Finite Group Databasebeta

← C31×#18996 #21475755

Subgroups — C31×#18996 #21475755

Order 7936.

948 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3529C211yesmaximal, characteristic
2C31×#3239C212yesmaximal, complemented
3C31×#3152C211yesmaximal, characteristic, complemented
4C31×#1661C214yesmaximal, complemented
5C31×#1660C212yesmaximal, complemented
6C31×#1659C212yesmaximal, complemented
7C31×#1536C211yesmaximal, characteristic, complemented
8C31×#1529C212yesmaximal, complemented
9C2×C31×#137—21no—
10C2×C31×#137—21no—
11C31×#563—21no—
12C31×#563C2211yescharacteristic
13C31×#562—22no—
14C31×#562C2212yes—
15C31×H(3,𝔽4)C2211yescharacteristic
16C31×#560—21no—
17C31×#560—22no—
18C31×#532—21no—
19C31×#532C2211yescharacteristic
20C2×C31×C4⋊Q8C2211yescharacteristic, complemented
21C31×#529—22no—
22C31×#529C2212yes—
23C31×#525—21no—
24C31×#525—22no—
25C31×#501C2214yescomplemented
26C31×#497C2212yes—
27C31×#497C2212yescomplemented
28C31×#494—21no—
29C31×#492—21no—
30C31×#489C2214yes—
31C31×#484—22no—
32C31×#481—22no—
33C31×#475—22no—
34C31×#474—22no—
35C31×#432C2212yescomplemented
36C4×M16×C31C2211yescharacteristic, complemented
37C31×#330C2214yescomplemented
38C31×#329C2212yes—
39C31×#329C2214yes—
40C31×#329C2212yescomplemented
41C2×#41×C31—21no—
42C2×#41×C31D411yescharacteristic
43C2×#41×C31D411yescharacteristic
44C22×Q8×C31—21no—
45C23×C4×C31—21no—
46C31×#137—44no—
47C31×#137—44no—
48C2×Q16×C31—22no—
49C2×Q16×C31—22no—
50C2×SD16×C31—22no—
51C2×SD16×C31—22no—
52C2×M16×C31—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8D411yescharacteristic
57C31×C4⋊Q8C2312yes—
58C31×#127—21no—
59C31×#127D411yescharacteristic
60C31×#126—22no—
61C31×#126—22no—
62C31×#126—22no—
63C31×#125—22no—
64C31×#125C2312yes—
65C31×#124—21no—
66C31×#124—22no—
67C31×#124—21no—
68C31×#124C2312yes—
69C31×#124C2311yescharacteristic
70C31×#124C2311yescharacteristic
71C31×#123—22no—
72C31×#123—21no—
73C31×#123—22no—
74C31×#123—21no—
75C31×#123—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×C22⋊Q8—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8D412yes—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121—22no—
89C31×#121—22no—
90C31×#121D412yes—
91C31×#120—21no—
92C31×#120—22no—
93C31×#120—21no—
94C31×#120—22no—
95C4×Q8×C31D412yes—
96C4×D4×C31—22no—
97C4×D4×C31—22no—
98C4×D4×C31—22no—
99C4×D4×C31—22no—
100C4×D4×C31D412yes—
101C31×#117—21no—
102C31×#117—22no—
103C2×C4⋊C4×C31—22no—
104C2×C42×C31—21no—
105C2×C42×C31C2311yescharacteristic
106C31×#107—24no—
107C31×#106—24no—
108C31×C4⋊C8—22no—
109C31×#103—24no—
110C31×#102—24no—
111C31×C22⋊C8—22no—
112C31×#97C2312yes—
113C31×#97C2314yes—
114C4×C8×C31—21no—
115C24×C31—21no—
116#41×C31—42no—
117#41×C31—42no—
118#41×C31—82no—
119C2×Q8×C31—21no—
120C2×Q8×C31—22no—
121C2×Q8×C31—42no—
122C2×Q8×C31—42no—
123C2×Q8×C31—41no—
124C2×Q8×C31C2×D412yes—
125C2×Q8×C31C2×D411yescharacteristic
126C2×Q8×C31C2×D411yescharacteristic
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—21no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—22no—
135C2×D4×C31C2×D412yes—
136C2×D4×C31C2×D411yescharacteristic
137C2×D4×C31C2×D411yescharacteristic
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—41no—
145C22×C4×C31—82no—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D412yes—
148Q16×C31—44no—
149Q16×C31—44no—
150SD16×C31—44no—
151SD16×C31—44no—
152M16×C31—24no—
153C2×C8×C31—22no—
