Finite Group Databasebeta

← C31×#8500 #21465259

Subgroups — C31×#8500 #21465259

Order 7936.

736 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3546C211yesmaximal, characteristic
2C31×#1553C211yesmaximal, characteristic, complemented
3C31×#1535C211yesmaximal, characteristic, complemented
4C31×#1521C211yesmaximal, characteristic, complemented
5C31×#1514C211yesmaximal, characteristic, complemented
6C31×#1514C211yesmaximal, characteristic, complemented
7C31×#1496C211yesmaximal, characteristic, complemented
8C31×#560—22no—
9C31×#556C411yescharacteristic, complemented
10C31×#553—21no—
11C31×#550—21no—
12C31×#550C411yescharacteristic, complemented
13C31×#547C411yescharacteristic, complemented
14C31×#546—21no—
15C31×#546—21no—
16D42×C31C411yescharacteristic, complemented
17C31×#542—21no—
18C31×#538—22no—
19C31×#537—21no—
20C31×#535—21no—
21C31×#535—21no—
22C31×#532C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C31×#432C2211yescharacteristic, complemented
26C31×#432C2211yescharacteristic, complemented
27C31×#431C2211yescharacteristic, complemented
28C31×#383C2211yescharacteristic, complemented
29C31×#332—21no—
30C31×#331—21no—
31C31×#330—21no—
32C31×#329—21no—
33C31×#328—21no—
34C31×#327—21no—
35C31×#327—21no—
36C31×#327—22no—
37C31×#327—21no—
38C31×D4⋊C8—22no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—21no—
42C2×#41×C31—21no—
43C22×D4×C31—21no—
44C22×D4×C31—21no—
45C31×C4⋊Q8C2×C411yescharacteristic
46C31×C4⋊Q8D411yescharacteristic
47C31×#127—22no—
48C31×#127—21no—
49C31×#127C2×C411yescharacteristic
50C31×#127D411yescharacteristic
51C31×#126—22no—
52C31×#126—22no—
53C31×#126—21no—
54C31×#125D411yescharacteristic
55C31×#124—22no—
56C31×#124—21no—
57C31×#124—21no—
58C31×#124—21no—
59C31×#124D411yescharacteristic
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—21no—
63C31×#123—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—21no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8D411yescharacteristic
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—22no—
72C31×#121—22no—
73C31×#121—21no—
74C31×#121—22no—
75C31×#121—21no—
76C31×#121—21no—
77C31×#121—21no—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121D411yescharacteristic
81C31×#121D411yescharacteristic
82C31×#121D411yescharacteristic
83C31×#121D412yes—
84C31×#120—22no—
85C31×#120—22no—
86C31×#120—22no—
87C31×#120—22no—
88C4×Q8×C31—21no—
89C4×Q8×C31C2×C411yescharacteristic
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31—21no—
95C4×D4×C31C2×C411yescharacteristic
96C4×D4×C31C2×C411yescharacteristic
97C4×D4×C31C2×C411yescharacteristic
98C4×D4×C31D412yes—
99C31×#117—21no—
100C31×#117—21no—
101C2×C4⋊C4×C31—21no—
102C2×C22⋊C4×C31—22no—
103C2×C42×C31C2311yescharacteristic
104C31×C4⋊C8—21no—
105C31×C4⋊C8—21no—
106C31×C4⋊C8—21no—
107C31×C22⋊C8—21no—
108C31×C22⋊C8—21no—
109C31×C22⋊C8—21no—
110C31×#97—21no—
111C31×#97—21no—
112C4×C8×C31—21no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—21no—
116C24×C31—22no—
117#41×C31—82no—
118#41×C31—82no—
119#41×C31—82no—
120#41×C31—82no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31C22⋊C411yescharacteristic
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31—21no—
136C2×D4×C31—21no—
137C2×D4×C31—82no—
138C2×D4×C31—82no—
139C2×D4×C31C22⋊C411yescharacteristic
140C2×D4×C31C22⋊C411yescharacteristic
141C2×D4×C31C22⋊C411yescharacteristic
142C2×D4×C31C22⋊C412yes—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—22no—
149C22×C4×C31—41no—
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D411yescharacteristic
152C22×C4×C31C2×D411yescharacteristic
