Finite Group Databasebeta

← C31×#8528 #21465287

Subgroups — C31×#8528 #21465287

Order 7936.

728 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3545C211yesmaximal, characteristic
2C31×#1958C211yesmaximal, characteristic, complemented
3C31×#1533C211yesmaximal, characteristic, complemented
4C31×#1512C212yesmaximal, complemented
5C31×#1494C212yesmaximal, complemented
6C31×#563C411yescharacteristic, complemented
7C31×#562C412yescomplemented
8C31×H(3,𝔽4)C411yescharacteristic, complemented
9C31×#555—21no—
10C31×#551—21no—
11C31×#551—22no—
12C31×#546—22no—
13C31×#542—22no—
14C31×#538—22no—
15C31×#536—21no—
16C31×#534—21no—
17C31×#529C2212yes—
18C2×C31×#124C2211yescharacteristic
19C31×#432C2212yescomplemented
20C2×C31×#103—21no—
21C2×C31×#102—21no—
22C2×C4×C8×C31C2211yescharacteristic, complemented
23C31×#382C2211yescharacteristic, complemented
24C31×#329—21no—
25C31×#329—21no—
26C31×#328—22no—
27C31×#327—22no—
28C31×Q8⋊C8—22no—
29C31×D4⋊C8—22no—
30C2×#41×C31—21no—
31C22×Q8×C31D411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C22×C8×C31—21no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8—21no—
37C31×#126—22no—
38C31×#126—22no—
39C31×#126—22no—
40C31×#125C2×C412yes—
41C31×#124—21no—
42C31×#124—22no—
43C31×#124—21no—
44C31×#124—22no—
45C31×#124—22no—
46C31×#124C2×C411yescharacteristic
47C31×#124C2×C411yescharacteristic
48C31×#124C2×C412yes—
49C31×#124D411yescharacteristic
50C31×#124D411yescharacteristic
51C31×#123—21no—
52C31×#123—22no—
53C31×#123—22no—
54C31×#123—21no—
55C31×#123—22no—
56C31×#123—22no—
57C31×#123—22no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8D412yes—
63C31×#121—22no—
64C31×#121—22no—
65C31×#121—22no—
66C31×#121—22no—
67C31×#121D412yes—
68C31×#120—21no—
69C31×#120—22no—
70C31×#120—21no—
71C31×#120—22no—
72C4×Q8×C31D412yes—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C4×D4×C31D412yes—
77C31×#117—21no—
78C31×#117—22no—
79C31×#117—22no—
80C2×C4⋊C4×C31—21no—
81C2×C22⋊C4×C31—21no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C42×C31C2311yescharacteristic
85C31×C4⋊C8—22no—
86C31×#103—42no—
87C31×#102—42no—
88C31×C22⋊C8—22no—
89C31×#97—22no—
90C4×C8×C31—22no—
91C31×#95—22no—
92C24×C31—21no—
93C24×C31—21no—
94#41×C31—81no—
95#41×C31—81no—
96C2×Q8×C31—21no—
97C2×Q8×C31—21no—
98C2×Q8×C31—41no—
99C2×Q8×C31—41no—
100C2×Q8×C31C22⋊C412yes—
101C2×Q8×C31C22⋊C411yescharacteristic
102C2×Q8×C31C22⋊C411yescharacteristic
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—21no—
107C2×D4×C31—22no—
108C2×D4×C31—21no—
109C2×D4×C31—41no—
110C2×D4×C31—41no—
111C2×D4×C31—81no—
112C2×D4×C31—81no—
113C2×D4×C31C22⋊C412yes—
114C2×D4×C31C22⋊C411yescharacteristic
115C2×D4×C31C22⋊C411yescharacteristic
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—21no—
120C22×C4×C31—22no—
121C22×C4×C31—41no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D412yes—
124C2×C8×C31—22no—
125C2×C8×C31—41no—
126C2×C8×C31—41no—
127C2×C8×C31—42no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31C22⋊C412yes—
138C4⋊C4×C31C22⋊C412yes—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—21no—
