Finite Group Databasebeta

← C31×#8449 #21465208

Subgroups — C31×#8449 #21465208

Order 7936.

788 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#556C211yesmaximal, characteristic
2C31×#1990C211yesmaximal, characteristic, complemented
3C31×#1803C211yesmaximal, characteristic, complemented
4C31×#1802C211yesmaximal, characteristic
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×Q8⋊C8C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic
8C31×#556—22no—
9C31×#556—22no—
10C31×#556—22no—
11C31×#556C414yescomplemented
12C2×C31×C4⋊Q8C2211yescharacteristic
13C2×C31×#126—22no—
14C2×C31×#124—21no—
15C2×C31×C22⋊Q8—21no—
16C2×C4×Q8×C31—21no—
17C2×C4×Q8×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C2×C31×#117—21no—
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×#103—21no—
23C2×C31×#102—21no—
24C2×C4×C8×C31C2211yescharacteristic
25C31×#384C2211yescharacteristic
26C31×#340—21no—
27C31×#340—21no—
28C31×#340—22no—
29C31×#332—22no—
30C31×Q8⋊C8—22no—
31C31×D4⋊C8—22no—
32C22×Q8×C31—21no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C22×C8×C31—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8C2×C412yes—
42C31×C4⋊Q8D412yes—
43C31×#126—28no—
44C31×#126—28no—
45C31×#124—24no—
46C31×#124—24no—
47C31×C22⋊Q8—24no—
48C31×C22⋊Q8—24no—
49C4×Q8×C31—24no—
50C4×Q8×C31—24no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31C2×C412yes—
54C4×Q8×C31D412yes—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31C2×C412yes—
58C4×D4×C31D412yes—
59C31×#117—24no—
60C31×#117—24no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31D411yescharacteristic
68C2×C4⋊C4×C31D411yescharacteristic
69C2×C4⋊C4×C31D412yes—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—21no—
73C2×C42×C31—22no—
74C2×C42×C31—21no—
75C2×C42×C31C2311yescharacteristic
76C31×C4⋊C8—21no—
77C31×C4⋊C8—21no—
78C31×C4⋊C8—21no—
79C31×C4⋊C8—21no—
80C31×#103—42no—
81C31×#102—42no—
82C4×C8×C31—21no—
83C4×C8×C31—21no—
84C24×C31—21no—
85C2×Q8×C31—22no—
86C2×Q8×C31—21no—
87C2×Q8×C31—22no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31—41no—
93C2×Q8×C31—41no—
94C2×Q8×C31—41no—
95C2×Q8×C31—41no—
96C2×Q8×C31C22⋊C412yes—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31—21no—
102C2×D4×C31C22⋊C412yes—
103C22×C4×C31—21no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C2×C8×C31—41no—
120C2×C8×C31—41no—
121C2×C8×C31—41no—
122C2×C8×C31—41no—
123C2×C8×C31—41no—
124C2×C8×C31—41no—
125C2×C8×C31—41no—
126C4⋊C4×C31—28no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—42no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31—41no—
138C4⋊C4×C31—41no—
139C4⋊C4×C31C22⋊C412yes—
140C4⋊C4×C31C22⋊C412yes—
141C4⋊C4×C31C22⋊C414yes—
142C22⋊C4×C31—28no—
143C22⋊C4×C31—28no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C42×C31—28no—
147C42×C31—22no—
148C42×C31—22no—
149C42×C31C22×C411yescharacteristic
150C42×C31C2×D411yescharacteristic
151C42×C31C2×D411yescharacteristic
152C42×C31C2×D411yescharacteristic
153#8449C3111yesmaximal, characteristic, direct product factor, Sylow
154C23×C31—22no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31—41no—
159C23×C31—41no—
160C23×C31—41no—
161C23×C31#12011yescharacteristic
162Q8×C31—22no—
163Q8×C31—22no—
164Q8×C31—42no—
165Q8×C31—42no—
166Q8×C31—41no—
167Q8×C31—42no—
168Q8×C31—41no—
169Q8×C31—42no—