154C2×C8×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—42no—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D412yes—
167C22⋊C4×C31—21no—
168C22⋊C4×C31—21no—
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—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—22no—
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D412yes—
185C42×C31C2411yescharacteristic
186#18996C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—21no—
188C23×C31—22no—
189C23×C31—21no—
190C23×C31—81no—
191C23×C31—82no—
192C23×C31#12011yescharacteristic
193Q8×C31—41no—
194Q8×C31—41no—
195Q8×C31—42no—
196Q8×C31—42no—
197Q8×C31—42no—
198Q8×C31—81no—
199Q8×C31—82no—
200D4×C31—41no—
201D4×C31—41no—
202D4×C31—42no—
203D4×C31—82no—
204D4×C31—82no—
205D4×C31—81no—
206D4×C31—81no—
207D4×C31—82no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31—81no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—22no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—44no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—82no—
223C2×C4×C31—82no—
224C2×C4×C31—81no—
225C2×C4×C31—82no—
226C2×C4×C31#12012yes—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31C22×D412yes—
229C2×C4×C31C22×D411yescharacteristic
230C8×C31—24no—
231C8×C31—44no—
232#3529C2×C3111yescharacteristic
233#3239C2×C3112yescomplemented
234#3152C2×C3111yescharacteristic, complemented
235#1661C2×C3114yescomplemented
236#1660C2×C3112yescomplemented
237#1659C2×C3112yescomplemented
238#1536C2×C3111yescharacteristic, complemented
239#1529C2×C3112yescomplemented
240C22×C31—21no—
241C22×C31—22no—
242C22×C31—41no—
243C22×C31—42no—
244C22×C31—41no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—42no—
249C22×C31—84no—
250C22×C31C2×#12011yescharacteristic
251C4×C31—21no—
252C4×C31—22no—
253C4×C31—22no—
254C4×C31—41no—
255C4×C31—42no—
256C4×C31—41no—
257C4×C31—42no—
258C4×C31—42no—
259C4×C31—82no—
260C4×C31#45312yes—
261C2×#137—21no—
262C2×#137—21no—
263#563—21no—
264#563C22×C3111yescharacteristic
265#562—22no—
266#562C22×C3112yes—
267H(3,𝔽4)C22×C3111yescharacteristic
268#560—21no—
269#560—22no—
270#532—21no—
271#532C22×C3111yescharacteristic
272C2×C4⋊Q8C22×C3111yescharacteristic, complemented
273#529—22no—
274#529C22×C3112yes—
275#525—21no—
276#525—22no—
277#501C22×C3114yescomplemented
278#497C22×C3112yes—
279#497C22×C3112yescomplemented
280#494—21no—
281#492—21no—
282#489C22×C3114yes—
283#484—22no—
284#481—22no—
285#475—22no—
286#474—22no—
287#432C22×C3112yescomplemented
288C4×M16C22×C3111yescharacteristic, complemented
289#330C22×C3114yescomplemented
290#329C22×C3112yes—
291#329C22×C3114yes—
292#329C22×C3112yescomplemented
293C2×C31—41no—
294C2×C31—81no—
295C2×C31—81no—
296C2×C31—82no—
297C2×C31C2×#45311yescharacteristic
298C2×C31#302212yes—
299C2×#41—21no—
300C2×#41D4×C3111yescharacteristic
301C2×#41D4×C3111yescharacteristic
302C22×Q8—21no—
303C23×C4—21no—
304#137—44no—
305#137—44no—
306C2×Q16—22no—
307C2×Q16—22no—
308C2×SD16—22no—
309C2×SD16—22no—
310C2×M16—21no—
311C4⋊Q8—21no—
312C4⋊Q8—22no—
313C4⋊Q8—21no—
314C4⋊Q8D4×C3111yescharacteristic
315C4⋊Q8C23×C3112yes—
316#127—21no—
317#127D4×C3111yescharacteristic
318#126—22no—
319#126—22no—
320#126—22no—
321#125—22no—
322#125C23×C3112yes—
323#124—21no—
324#124—22no—
325#124—21no—
326#124C23×C3112yes—
327#124C23×C3111yescharacteristic
328#124C23×C3111yescharacteristic
329#123—22no—
330#123—21no—
