4×C3111yescharacteristic
416C2×D4C2×D4×C3114yes—
417C2×D4C2×D4×C3111yescharacteristic
418C2×D4C2×D4×C3112yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3112yes—
426Q16—42no—
427SD16—42no—
428SD16—48no—
429SD16—42no—
430D8—48no—
431D8—42no—
432M16—24no—
433M16—22no—
434M16—42no—
435M16—41no—
436C2×C8—42no—
437C2×C8—41no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C42—22no—
450C42—21no—
451C42C2×D4×C3111yescharacteristic
452C42C2×D4×C3111yescharacteristic
453C42C2×D4×C3114yes—
454C23—24no—
455C23—21no—
456C23—41no—
457C23—42no—
458C23—82no—
459C23—84no—
460C23—84no—
461C23—82no—
462C23C31×#12012yes—
463C23C31×#12011yescharacteristic
464Q8—24no—
465Q8—22no—
466Q8—41no—
467Q8—41no—
468Q8—41no—
469Q8—82no—
470D4—24no—
471D4—22no—
472D4—42no—
473D4—41no—
474D4—44no—
475D4—41no—
476D4—44no—
477D4—44no—
478D4—42no—
479D4—44no—
480D4—42no—
481D4—41no—
482D4—42no—
483D4—42no—
484D4—81no—
485D4—84no—
486D4—84no—
487D4—82no—
488D4—84no—
489D4—82no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4C31×#12011yescharacteristic
508C2×C4C22×D4×C3111yescharacteristic
509C2×C4C22×D4×C3112yes—
510C8—44no—
511C8—42no—
512C22—21no—
513C22—22no—
514C22—41no—
515C22—42no—
516C22—44no—
517C22—44no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—44no—
522C22—162no—
523C22—161no—
524C22—164no—
525C22—164no—
526C22C2×C31×#12011yescharacteristic
527C4—22no—
528C4—22no—
529C4—21no—
530C4—44no—
531C4—41no—
532C4—41no—
533C4—42no—
534C4C31×B(4,𝔽2)11yescharacteristic
535C4C31×B(4,𝔽2)11yescharacteristic
536C2—21no—
537C2—42no—
538C2—41no—
539C2—84no—
540C2—81no—
541C2—82no—
542C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C31C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C31
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3549C2semidirect
2. C31×#3021C2semidirect
3. C31×#3018C2semidirect
4. C31×#2204C2semidirect
5. C31×#2202C2semidirect
6. C31×#2134C2semidirect
7. C31×#2132C2semidirect
8. C31×#585C22semidirect
12. C31×H(3,𝔽4)C22semidirect
14. C31×#555C22semidirect
22. C31×#456C22semidirect
24. C31×#454C22semidirect
25. C31×#454C22semidirect
30. C31×#413C22semidirect
31. C31×#413C22semidirect
32. C31×#356C22semidirect
33. C31×#355C22semidirect
34. C31×#355C22semidirect
182. #33570C31direct (N × H)
241. #3549C2×C31semidirect
242. #3021C2×C31semidirect
243. #3018C2×C31semidirect
244. #2204C2×C31semidirect
245. #2202C2×C31semidirect
246. #2134C2×C31semidirect
247. #2132C2×C31semidirect
272. #585C22×C31semidirect
276. H(3,𝔽4)C22×C31semidirect
278. #555C22×C31semidirect
286. #456C22×C31semidirect
288. #454C22×C31semidirect
289. #454C22×C31semidirect
294. #413C22×C31semidirect
295. #413C22×C31semidirect
296. #356C22×C31semidirect
297. #355C22×C31semidirect
298. #355C22×C31semidirect
371. C31#33570direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35492
C31×#30212
C31×#30182
C31×#22042
C31×#22022
C31×#21342
C31×#21322
#3357031