153C2×C8×C31—41no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31C22⋊C411yescharacteristic
165C4⋊C4×C31C22⋊C411yescharacteristic
166C4⋊C4×C31C22⋊C411yescharacteristic
167C4⋊C4×C31C22⋊C411yescharacteristic
168C4⋊C4×C31C22⋊C412yes—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—21no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C42×C31—22no—
186C42×C31—21no—
187C42×C31C22×C411yescharacteristic
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191#8500C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—22no—
195C23×C31—21no—
196C23×C31—82no—
197C23×C31—82no—
198C23×C31—82no—
199C23×C31#12011yescharacteristic
200Q8×C31—41no—
201Q8×C31—42no—
202Q8×C31—81no—
203D4×C31—41no—
204D4×C31—41no—
205D4×C31—42no—
206D4×C31—41no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—81no—
210D4×C31—81no—
211D4×C31—81no—
212D4×C31—81no—
213D4×C31—81no—
214D4×C31—82no—
215D4×C31—82no—
216D4×C31—82no—
217D4×C31—82no—
218D4×C31—82no—
219D4×C31—82no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—22no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—82no—
233C2×C4×C31—82no—
234C2×C4×C31—82no—
235C2×C4×C31—81no—
236C2×C4×C31—81no—
237C2×C4×C31—81no—
238C2×C4×C31C2×C22⋊C411yescharacteristic
239C2×C4×C31C2×C22⋊C411yescharacteristic
240C2×C4×C31C2×C22⋊C411yescharacteristic
241C2×C4×C31#12011yescharacteristic
242C2×C4×C31#12011yescharacteristic
243C2×C4×C31#12011yescharacteristic
244C8×C31—81no—
245C8×C31—81no—
246C8×C31—81no—
247#3546C2×C3111yescharacteristic
248#1553C2×C3111yescharacteristic, complemented
249#1535C2×C3111yescharacteristic, complemented
250#1521C2×C3111yescharacteristic, complemented
251#1514C2×C3111yescharacteristic, complemented
252#1514C2×C3111yescharacteristic, complemented
253#1496C2×C3111yescharacteristic, complemented
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—42no—
258C22×C31—41no—
259C22×C31—41no—
260C22×C31—41no—
261C22×C31—41no—
262C22×C31—41no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—42no—
266C22×C31—42no—
267C22×C31—41no—
268C22×C31—41no—
269C22×C31—164no—
270C22×C31#37911yescharacteristic
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—21no—
277C4×C31—41no—
278C4×C31—41no—
279C4×C31—41no—
280C4×C31—41no—
281C4×C31—41no—
282C4×C31—42no—
283C4×C31—161no—
284#560—22no—
285#556C4×C3111yescharacteristic, complemented
286#553—21no—
287#550—21no—
288#550C4×C3111yescharacteristic, complemented
289#547C4×C3111yescharacteristic, complemented
290#546—21no—
291#546—21no—
292D42C4×C3111yescharacteristic, complemented
293#542—21no—
294#538—22no—
295#537—21no—
296#535—21no—
297#535—21no—
298#532C22×C3111yescharacteristic
299#532C22×C3111yescharacteristic
300#529C22×C3111yescharacteristic
301#432C22×C3111yescharacteristic, complemented
302#432C22×C3111yescharacteristic, complemented
303#431C22×C3111yescharacteristic, complemented
304#383C22×C3111yescharacteristic, complemented
305#332—21no—
306#331—21no—
307#330—21no—
308#329—21no—
309#328—21no—
310#327—21no—
311#327—21no—
312#327—22no—
313#327—21no—
314D4⋊C8—22no—
315C2×C31—41no—
316C2×C31—81no—
317C2×C31—82no—
318C2×C31—81no—
319C2×C31—81no—
320C2×C31#179411yescharacteristic
321C2×C31#179511yescharacteristic
322C2×C31#179511yescharacteristic
323C2×#41—21no—
324C2×#41—21no—
325C2×#41—21no—
326C2×#41—21no—
327C22×D4—21no—
328C22×D4—21no—
329C4⋊Q8C2×C4×C3111yescharacteristic
330C4⋊Q8D4×C3111yescharacteristic
331#127—22no—
332#127—21no—
333#127C2×C4×C3111yescharacteristic
334#127D4×C3111yescharacteristic
335#126—22no—
336#126—22no—
337#126—21no—
338#125D4×C3111yescharacteristic