331#123—22no—
332#123—21no—
333#123—22no—
334#123—22no—
335#123—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8D4×C3112yes—
341#121—22no—
342#121—22no—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121D4×C3112yes—
349#120—21no—
350#120—22no—
351#120—21no—
352#120—22no—
353C4×Q8D4×C3112yes—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4D4×C3112yes—
359#117—21no—
360#117—22no—
361C2×C4⋊C4—22no—
362C2×C42—21no—
363C2×C42C23×C3111yescharacteristic
364#107—24no—
365#106—24no—
366C4⋊C8—22no—
367#103—24no—
368#102—24no—
369C22⋊C8—22no—
370#97C23×C3114yes—
371#97C23×C3112yes—
372C4×C8—21no—
373C31#1899611yescharacteristic, direct product factor, Sylow
374C24—21no—
375#41—42no—
376#41—42no—
377#41—82no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—41no—
383C2×Q8C2×D4×C3112yes—
384C2×Q8C2×D4×C3111yescharacteristic
385C2×Q8C2×D4×C3111yescharacteristic
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4C2×D4×C3112yes—
395C2×D4C2×D4×C3111yescharacteristic
396C2×D4C2×D4×C3111yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—82no—
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3112yes—
407Q16—44no—
408Q16—44no—
409SD16—44no—
410SD16—44no—
411M16—24no—
412C2×C8—22no—
413C2×C8—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4C2×D4×C3112yes—
425C4⋊C4C2×D4×C3112yes—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C42—21no—
440C42—21no—
441C42—22no—
442C42C2×D4×C3111yescharacteristic
443C42C2×D4×C3112yes—
444C42C24×C3111yescharacteristic
445C23—21no—
446C23—22no—
447C23—21no—
448C23—81no—
449C23—82no—
450C23C31×#12011yescharacteristic
451Q8—41no—
452Q8—41no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—81no—
457Q8—82no—
458D4—41no—
459D4—41no—
460D4—42no—
461D4—82no—
462D4—82no—
463D4—81no—
464D4—81no—
465D4—82no—
466D4—82no—
467D4—82no—
468D4—81no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4C31×#12012yes—
485C2×C4C31×#12011yescharacteristic
486C2×C4C22×D4×C3112yes—
487C2×C4C22×D4×C3111yescharacteristic
488C8—24no—
489C8—44no—
490C22—21no—
491C22—22no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—84no—
500C22C2×C31×#12011yescharacteristic
501C4—21no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4C31×#45312yes—
511C2—41no—
512C2—81no—
513C2—82no—
514C2—81no—
515C2C2×C31×#45311yescharacteristic
516C2C31×#302212yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3529C2
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×#3239C2semidirect
3. C31×#3152C2semidirect
4. C31×#1661C2semidirect
5. C31×#1660C2semidirect
6. C31×#1659C2semidirect
7. C31×#1536C2semidirect
8. C31×#1529C2semidirect
20. C2×C31×C4⋊Q8C22semidirect
25. C31×#501C22semidirect
27. C31×#497C22semidirect
35. C31×#432C22semidirect
36. C4×M16×C31C22semidirect
37. C31×#330C22semidirect
40. C31×#329C22semidirect
186. #18996C31direct (N × H)
233. #3239C2×C31semidirect
234. #3152C2×C31semidirect
235. #1661C2×C31semidirect
236. #1660C2×C31semidirect
237. #1659C2×C31semidirect
238. #1536C2×C31semidirect
239. #1529C2×C31semidirect
272. C2×C4⋊Q8C22×C31semidirect
277. #501C22×C31semidirect
279. #497C22×C31semidirect
287. #432C22×C31semidirect
288. C4×M16C22×C31semidirect
289. #330C22×C31semidirect
292. #329C22×C31semidirect
373. C31#18996direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35292
C31×#32392
C31×#31522
C31×#16612
C31×#16602
C31×#16592
C31×#15362
C31×#15292
#1899631