170D4×C31—22no—
171D4×C31—22no—
172D4×C31—41no—
173D4×C31—41no—
174C2×C4×C31—21no—
175C2×C4×C31—21no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—24no—
180C2×C4×C31—41no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—41no—
184C2×C4×C31—41no—
185C2×C4×C31—42no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—42no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—42no—
192C2×C4×C31—81no—
193C2×C4×C31—81no—
194C2×C4×C31—81no—
195C2×C4×C31C2×C22⋊C411yescharacteristic
196C2×C4×C31C2×C22⋊C411yescharacteristic
197C2×C4×C31C2×C22⋊C411yescharacteristic
198C2×C4×C31#12011yescharacteristic
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31#12011yescharacteristic
201C2×C4×C31#13511yescharacteristic
202C2×C4×C31#13511yescharacteristic
203C2×C4×C31#13511yescharacteristic
204C2×C4×C31#13511yescharacteristic
205C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
206C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
207C2×C4×C31#13711yescharacteristic
208C2×C4×C31#13711yescharacteristic
209C8×C31—42no—
210C8×C31—81no—
211C8×C31—81no—
212C2×#556C2×C3111yescharacteristic
213#1990C2×C3111yescharacteristic, complemented
214#1803C2×C3111yescharacteristic, complemented
215#1802C2×C3111yescharacteristic
216C2×#332C2×C3111yescharacteristic, complemented
217C2×Q8⋊C8C2×C3111yescharacteristic, complemented
218C2×D4⋊C8C2×C3111yescharacteristic
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—42no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31—81no—
226C22×C31#37911yescharacteristic
227C22×C31#44911yescharacteristic
228C22×C31#44911yescharacteristic
229C22×C31#45211yescharacteristic
230C22×C31#45211yescharacteristic
231C22×C31#45411yescharacteristic
232C22×C31#45611yescharacteristic
233C4×C31—21no—
234C4×C31—21no—
235C4×C31—42no—
236C4×C31—42no—
237C4×C31—42no—
238C4×C31—44no—
239C4×C31—161no—
240C4×C31#41611yescharacteristic
241C4×C31#41611yescharacteristic
242C4×C31#41711yescharacteristic
243C4×C31#41711yescharacteristic
244C4×C31#44911yescharacteristic
245C4×C31#44911yescharacteristic
246C4×C31#45211yescharacteristic
247C4×C31#45211yescharacteristic
248#556—22no—
249#556—22no—
250#556—22no—
251#556C4×C3114yescomplemented
252C2×C4⋊Q8C22×C3111yescharacteristic
253C2×#126—22no—
254C2×#124—21no—
255C2×C22⋊Q8—21no—
256C2×C4×Q8—21no—
257C2×C4×Q8C22×C3111yescharacteristic
258C2×C4×D4C22×C3111yescharacteristic
259C2×#117—21no—
260C2×C4⋊C8C22×C3111yescharacteristic
261C2×C4⋊C8C22×C3111yescharacteristic
262C2×#103—21no—
263C2×#102—21no—
264C2×C4×C8C22×C3111yescharacteristic
265#384C22×C3111yescharacteristic
266#340—21no—
267#340—21no—
268#340—22no—
269#332—22no—
270Q8⋊C8—22no—
271D4⋊C8—22no—
272C2×C31—82no—
273C2×C31#162811yescharacteristic
274C2×C31#163811yescharacteristic
275C2×C31#165011yescharacteristic
276C2×C31#165211yescharacteristic
277C2×C31#178811yescharacteristic
278C2×C31#178811yescharacteristic
279C2×C31#179311yescharacteristic
280C22×Q8—21no—
281C22×Q8D4×C3111yescharacteristic
282C22×D4D4×C3111yescharacteristic
283C23×C4—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8C2×C4×C3112yes—
290C4⋊Q8D4×C3112yes—
291#126—28no—
292#126—28no—
293#124—24no—
294#124—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8C2×C4×C3112yes—
302C4×Q8D4×C3112yes—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4C2×C4×C3112yes—
