Finite Group Databasebeta

← C31×#8484 #21465243

Subgroups — C31×#8484 #21465243

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3561C211yesmaximal, characteristic
2C31×#1967C211yesmaximal, characteristic, complemented
3C31×#1559C211yesmaximal, characteristic, complemented
4C31×#1529C211yesmaximal, characteristic, complemented
5C31×#1495C211yesmaximal, characteristic, complemented
6C31×#1485C211yesmaximal, characteristic, complemented
7C31×#1469C211yesmaximal, characteristic, complemented
8C31×#563—21no—
9C31×#562—21no—
10C31×#556—21no—
11C31×#551—22no—
12D4×Q8×C31—21no—
13C31×#548—21no—
14C31×#546—21no—
15C31×#543C412yes—
16C31×#542—21no—
17C31×#542—22no—
18C31×#538C412yes—
19C31×#536—21no—
20C31×#533C2211yescharacteristic
21C31×#529C2211yescharacteristic
22C2×C31×#124C2211yescharacteristic
23C31×#432C2211yescharacteristic
24C31×#431C2211yescharacteristic
25C31×#423C2211yescharacteristic
26C2×C31×#101—21no—
27C2×C31×#100—21no—
28C4×M16×C31C2211yescharacteristic
29C31×#333—21no—
30C31×#332—21no—
31C31×#330—21no—
32C31×#329—21no—
33C31×Q8⋊C8—22no—
34C31×Q8⋊C8—21no—
35C31×D4⋊C8—21no—
36C31×#323—22no—
37C2×#41×C31—21no—
38C22×Q8×C31—21no—
39C22×Q8×C31D411yescharacteristic
40C22×D4×C31D411yescharacteristic
41C2×M16×C31—21no—
42C2×M16×C31—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8D411yescharacteristic
46C31×C4⋊Q8D411yescharacteristic
47C31×#126—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—21no—
51C31×#125—21no—
52C31×#125D411yescharacteristic
53C31×#124—21no—
54C31×#124—22no—
55C31×#124—22no—
56C31×#124—21no—
57C31×#124—21no—
58C31×#124—21no—
59C31×#124—21no—
60C31×#124D411yescharacteristic
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123—21no—
65C31×#123—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8C2×C411yescharacteristic
76C31×C22⋊Q8C2×C412yes—
77C31×#121—21no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121C2×C411yescharacteristic
81C31×#120—22no—
82C4×Q8×C31—21no—
83C4×Q8×C31—21no—
84C4×Q8×C31D411yescharacteristic
85C4×Q8×C31D412yes—
86C4×D4×C31—21no—
87C4×D4×C31—22no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31D411yescharacteristic
91C31×#117—22no—
92C31×#117—21no—
93C31×#117—21no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31C2×C412yes—
96C2×C22⋊C4×C31D412yes—
97C2×C42×C31C2311yescharacteristic
98C31×C4⋊C8—21no—
99C31×C4⋊C8—21no—
100C31×C4⋊C8—21no—
101C31×C4⋊C8—21no—
102C31×#101—42no—
103C31×#100—42no—
104C31×C22⋊C8—21no—
105C31×C22⋊C8—21no—
106C31×#97—21no—
107C31×#97—21no—
108C4×C8×C31—21no—
109C4×C8×C31—21no—
110C24×C31C22⋊C412yes—
111#41×C31—81no—
112#41×C31—81no—
113C2×Q8×C31—22no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31—81no—
120C2×Q8×C31—81no—
121C2×Q8×C31C22⋊C411yescharacteristic
122C2×Q8×C31C22⋊C412yes—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—41no—
129C2×D4×C31—41no—
130C2×D4×C31C22⋊C411yescharacteristic
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31C22⋊C412yes—
136C22×C4×C31C22×C411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139M16×C31—22no—
140M16×C31—41no—
141M16×C31—41no—
142M16×C31—41no—
143C2×C8×C31—41no—
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—22no—
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×C31C22⋊C411yescharacteristic
161C4⋊C4×C31C22⋊C411yescharacteristic
162C4⋊C4×C31C22⋊C412yes—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—21no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—21no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C42×C31—22no—
179C42×C31—22no—
180C42×C31—21no—
181C42×C31C2×D411yescharacteristic
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D411yescharacteristic
185#8484C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—22no—
187C23×C31—21no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31C2×C22⋊C411yescharacteristic
192Q8×C31—41no—
193Q8×C31—42no—
194Q8×C31—41no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—41no—
198Q8×C31—81no—
199Q8×C31—82no—
200D4×C31—41no—
201D4×C31—42no—
202D4×C31—41no—
203D4×C31—41no—
204D4×C31—81no—
205D4×C31—82no—
206D4×C31—82no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—22no—
211C2×C4×C31—21no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—82no—
221C2×C4×C31—82no—
222C2×C4×C31—82no—
223C2×C4×C31C2×C22⋊C411yescharacteristic
224C2×C4×C31C2×C22⋊C411yescharacteristic
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C8×C31—42no—
230C8×C31—81no—
231C8×C31—81no—
232C8×C31—81no—
233#3561C2×C3111yescharacteristic
234#1967C2×C3111yescharacteristic, complemented
235#1559C2×C3111yescharacteristic, complemented
236#1529C2×C3111yescharacteristic, complemented
237#1495C2×C3111yescharacteristic, complemented
238#1485C2×C3111yescharacteristic, complemented
239#1469C2×C3111yescharacteristic, complemented
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—41no—
