Finite Group Databasebeta

← C31×#8501 #21465260

Subgroups — C31×#8501 #21465260

Order 7936.

616 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3555C211yesmaximal, characteristic
2C31×#1554C211yesmaximal, characteristic, complemented
3C31×#1535C211yesmaximal, characteristic
4C31×#1521C211yesmaximal, characteristic
5C31×#1515C211yesmaximal, characteristic, complemented
6C31×#1515C211yesmaximal, characteristic, complemented
7C31×#1497C211yesmaximal, characteristic, complemented
8C31×#563—22no—
9Q82×C31C411yescharacteristic, complemented
10C31×#557—21no—
11C31×#557C411yescharacteristic, complemented
12C31×#556—21no—
13C31×#556—21no—
14C31×#556C411yescharacteristic, complemented
15C31×#554C411yescharacteristic, complemented
16C31×#552—21no—
17C31×#544—21no—
18C31×#542—21no—
19C31×#541—21no—
20C31×#541—21no—
21C31×#541—22no—
22C31×#533C2211yescharacteristic
23C31×#533C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C31×#432C2211yescharacteristic
26C31×#432C2211yescharacteristic
27C31×#431C2211yescharacteristic
28C31×#383C2211yescharacteristic
29C31×#333—21no—
30C31×#332—21no—
31C31×#330—21no—
32C31×#329—21no—
33C31×#328—21no—
34C31×#328—21no—
35C31×#328—22no—
36C31×#328—21no—
37C31×#327—21no—
38C31×Q8⋊C8—22no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—22no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8C2×C411yescharacteristic
45C31×C4⋊Q8C2×C411yescharacteristic
46C31×C4⋊Q8D411yescharacteristic
47C31×C4⋊Q8D411yescharacteristic
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#126—22no—
54C31×#126—21no—
55C31×#125—22no—
56C31×#125—22no—
57C31×#125—22no—
58C31×#125—22no—
59C31×#125—21no—
60C31×#125—22no—
61C31×#125—21no—
62C31×#125—21no—
63C31×#125—21no—
64C31×#125D411yescharacteristic
65C31×#124—22no—
66C31×#124—21no—
67C31×#124D411yescharacteristic
68C31×#123—22no—
69C31×#123—21no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—21no—
76C31×C22⋊Q8—21no—
77C31×C22⋊Q8D411yescharacteristic
78C31×C22⋊Q8D411yescharacteristic
79C31×C22⋊Q8D411yescharacteristic
80C31×C22⋊Q8D412yes—
81C31×#121D411yescharacteristic
82C4×Q8×C31—22no—
83C4×Q8×C31—21no—
84C4×Q8×C31—21no—
85C4×Q8×C31—21no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—21no—
89C4×Q8×C31—21no—
90C4×Q8×C31C2×C411yescharacteristic
91C4×Q8×C31C2×C411yescharacteristic
92C4×Q8×C31C2×C411yescharacteristic
93C4×Q8×C31D412yes—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31C2×C411yescharacteristic
97C31×#117—22no—
98C31×#117—21no—
99C31×#117—21no—
100C31×#117—21no—
101C31×#117—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—
116C2×Q8×C31—22no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31C22⋊C411yescharacteristic
120C2×Q8×C31C22⋊C411yescharacteristic
121C2×Q8×C31C22⋊C411yescharacteristic
122C2×Q8×C31C22⋊C412yes—
123C2×D4×C31—21no—
124C2×D4×C31C22⋊C411yescharacteristic
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—41no—
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C2×C8×C31—41no—
132C2×C8×C31—41no—
133C2×C8×C31—41no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31C22⋊C411yescharacteristic
159C4⋊C4×C31C22⋊C411yescharacteristic
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—21no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—21no—
169C22⋊C4×C31—21no—
170C22⋊C4×C31—21no—
171C22⋊C4×C31—21no—
172C22⋊C4×C31—21no—
173C22⋊C4×C31—21no—
174C42×C31—22no—
175C42×C31—22no—
176C42×C31—22no—
177C42×C31—21no—
178C42×C31—21no—
179C42×C31—21no—
180C42×C31—22no—
181C42×C31C22×C411yescharacteristic
182C42×C31C2×D411yescharacteristic
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D411yescharacteristic
185#8501C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—21no—
187C23×C31#12011yescharacteristic
188Q8×C31—42no—
189Q8×C31—41no—
190Q8×C31—41no—
191Q8×C31—41no—
192Q8×C31—42no—
193Q8×C31—81no—
194Q8×C31—82no—
195D4×C31—41no—
196D4×C31—81no—
197C2×C4×C31—21no—
198C2×C4×C31—21no—
199C2×C4×C31—21no—
200C2×C4×C31—22no—
201C2×C4×C31—21no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—82no—
211C2×C4×C31—81no—
212C2×C4×C31—81no—
213C2×C4×C31—81no—
214C2×C4×C31C2×C22⋊C411yescharacteristic
215C2×C4×C31C2×C22⋊C411yescharacteristic
216C2×C4×C31C2×C22⋊C411yescharacteristic
217C2×C4×C31#12011yescharacteristic
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12011yescharacteristic
220C8×C31—81no—
221C8×C31—81no—
222C8×C31—81no—
223#3555C2×C3111yescharacteristic
224#1554C2×C3111yescharacteristic, complemented
225#1535C2×C3111yescharacteristic
226#1521C2×C3111yescharacteristic
227#1515C2×C3111yescharacteristic, complemented
228#1515C2×C3111yescharacteristic, complemented
229#1497C2×C3111yescharacteristic, complemented
230C22×C31—21no—
231C22×C31—21no—
232C22×C31—21no—
233C22×C31—41no—
234C22×C31—41no—
235C22×C31—41no—
236C22×C31#37911yescharacteristic
