Finite Group Databasebeta

← C31×#8502 #21465261

Subgroups — C31×#8502 #21465261

Order 7936.

672 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×#3547C211yesmaximal, characteristic
2C31×#1553C211yesmaximal, characteristic, complemented
3C31×#1535C211yesmaximal, characteristic, complemented
4C31×#1521C211yesmaximal, characteristic, complemented
5C31×#1515C211yesmaximal, characteristic, complemented
6C31×#1514C211yesmaximal, characteristic, complemented
7C31×#1497C211yesmaximal, characteristic, complemented
8C31×#562—22no—
9C31×#555—21no—
10C31×#555C411yescharacteristic, complemented
11C31×#554C411yescharacteristic, complemented
12C31×#553—21no—
13C31×#552—21no—
14D4×Q8×C31C411yescharacteristic, complemented
15C31×#548—21no—
16C31×#547C411yescharacteristic, complemented
17C31×#544—21no—
18C31×#542—22no—
19C31×#540—21no—
20C31×#537—21no—
21C31×#536—21no—
22C31×#533C2211yescharacteristic
23C31×#532C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C31×#432C2211yescharacteristic
26C31×#432C2211yescharacteristic
27C31×#431C2211yescharacteristic
28C31×#383C2211yescharacteristic
29C31×#332—21no—
30C31×#331—21no—
31C31×#330—21no—
32C31×#329—21no—
33C31×#328—22no—
34C31×#328—21no—
35C31×#327—21no—
36C31×#327—21no—
37C31×#327—21no—
38C31×Q8⋊C8—22no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C22×Q8×C31—21no—
42C31×C4⋊Q8—22no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8C2×C411yescharacteristic
45C31×C4⋊Q8C2×C411yescharacteristic
46C31×C4⋊Q8D411yescharacteristic
47C31×#127D411yescharacteristic
48C31×#126—22no—
49C31×#126—21no—
50C31×#125—22no—
51C31×#125—21no—
52C31×#125—21no—
53C31×#125D411yescharacteristic
54C31×#124—22no—
55C31×#124—22no—
56C31×#124—22no—
57C31×#124—21no—
58C31×#124D411yescharacteristic
59C31×#123—22no—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—21no—
63C31×#123—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—21no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8D411yescharacteristic
73C31×C22⋊Q8D412yes—
74C31×#121—22no—
75C31×#121—22no—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121—21no—
79C31×#121—21no—
80C31×#121—21no—
81C31×#121—21no—
82C31×#121D411yescharacteristic
83C31×#121D411yescharacteristic
84C31×#121D411yescharacteristic
85C4×Q8×C31—21no—
86C4×Q8×C31C2×C411yescharacteristic
87C4×Q8×C31D412yes—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31—21no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31C2×C411yescharacteristic
94C4×D4×C31C2×C411yescharacteristic
95C4×D4×C31C2×C411yescharacteristic
96C31×#117—22no—
97C31×#117—21no—
98C31×#117—21no—
99C31×#117—21no—
100C31×#117—21no—
101C2×C4⋊C4×C31—21no—
102C2×C4⋊C4×C31—21no—
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—
116#41×C31—82no—
117#41×C31—82no—
118C2×Q8×C31—22no—
119C2×Q8×C31—21no—
120C2×Q8×C31—21no—
121C2×Q8×C31—82no—
122C2×Q8×C31C22⋊C411yescharacteristic
123C2×Q8×C31C22⋊C412yes—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31C22⋊C411yescharacteristic
132C2×D4×C31C22⋊C411yescharacteristic
133C2×D4×C31C22⋊C411yescharacteristic
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—41no—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C2×C8×C31—41no—
145C2×C8×C31—41no—
146C2×C8×C31—41no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31C22⋊C411yescharacteristic
162C4⋊C4×C31C22⋊C411yescharacteristic
163C4⋊C4×C31C22⋊C411yescharacteristic
164C4⋊C4×C31C22⋊C411yescharacteristic
165C4⋊C4×C31C22⋊C412yes—
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—21no—
173C22⋊C4×C31—21no—
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—
181C42×C31—22no—
182C42×C31—21no—
183C42×C31—22no—
184C42×C31C22×C411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188#8502C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—21no—
190C23×C31—21no—
191C23×C31—21no—
192C23×C31#12011yescharacteristic
193Q8×C31—41no—
194Q8×C31—42no—
195Q8×C31—42no—
196Q8×C31—81no—
197Q8×C31—82no—
198D4×C31—41no—
199D4×C31—41no—
200D4×C31—41no—
201D4×C31—81no—
202D4×C31—81no—
203D4×C31—81no—
204D4×C31—81no—
205D4×C31—81no—
206D4×C31—82no—
207D4×C31—82no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—82no—
222C2×C4×C31—82no—
223C2×C4×C31—82no—
224C2×C4×C31—81no—
225C2×C4×C31—81no—
226C2×C4×C31—81no—
227C2×C4×C31C2×C22⋊C411yescharacteristic
228C2×C4×C31C2×C22⋊C411yescharacteristic
229C2×C4×C31C2×C22⋊C411yescharacteristic
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31#12011yescharacteristic
233C8×C31—81no—
234C8×C31—81no—
235C8×C31—81no—
236#3547C2×C3111yescharacteristic
237#1553C2×C3111yescharacteristic, complemented
238#1535C2×C3111yescharacteristic, complemented
239#1521C2×C3111yescharacteristic, complemented
240#1515C2×C3111yescharacteristic, complemented
241#1514C2×C3111yescharacteristic, complemented
242#1497C2×C3111yescharacteristic, complemented
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—41no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31—41no—
