Finite Group Databasebeta

← C31×#18345 #21475104

Subgroups — C31×#18345 #21475104

Order 7936.

888 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×#3461C211yesmaximal, characteristic
2C31×#3239C212yesmaximal, complemented
3C31×#2568C211yesmaximal, characteristic, complemented
4C31×#1617C212yesmaximal, complemented
5C31×#1616C212yesmaximal
6C31×#1615C214yesmaximal, complemented
7C31×#1509C212yesmaximal, complemented
8C31×#1467C211yesmaximal, characteristic, complemented
9C31×#544C2211yescharacteristic
10C31×#542—22no—
11C31×#541—22no—
12C31×#539—21no—
13C31×#539C2212yes—
14C31×#537C2211yescharacteristic
15C31×#536—21no—
16C31×#533—22no—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C31×#529—22no—
19C31×#529—21no—
20C31×#529C2212yes—
21C4×#41×C31—21no—
22C2×C4×Q8×C31C2211yescharacteristic
23C31×#501—22no—
24C31×#489—22no—
25C31×#484C2214yes—
26C31×#481C2214yes—
27C31×#475—22no—
28C31×#474—22no—
29C31×#432C2212yes—
30C31×#400C2212yes—
31C31×#400C2212yes—
32C31×#391—21no—
33C31×#391—21no—
34C31×#391—21no—
35C31×#382—21no—
36C4×C4⋊C4×C31C2211yescharacteristic
37C31×#328C2214yes—
38C31×#327C2214yes—
39C31×#324C2212yes—
40C31×#324C2212yes—
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8—22no—
46C31×C4⋊Q8—22no—
47C31×C4⋊Q8—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#125—21no—
53C31×#125—22no—
54C31×#125—22no—
55C31×#125—22no—
56C31×#125D412yes—
57C31×#124—22no—
58C31×#124—21no—
59C31×#124D412yes—
60C31×#123—22no—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8C2312yes—
71C31×#121—22no—
72C31×#121—21no—
73C31×#121C2312yes—
74C4×Q8×C31—21no—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31—22no—
78C4×Q8×C31—22no—
79C4×Q8×C31D412yes—
80C4×D4×C31—21no—
81C4×D4×C31—21no—
82C4×D4×C31—21no—
83C4×D4×C31—22no—
84C4×D4×C31—22no—
85C4×D4×C31D412yes—
86C31×#117—22no—
87C31×#117—22no—
88C31×#117—21no—
89C31×#117—22no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C4⋊C4×C31C2312yes—
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C4⋊C4×C31C2311yescharacteristic
96C2×C42×C31—22no—
97C2×C42×C31—21no—
98C2×C42×C31D411yescharacteristic
99C2×C42×C31D411yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#106—24no—
103C31×C4⋊C8—22no—
104C31×#103—24no—
105C31×#102—24no—
106C31×C22⋊C8C2314yes—
107C31×#97—22no—
108C31×#95—22no—
109C31×#95—22no—
110C31×#95—22no—
111C31×#95—22no—
112C31×#95C2312yes—
113#41×C31—42no—
114#41×C31—42no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—41no—
120C2×Q8×C31—44no—
121C2×Q8×C31—41no—
122C2×Q8×C31—41no—
123C2×Q8×C31C2×D412yes—
124C2×D4×C31—21no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31C2×D412yes—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2411yescharacteristic
140C2×C8×C31—24no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—42no—
159C4⋊C4×C31C2×D412yes—
160C4⋊C4×C31C2×D412yes—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C42×C31—22no—
171C42×C31—22no—
172C42×C31—21no—
173C42×C31—22no—
174C42×C31—21no—
175C42×C31—21no—
176C42×C31—21no—
177C42×C31—42no—
178C42×C31C2×D412yes—
179C42×C31C2×D412yes—
180#18345C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—22no—
182C23×C31C22×D411yescharacteristic
183Q8×C31—44no—
184Q8×C31—42no—
185Q8×C31—42no—
186Q8×C31—82no—
187Q8×C31—81no—
188Q8×C31—82no—
189D4×C31—42no—
190D4×C31—42no—
191D4×C31—42no—
192D4×C31—42no—
193D4×C31—81no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—21no—
198C2×C4×C31—22no—
199C2×C4×C31—21no—
200C2×C4×C31—22no—
201C2×C4×C31—22no—
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—41no—
208C2×C4×C31—44no—
209C2×C4×C31—44no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—44no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—82no—
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12012yes—
220C2×C4×C31#12011yescharacteristic
221C2×C4×C31#13712yes—
222C2×C4×C31C22×D412yes—
223C8×C31—44no—
224#3461C2×C3111yescharacteristic
225#3239C2×C3112yescomplemented
226#2568C2×C3111yescharacteristic, complemented
227#1617C2×C3112yescomplemented
228#1616C2×C3112yes—
229#1615C2×C3114yescomplemented
230#1509C2×C3112yescomplemented
231#1467C2×C3111yescharacteristic, complemented
232C22×C31—21no—
233C22×C31—21no—
234C22×C31—42no—
235C22×C31—42no—
236C22×C31—42no—
237C22×C31C2×#12011yescharacteristic