339#124—22no—
340#124—21no—
341#124—21no—
342#124—21no—
343#124D4×C3111yescharacteristic
344#123—22no—
345#123—22no—
346#123—21no—
347#123—22no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8D4×C3111yescharacteristic
352#121—22no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121—22no—
359#121—21no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121D4×C3111yescharacteristic
365#121D4×C3111yescharacteristic
366#121D4×C3111yescharacteristic
367#121D4×C3112yes—
368#120—22no—
369#120—22no—
370#120—22no—
371#120—22no—
372C4×Q8—21no—
373C4×Q8C2×C4×C3111yescharacteristic
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4C2×C4×C3111yescharacteristic
380C4×D4C2×C4×C3111yescharacteristic
381C4×D4C2×C4×C3111yescharacteristic
382C4×D4D4×C3112yes—
383#117—21no—
384#117—21no—
385C2×C4⋊C4—21no—
386C2×C22⋊C4—22no—
387C2×C42C23×C3111yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394#97—21no—
395#97—21no—
396C4×C8—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400C31#850011yescharacteristic, direct product factor, Sylow
401C24—22no—
402#41—82no—
403#41—82no—
404#41—82no—
405#41—82no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C22⋊C4×C3111yescharacteristic
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C22⋊C4×C3111yescharacteristic
425C2×D4C22⋊C4×C3111yescharacteristic
426C2×D4C22⋊C4×C3111yescharacteristic
427C2×D4C22⋊C4×C3112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4C22⋊C4×C3111yescharacteristic
450C4⋊C4C22⋊C4×C3111yescharacteristic
451C4⋊C4C22⋊C4×C3111yescharacteristic
452C4⋊C4C22⋊C4×C3111yescharacteristic
453C4⋊C4C22⋊C4×C3112yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—22no—
471C42—21no—
472C42C22×C4×C3111yescharacteristic
473C42C2×D4×C3111yescharacteristic
474C42C2×D4×C3111yescharacteristic
475C42C2×D4×C3111yescharacteristic
476C23—21no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23C31×#12011yescharacteristic
484Q8—41no—
485Q8—42no—
486Q8—81no—
487D4—41no—
488D4—41no—
489D4—42no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4C2×C22⋊C4×C3111yescharacteristic
523C2×C4C2×C22⋊C4×C3111yescharacteristic
524C2×C4C2×C22⋊C4×C3111yescharacteristic
525C2×C4C31×#12011yescharacteristic
526C2×C4C31×#12011yescharacteristic
527C2×C4C31×#12011yescharacteristic
528C8—81no—
529C8—81no—
530C8—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—164no—
547C22C31×#37911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—161no—
561C2—41no—
562C2—81no—
563C2—81no—
564C2—82no—
565C2—81no—
566C2C31×#179411yescharacteristic
567C2C31×#179511yescharacteristic
568C2C31×#179511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3546C2
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×#1553C2semidirect
3. C31×#1535C2semidirect
4. C31×#1521C2semidirect
5. C31×#1514C2semidirect
6. C31×#1514C2semidirect
7. C31×#1496C2semidirect
9. C31×#556C4semidirect
12. C31×#550C4semidirect
13. C31×#547C4semidirect
16. D42×C31C4semidirect
25. C31×#432C22semidirect
26. C31×#432C22semidirect
27. C31×#431C22semidirect
28. C31×#383C22semidirect
191. #8500C31direct (N × H)
248. #1553C2×C31semidirect
249. #1535C2×C31semidirect
250. #1521C2×C31semidirect
251. #1514C2×C31semidirect
252. #1514C2×C31semidirect
253. #1496C2×C31semidirect
285. #556C4×C31semidirect
288. #550C4×C31semidirect
289. #547C4×C31semidirect
292. D42C4×C31semidirect
301. #432C22×C31semidirect
302. #432C22×C31semidirect
303. #431C22×C31semidirect
304. #383C22×C31semidirect
400. C31#8500direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35462
C31×#15532
C31×#15352
C31×#15212
C31×#15142
C31×#14962
#850031