306C4×D4D4×C3112yes—
307#117—24no—
308#117—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4D4×C3111yescharacteristic
316C2×C4⋊C4D4×C3111yescharacteristic
317C2×C4⋊C4D4×C3112yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42C23×C3111yescharacteristic
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328#103—42no—
329#102—42no—
330C4×C8—21no—
331C4×C8—21no—
332C31#844911yescharacteristic, direct product factor, Sylow
333C24—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8C22⋊C4×C3112yes—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4C22⋊C4×C3112yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C2×D4×C3111yescharacteristic
364C22×C4C2×D4×C3111yescharacteristic
365C22×C4C2×D4×C3111yescharacteristic
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C22⋊C4×C3112yes—
389C4⋊C4C22⋊C4×C3112yes—
390C4⋊C4C22⋊C4×C3114yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C42—28no—
396C42—22no—
397C42—22no—
398C42C22×C4×C3111yescharacteristic
399C42C2×D4×C3111yescharacteristic
400C42C2×D4×C3111yescharacteristic
401C42C2×D4×C3111yescharacteristic
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23C31×#12011yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—42no—
413Q8—42no—
414Q8—41no—
415Q8—42no—
416Q8—41no—
417Q8—42no—
418D4—22no—
419D4—22no—
420D4—41no—
421D4—41no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4C2×C22⋊C4×C3111yescharacteristic
444C2×C4C2×C22⋊C4×C3111yescharacteristic
445C2×C4C2×C22⋊C4×C3111yescharacteristic
446C2×C4C31×#12011yescharacteristic
447C2×C4C31×#12011yescharacteristic
448C2×C4C31×#12011yescharacteristic
449C2×C4C31×#13511yescharacteristic
450C2×C4C31×#13511yescharacteristic
451C2×C4C31×#13511yescharacteristic
452C2×C4C31×#13511yescharacteristic
453C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
455C2×C4C31×#13711yescharacteristic
456C2×C4C31×#13711yescharacteristic
457C8—42no—
458C8—81no—
459C8—81no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22C31×#37911yescharacteristic
468C22C31×#44911yescharacteristic
469C22C31×#44911yescharacteristic
470C22C31×#45211yescharacteristic
471C22C31×#45211yescharacteristic
472C22C31×#45411yescharacteristic
473C22C31×#45611yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—161no—
481C4C31×#41611yescharacteristic
482C4C31×#41611yescharacteristic
483C4C31×#41711yescharacteristic
484C4C31×#41711yescharacteristic
485C4C31×#44911yescharacteristic
486C4C31×#44911yescharacteristic
487C4C31×#45211yescharacteristic
488C4C31×#45211yescharacteristic
489C2—82no—
490C2C31×#162811yescharacteristic
491C2C31×#163811yescharacteristic
492C2C31×#165011yescharacteristic
493C2C31×#165211yescharacteristic
494C2C31×#178811yescharacteristic
495C2C31×#178811yescharacteristic
496C2C31×#179311yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#1990C2semidirect
3. C31×#1803C2semidirect
5. C2×C31×#332C2semidirect
6. C2×C31×Q8⋊C8C2semidirect
11. C31×#556C4semidirect
153. #8449C31direct (N × H)
213. #1990C2×C31semidirect
214. #1803C2×C31semidirect
216. C2×#332C2×C31semidirect
217. C2×Q8⋊C8C2×C31semidirect
251. #556C4×C31semidirect
332. C31#8449direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5562
C31×#19902
C31×#18032
C31×#18022
C2×C31×#3322
C2×C31×Q8⋊C82
C2×C31×D4⋊C82
#844931