238C22×C31C2×#13711yescharacteristic
239C22×C31#57511yescharacteristic
240C4×C31—22no—
241C4×C31—22no—
242C4×C31—21no—
243C4×C31—22no—
244C4×C31—41no—
245C4×C31—42no—
246C4×C31—42no—
247C4×C31—41no—
248C4×C31—41no—
249C4×C31—42no—
250C4×C31—82no—
251C4×C31—82no—
252C4×C31#45412yes—
253#544C22×C3111yescharacteristic
254#542—22no—
255#541—22no—
256#539—21no—
257#539C22×C3112yes—
258#537C22×C3111yescharacteristic
259#536—21no—
260#533—22no—
261C2×C4⋊Q8C22×C3111yescharacteristic
262#529—22no—
263#529—21no—
264#529C22×C3112yes—
265C4×#41—21no—
266C2×C4×Q8C22×C3111yescharacteristic
267#501—22no—
268#489—22no—
269#484C22×C3114yes—
270#481C22×C3114yes—
271#475—22no—
272#474—22no—
273#432C22×C3112yes—
274#400C22×C3112yes—
275#400C22×C3112yes—
276#391—21no—
277#391—21no—
278#391—21no—
279#382—21no—
280C4×C4⋊C4C22×C3111yescharacteristic
281#328C22×C3114yes—
282#327C22×C3114yes—
283#324C22×C3112yes—
284#324C22×C3112yes—
285C2×C31—21no—
286C2×C31—21no—
287C2×C31—82no—
288C2×C31#300711yescharacteristic
289C2×C31C2×#45411yescharacteristic
290C2×C31#302811yescharacteristic
291C2×#41—21no—
292C22×Q8—21no—
293C22×Q8D4×C3111yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298#126—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#125—21no—
303#125—22no—
304#125—22no—
305#125—22no—
306#125D4×C3112yes—
307#124—22no—
308#124—21no—
309#124D4×C3112yes—
310#123—22no—
311#123—22no—
312#123—22no—
313#123—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—21no—
319C22⋊Q8—22no—
320C22⋊Q8C23×C3112yes—
321#121—22no—
322#121—21no—
323#121C23×C3112yes—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8D4×C3112yes—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4D4×C3112yes—
336#117—22no—
337#117—22no—
338#117—21no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4D4×C3111yescharacteristic
343C2×C4⋊C4C23×C3112yes—
344C2×C4⋊C4C23×C3111yescharacteristic
345C2×C4⋊C4C23×C3111yescharacteristic
346C2×C42—22no—
347C2×C42—21no—
348C2×C42D4×C3111yescharacteristic
349C2×C42D4×C3111yescharacteristic
350C2×C42C23×C3111yescharacteristic
351#107—24no—
352#106—24no—
353C4⋊C8—22no—
354#103—24no—
355#102—24no—
356C22⋊C8C23×C3114yes—
357#97—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C23×C3112yes—
363C31#1834511yescharacteristic, direct product factor, Sylow
364#41—42no—
365#41—42no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8C2×D4×C3112yes—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4C2×D4×C3112yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387C22×C4C2×D4×C3111yescharacteristic
388C22×C4C2×D4×C3111yescharacteristic
389C22×C4C2×D4×C3111yescharacteristic
390C22×C4C24×C3111yescharacteristic
391C2×C8—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×D4×C3112yes—
411C4⋊C4C2×D4×C3112yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—42no—
429C42C2×D4×C3112yes—
430C42C2×D4×C3112yes—
431C23—22no—
432C23C22×D4×C3111yescharacteristic
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8—81no—
438Q8—82no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—81no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4C31×#12011yescharacteristic
469C2×C4C31×#12012yes—
470C2×C4C31×#12011yescharacteristic
471C2×C4C31×#13712yes—
472C2×C4C22×D4×C3112yes—
473C8—44no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22C2×C31×#12011yescharacteristic
480C22C2×C31×#13711yescharacteristic
481C22C31×#57511yescharacteristic
482C4—22no—
483C4—22no—
484C4—21no—
485C4—22no—
486C4—41no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—82no—
493C4—82no—
494C4C31×#45412yes—
495C2—21no—
496C2—21no—
497C2—82no—
498C2C31×#300711yescharacteristic
499C2C2×C31×#45411yescharacteristic
500C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#537C22
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×#3239C2semidirect
3. C31×#2568C2semidirect
4. C31×#1617C2semidirect
6. C31×#1615C2semidirect
7. C31×#1509C2semidirect
8. C31×#1467C2semidirect
180. #18345C31direct (N × H)
225. #3239C2×C31semidirect
226. #2568C2×C31semidirect
227. #1617C2×C31semidirect
229. #1615C2×C31semidirect
230. #1509C2×C31semidirect
231. #1467C2×C31semidirect
363. C31#18345direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34612
C31×#32392
C31×#25682
C31×#16172
C31×#16162
C31×#16152
C31×#15092
C31×#14672
#1834531