141C22⋊C4×C31—21no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C22⋊C4×C31—22no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—22no—
152C22⋊C4×C31—22no—
153C22⋊C4×C31—22no—
154C42×C31—21no—
155C42×C31—21no—
156C42×C31—22no—
157C42×C31C22×C411yescharacteristic
158C42×C31C2×D412yes—
159C42×C31C2×D411yescharacteristic
160#8528C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—21no—
162C23×C31—22no—
163C23×C31—21no—
164C23×C31—41no—
165C23×C31—41no—
166C23×C31—82no—
167C23×C31—81no—
168C23×C31—82no—
169C23×C31#12011yescharacteristic
170Q8×C31—41no—
171Q8×C31—41no—
172Q8×C31—42no—
173Q8×C31—81no—
174Q8×C31—81no—
175D4×C31—41no—
176D4×C31—41no—
177D4×C31—42no—
178D4×C31—42no—
179D4×C31—42no—
180D4×C31—81no—
181D4×C31—82no—
182D4×C31—82no—
183D4×C31—82no—
184D4×C31—81no—
185C2×C4×C31—21no—
186C2×C4×C31—21no—
187C2×C4×C31—22no—
188C2×C4×C31—21no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—21no—
192C2×C4×C31—42no—
193C2×C4×C31—41no—
194C2×C4×C31—41no—
195C2×C4×C31—82no—
196C2×C4×C31—82no—
197C2×C4×C31—81no—
198C2×C4×C31—82no—
199C2×C4×C31—82no—
200C2×C4×C31—81no—
201C2×C4×C31C2×C22⋊C412yes—
202C2×C4×C31C2×C22⋊C411yescharacteristic
203C2×C4×C31#12012yes—
204C2×C4×C31#12011yescharacteristic
205C8×C31—42no—
206C8×C31—82no—
207#3545C2×C3111yescharacteristic
208#1958C2×C3111yescharacteristic, complemented
209#1533C2×C3111yescharacteristic, complemented
210#1512C2×C3112yescomplemented
211#1494C2×C3112yescomplemented
212C22×C31—22no—
213C22×C31—42no—
214C22×C31—41no—
215C22×C31—41no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—161no—
221C22×C31—161no—
222C22×C31—162no—
223C22×C31#37911yescharacteristic
224C22×C31#45311yescharacteristic
225C22×C31#45511yescharacteristic
226C4×C31—21no—
227C4×C31—21no—
228C4×C31—22no—
229C4×C31—22no—
230C4×C31—41no—
231C4×C31—41no—
232C4×C31—42no—
233C4×C31—42no—
234C4×C31—42no—
235C4×C31—161no—
236#563C4×C3111yescharacteristic, complemented
237#562C4×C3112yescomplemented
238H(3,𝔽4)C4×C3111yescharacteristic, complemented
239#555—21no—
240#551—21no—
241#551—22no—
242#546—22no—
243#542—22no—
244#538—22no—
245#536—21no—
246#534—21no—
247#529C22×C3112yes—
248C2×#124C22×C3111yescharacteristic
249#432C22×C3112yescomplemented
250C2×#103—21no—
251C2×#102—21no—
252C2×C4×C8C22×C3111yescharacteristic, complemented
253#382C22×C3111yescharacteristic, complemented
254#329—21no—
255#329—21no—
256#328—22no—
257#327—22no—
258Q8⋊C8—22no—
259D4⋊C8—22no—
260C2×C31—21no—
261C2×C31—21no—
262C2×C31—82no—
263C2×C31—81no—
264C2×C31—81no—
265C2×C31#179211yescharacteristic
266C2×C31#179412yes—
267C2×#41—21no—
268C22×Q8D4×C3111yescharacteristic
269C22×D4—21no—
270C22×D4D4×C3111yescharacteristic
271C22×C8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274#126—22no—
275#126—22no—
276#126—22no—
277#125C2×C4×C3112yes—
278#124—21no—
279#124—22no—
280#124—21no—
281#124—22no—
282#124—22no—
283#124C2×C4×C3111yescharacteristic
284#124C2×C4×C3111yescharacteristic
285#124C2×C4×C3112yes—
286#124D4×C3111yescharacteristic
287#124D4×C3111yescharacteristic