237C4×C31—21no—
238C4×C31—21no—
239C4×C31—21no—
240C4×C31—21no—
241C4×C31—21no—
242C4×C31—21no—
243C4×C31—41no—
244C4×C31—42no—
245C4×C31—41no—
246C4×C31—41no—
247C4×C31—41no—
248C4×C31—41no—
249C4×C31—41no—
250C4×C31—41no—
251C4×C31—42no—
252C4×C31—161no—
253#563—22no—
254Q82C4×C3111yescharacteristic, complemented
255#557—21no—
256#557C4×C3111yescharacteristic, complemented
257#556—21no—
258#556—21no—
259#556C4×C3111yescharacteristic, complemented
260#554C4×C3111yescharacteristic, complemented
261#552—21no—
262#544—21no—
263#542—21no—
264#541—21no—
265#541—21no—
266#541—22no—
267#533C22×C3111yescharacteristic
268#533C22×C3111yescharacteristic
269#529C22×C3111yescharacteristic
270#432C22×C3111yescharacteristic
271#432C22×C3111yescharacteristic
272#431C22×C3111yescharacteristic
273#383C22×C3111yescharacteristic
274#333—21no—
275#332—21no—
276#330—21no—
277#329—21no—
278#328—21no—
279#328—21no—
280#328—22no—
281#328—21no—
282#327—21no—
283Q8⋊C8—22no—
284C2×C31—41no—
285C2×C31—81no—
286C2×C31#179411yescharacteristic
287C2×C31#179611yescharacteristic
288C2×C31#179611yescharacteristic
289C4⋊Q8—22no—
290C4⋊Q8—21no—
291C4⋊Q8—22no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C2×C4×C3111yescharacteristic
295C4⋊Q8C2×C4×C3111yescharacteristic
296C4⋊Q8D4×C3111yescharacteristic
297C4⋊Q8D4×C3111yescharacteristic
298#126—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#126—22no—
304#126—21no—
305#125—22no—
306#125—22no—
307#125—22no—
308#125—22no—
309#125—21no—
310#125—22no—
311#125—21no—
312#125—21no—
313#125—21no—
314#125D4×C3111yescharacteristic
315#124—22no—
316#124—21no—
317#124D4×C3111yescharacteristic
318#123—22no—
319#123—21no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—22no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8D4×C3111yescharacteristic
328C22⋊Q8D4×C3111yescharacteristic
329C22⋊Q8D4×C3111yescharacteristic
330C22⋊Q8D4×C3112yes—
331#121D4×C3111yescharacteristic
332C4×Q8—22no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8C2×C4×C3111yescharacteristic
341C4×Q8C2×C4×C3111yescharacteristic
342C4×Q8C2×C4×C3111yescharacteristic
343C4×Q8D4×C3112yes—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C2×C4×C3111yescharacteristic
347#117—22no—
348#117—21no—
349#117—21no—
350#117—21no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C42C23×C3111yescharacteristic
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360#97—21no—
361#97—21no—
362C4×C8—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366C31#850111yescharacteristic, direct product factor, Sylow
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8C22⋊C4×C3111yescharacteristic
371C2×Q8C22⋊C4×C3111yescharacteristic
372C2×Q8C22⋊C4×C3111yescharacteristic
373C2×Q8C22⋊C4×C3112yes—
374C2×D4—21no—
375C2×D4C22⋊C4×C3111yescharacteristic
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4C2×D4×C3111yescharacteristic
380C22×C4C2×D4×C3111yescharacteristic
381C22×C4C2×D4×C3111yescharacteristic
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4C22⋊C4×C3111yescharacteristic
410C4⋊C4C22⋊C4×C3111yescharacteristic
411C4⋊C4C22⋊C4×C3111yescharacteristic
412C4⋊C4C22⋊C4×C3111yescharacteristic
413C4⋊C4C22⋊C4×C3112yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42C22×C4×C3111yescharacteristic
433C42C2×D4×C3111yescharacteristic
434C42C2×D4×C3111yescharacteristic
435C42C2×D4×C3111yescharacteristic
436C23—21no—
437C23C31×#12011yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—41no—
441Q8—42no—
442Q8—42no—
443Q8—81no—
444Q8—82no—
445D4—41no—
446D4—81no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4C2×C22⋊C4×C3111yescharacteristic
465C2×C4C2×C22⋊C4×C3111yescharacteristic
466C2×C4C2×C22⋊C4×C3111yescharacteristic
467C2×C4C31×#12011yescharacteristic
468C2×C4C31×#12011yescharacteristic
469C2×C4C31×#12011yescharacteristic
470C8—81no—
471C8—81no—
472C8—81no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22C31×#37911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—161no—
496C2—41no—
497C2—81no—
498C2C31×#179411yescharacteristic
499C2C31×#179611yescharacteristic
500C2C31×#179611yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#121D4
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×#1554C2semidirect
5. C31×#1515C2semidirect
6. C31×#1515C2semidirect
7. C31×#1497C2semidirect
9. Q82×C31C4semidirect
11. C31×#557C4semidirect
14. C31×#556C4semidirect
15. C31×#554C4semidirect
185. #8501C31direct (N × H)
224. #1554C2×C31semidirect
227. #1515C2×C31semidirect
228. #1515C2×C31semidirect
229. #1497C2×C31semidirect
254. Q82C4×C31semidirect
256. #557C4×C31semidirect
259. #556C4×C31semidirect
260. #554C4×C31semidirect
366. C31#8501direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35552
C31×#15542
C31×#15352
C31×#15212
C31×#15152
C31×#14972
#850131