244C22×C31—42no—
245C22×C31—41no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—41no—
249C22×C31—162no—
250C22×C31#37911yescharacteristic
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—21no—
255C4×C31—21no—
256C4×C31—41no—
257C4×C31—42no—
258C4×C31—42no—
259C4×C31—41no—
260C4×C31—41no—
261C4×C31—42no—
262C4×C31#45311yescharacteristic
263C4×C31#45511yescharacteristic
264#563—21no—
265#562—21no—
266#556—21no—
267#551—22no—
268D4×Q8—21no—
269#548—21no—
270#546—21no—
271#543C4×C3112yes—
272#542—21no—
273#542—22no—
274#538C4×C3112yes—
275#536—21no—
276#533C22×C3111yescharacteristic
277#529C22×C3111yescharacteristic
278C2×#124C22×C3111yescharacteristic
279#432C22×C3111yescharacteristic
280#431C22×C3111yescharacteristic
281#423C22×C3111yescharacteristic
282C2×#101—21no—
283C2×#100—21no—
284C4×M16C22×C3111yescharacteristic
285#333—21no—
286#332—21no—
287#330—21no—
288#329—21no—
289Q8⋊C8—22no—
290Q8⋊C8—21no—
291D4⋊C8—21no—
292#323—22no—
293C2×C31—41no—
294C2×C31—81no—
295C2×C31—82no—
296C2×C31#179111yescharacteristic
297C2×C31#179211yescharacteristic
298C2×C31#179611yescharacteristic
299C2×#41—21no—
300C22×Q8—21no—
301C22×Q8D4×C3111yescharacteristic
302C22×D4D4×C3111yescharacteristic
303C2×M16—21no—
304C2×M16—21no—
305C4⋊Q8—21no—
306C4⋊Q8—22no—
307C4⋊Q8D4×C3111yescharacteristic
308C4⋊Q8D4×C3111yescharacteristic
309#126—22no—
310#126—22no—
311#126—22no—
312#126—21no—
313#125—21no—
314#125D4×C3111yescharacteristic
315#124—21no—
316#124—22no—
317#124—22no—
318#124—21no—
319#124—21no—
320#124—21no—
321#124—21no—
322#124D4×C3111yescharacteristic
323#123—22no—
324#123—22no—
325#123—22no—
326#123—21no—
327#123—22no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8C2×C4×C3111yescharacteristic
338C22⋊Q8C2×C4×C3112yes—
339#121—21no—
340#121—22no—
341#121—22no—
342#121C2×C4×C3111yescharacteristic
343#120—22no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8D4×C3111yescharacteristic
347C4×Q8D4×C3112yes—
348C4×D4—21no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4D4×C3111yescharacteristic
353#117—22no—
354#117—21no—
355#117—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×C4×C3112yes—
358C2×C22⋊C4D4×C3112yes—
359C2×C42C23×C3111yescharacteristic
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364#101—42no—
365#100—42no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368#97—21no—
369#97—21no—
370C4×C8—21no—
371C4×C8—21no—
372C31#848411yescharacteristic, direct product factor, Sylow
373C24C22⋊C4×C3112yes—
374#41—81no—
375#41—81no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8C22⋊C4×C3111yescharacteristic
385C2×Q8C22⋊C4×C3112yes—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4C22⋊C4×C3111yescharacteristic
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4C22⋊C4×C3112yes—
399C22×C4C22×C4×C3111yescharacteristic
400C22×C4C2×D4×C3111yescharacteristic
401C22×C4C2×D4×C3111yescharacteristic
402M16—22no—
403M16—41no—
404M16—41no—
405M16—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4C22⋊C4×C3111yescharacteristic
424C4⋊C4C22⋊C4×C3111yescharacteristic
425C4⋊C4C22⋊C4×C3112yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42C2×D4×C3111yescharacteristic
445C42C2×D4×C3111yescharacteristic
446C42C2×D4×C3111yescharacteristic
447C42C2×D4×C3111yescharacteristic
448C23—22no—
449C23—21no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C2×C22⋊C4×C3111yescharacteristic
454Q8—41no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461Q8—82no—
462D4—41no—
463D4—42no—
464D4—41no—
465D4—41no—
466D4—81no—
467D4—82no—
468D4—82no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C2×C22⋊C4×C3111yescharacteristic
486C2×C4C2×C22⋊C4×C3111yescharacteristic
487C2×C4C31×#12011yescharacteristic
488C2×C4C31×#12011yescharacteristic
489C2×C4C31×#12011yescharacteristic
490C2×C4C31×#12011yescharacteristic
491C8—42no—
492C8—81no—
493C8—81no—
494C8—81no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—162no—
505C22C31×#37911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4C31×#45311yescharacteristic
518C4C31×#45511yescharacteristic
519C2—41no—
520C2—82no—
521C2—81no—
522C2C31×#179111yescharacteristic
523C2C31×#179211yescharacteristic
524C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3561C2
LCSG ▷ C22×C4 ▷ 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×#1967C2semidirect
3. C31×#1559C2semidirect
4. C31×#1529C2semidirect
5. C31×#1495C2semidirect
6. C31×#1485C2semidirect
7. C31×#1469C2semidirect
185. #8484C31direct (N × H)
234. #1967C2×C31semidirect
235. #1559C2×C31semidirect
236. #1529C2×C31semidirect
237. #1495C2×C31semidirect
238. #1485C2×C31semidirect
239. #1469C2×C31semidirect
372. C31#8484direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#19672
C31×#15592
C31×#15292
C31×#14952
C31×#14852
C31×#14692
#848431