288#123—21no—
289#123—22no—
290#123—22no—
291#123—21no—
292#123—22no—
293#123—22no—
294#123—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8D4×C3112yes—
300#121—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121D4×C3112yes—
305#120—21no—
306#120—22no—
307#120—21no—
308#120—22no—
309C4×Q8D4×C3112yes—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4D4×C3112yes—
314#117—21no—
315#117—22no—
316#117—22no—
317C2×C4⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C42C23×C3111yescharacteristic
322C4⋊C8—22no—
323#103—42no—
324#102—42no—
325C22⋊C8—22no—
326#97—22no—
327C4×C8—22no—
328#95—22no—
329C31#852811yescharacteristic, direct product factor, Sylow
330C24—21no—
331C24—21no—
332#41—81no—
333#41—81no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8C22⋊C4×C3112yes—
339C2×Q8C22⋊C4×C3111yescharacteristic
340C2×Q8C22⋊C4×C3111yescharacteristic
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—81no—
350C2×D4—81no—
351C2×D4C22⋊C4×C3112yes—
352C2×D4C22⋊C4×C3111yescharacteristic
353C2×D4C22⋊C4×C3111yescharacteristic
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4C2×D4×C3111yescharacteristic
361C22×C4C2×D4×C3112yes—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4C22⋊C4×C3112yes—
376C4⋊C4C22⋊C4×C3112yes—
377C22⋊C4—22no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C42—21no—
393C42—21no—
394C42—22no—
395C42C22×C4×C3111yescharacteristic
396C42C2×D4×C3112yes—
397C42C2×D4×C3111yescharacteristic
398C23—21no—
399C23—22no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—82no—
404C23—81no—
405C23—82no—
406C23C31×#12011yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—42no—
410Q8—81no—
411Q8—81no—
412D4—41no—
413D4—41no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4C2×C22⋊C4×C3112yes—
439C2×C4C2×C22⋊C4×C3111yescharacteristic
440C2×C4C31×#12012yes—
441C2×C4C31×#12011yescharacteristic
442C8—42no—
443C8—82no—
444C22—22no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—161no—
453C22—161no—
454C22—162no—
455C22C31×#37911yescharacteristic
456C22C31×#45311yescharacteristic
457C22C31×#45511yescharacteristic
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—161no—
468C2—21no—
469C2—21no—
470C2—81no—
471C2—82no—
472C2—81no—
473C2C31×#179211yescharacteristic
474C2C31×#179412yes—
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3545C2
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×#1958C2semidirect
3. C31×#1533C2semidirect
4. C31×#1512C2semidirect
5. C31×#1494C2semidirect
6. C31×#563C4semidirect
7. C31×#562C4semidirect
8. C31×H(3,𝔽4)C4semidirect
19. C31×#432C22semidirect
22. C2×C4×C8×C31C22semidirect
23. C31×#382C22semidirect
160. #8528C31direct (N × H)
208. #1958C2×C31semidirect
209. #1533C2×C31semidirect
210. #1512C2×C31semidirect
211. #1494C2×C31semidirect
236. #563C4×C31semidirect
237. #562C4×C31semidirect
238. H(3,𝔽4)C4×C31semidirect
249. #432C22×C31semidirect
252. C2×C4×C8C22×C31semidirect
253. #382C22×C31semidirect
329. C31#8528direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35452
C31×#19582
C31×#15332
C31×#15122
C31×#14942
#852831