250C22×C31—41no—
251C22×C31—41no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31—41no—
255C22×C31#37911yescharacteristic
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—21no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—41no—
265C4×C31—41no—
266C4×C31—42no—
267C4×C31—41no—
268C4×C31—41no—
269C4×C31—161no—
270#562—22no—
271#555—21no—
272#555C4×C3111yescharacteristic, complemented
273#554C4×C3111yescharacteristic, complemented
274#553—21no—
275#552—21no—
276D4×Q8C4×C3111yescharacteristic, complemented
277#548—21no—
278#547C4×C3111yescharacteristic, complemented
279#544—21no—
280#542—22no—
281#540—21no—
282#537—21no—
283#536—21no—
284#533C22×C3111yescharacteristic
285#532C22×C3111yescharacteristic
286#529C22×C3111yescharacteristic
287#432C22×C3111yescharacteristic
288#432C22×C3111yescharacteristic
289#431C22×C3111yescharacteristic
290#383C22×C3111yescharacteristic
291#332—21no—
292#331—21no—
293#330—21no—
294#329—21no—
295#328—22no—
296#328—21no—
297#327—21no—
298#327—21no—
299#327—21no—
300Q8⋊C8—22no—
301C2×C31—41no—
302C2×C31—81no—
303C2×C31—81no—
304C2×C31—81no—
305C2×C31#179411yescharacteristic
306C2×C31#179511yescharacteristic
307C2×C31#179611yescharacteristic
308C2×#41—21no—
309C2×#41—21no—
310C22×Q8—21no—
311C4⋊Q8—22no—
312C4⋊Q8—21no—
313C4⋊Q8C2×C4×C3111yescharacteristic
314C4⋊Q8C2×C4×C3111yescharacteristic
315C4⋊Q8D4×C3111yescharacteristic
316#127D4×C3111yescharacteristic
317#126—22no—
318#126—21no—
319#125—22no—
320#125—21no—
321#125—21no—
322#125D4×C3111yescharacteristic
323#124—22no—
324#124—22no—
325#124—22no—
326#124—21no—
327#124D4×C3111yescharacteristic
328#123—22no—
329#123—22no—
330#123—22no—
331#123—21no—
332#123—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8D4×C3111yescharacteristic
342C22⋊Q8D4×C3112yes—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—21no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121D4×C3111yescharacteristic
352#121D4×C3111yescharacteristic
353#121D4×C3111yescharacteristic
354C4×Q8—21no—
355C4×Q8C2×C4×C3111yescharacteristic
356C4×Q8D4×C3112yes—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4C2×C4×C3111yescharacteristic
363C4×D4C2×C4×C3111yescharacteristic
364C4×D4C2×C4×C3111yescharacteristic
365#117—22no—
366#117—21no—
367#117—21no—
368#117—21no—
369#117—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C42C23×C3111yescharacteristic
373C4⋊C8—21no—
374C4⋊C8—21no—
375C4⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379#97—21no—
380#97—21no—
381C4×C8—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385C31#850211yescharacteristic, direct product factor, Sylow
386#41—82no—
387#41—82no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—82no—
392C2×Q8C22⋊C4×C3111yescharacteristic
393C2×Q8C22⋊C4×C3112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C22⋊C4×C3111yescharacteristic
402C2×D4C22⋊C4×C3111yescharacteristic
403C2×D4C22⋊C4×C3111yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4C22⋊C4×C3111yescharacteristic
432C4⋊C4C22⋊C4×C3111yescharacteristic
433C4⋊C4C22⋊C4×C3111yescharacteristic
434C4⋊C4C22⋊C4×C3111yescharacteristic
435C4⋊C4C22⋊C4×C3112yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—22no—
452C42—21no—
453C42—22no—
454C42C22×C4×C3111yescharacteristic
455C42C2×D4×C3111yescharacteristic
456C42C2×D4×C3111yescharacteristic
457C42C2×D4×C3111yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23C31×#12011yescharacteristic
462Q8—41no—
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—82no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—82no—
476D4—82no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4C2×C22⋊C4×C3111yescharacteristic
497C2×C4C2×C22⋊C4×C3111yescharacteristic
498C2×C4C2×C22⋊C4×C3111yescharacteristic
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#12011yescharacteristic
502C8—81no—
503C8—81no—
504C8—81no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22C31×#37911yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—161no—
532C2—41no—
533C2—81no—
534C2—81no—
535C2—81no—
536C2C31×#179411yescharacteristic
537C2C31×#179511yescharacteristic
538C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3547C2
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×#1515C2semidirect
6. C31×#1514C2semidirect
7. C31×#1497C2semidirect
10. C31×#555C4semidirect
11. C31×#554C4semidirect
14. D4×Q8×C31C4semidirect
16. C31×#547C4semidirect
188. #8502C31direct (N × H)
237. #1553C2×C31semidirect
238. #1535C2×C31semidirect
239. #1521C2×C31semidirect
240. #1515C2×C31semidirect
241. #1514C2×C31semidirect
242. #1497C2×C31semidirect
272. #555C4×C31semidirect
273. #554C4×C31semidirect
276. D4×Q8C4×C31semidirect
278. #547C4×C31semidirect
385. C31#8502direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35472
C31×#15532
C31×#15352
C31×#15212
C31×#15152
C31×#15142
C31×#14972
#850231