Finite Group Databasebeta

← C31×#12781 #21469540

Subgroups — C31×#12781 #21469540

Order 7936.

758 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3545C211yesmaximal, characteristic
2C31×#2133C212yesmaximal, complemented
3C2×C31×#355C211yesmaximal, characteristic, complemented
4C31×#1912C212yesmaximal, complemented
5C31×#1888C211yesmaximal, characteristic, complemented
6C31×#563C411yescharacteristic, complemented
7C31×#562—21no—
8C31×H(3,𝔽4)C411yescharacteristic, complemented
9C31×#555C412yescomplemented
10C31×#551—21no—
11C31×#551—22no—
12C31×#546—22no—
13C31×#542—22no—
14C31×#538—22no—
15C31×#536C2212yes—
16C31×#534—21no—
17C31×#529—21no—
18C2×C31×#124C2211yescharacteristic
19C31×#519C2211yescharacteristic, complemented
20C31×#421—22no—
21C31×#418—21no—
22C2×C31×#101C2212yescomplemented
23C2×C31×#101C2211yescharacteristic, complemented
24C31×#356—22no—
25C31×#355—21no—
26C31×#355—21no—
27C31×#355—22no—
28C31×#339—21no—
29C31×H(3,ℤ/4)—22no—
30C2×#41×C31D412yes—
31C22×Q8×C31C2311yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31D411yescharacteristic
34C2×M16×C31—21no—
35C2×M16×C31—22no—
36C31×C4⋊Q8C2×C412yes—
37C31×C4⋊Q8D412yes—
38C31×#126—22no—
39C31×#126—22no—
40C31×#126—22no—
41C31×#125—21no—
42C31×#124—22no—
43C31×#124—22no—
44C31×#124—21no—
45C31×#124—22no—
46C31×#124C2×C411yescharacteristic
47C31×#124C2×C411yescharacteristic
48C31×#124C2×C412yes—
49C31×#124D411yescharacteristic
50C31×#124D411yescharacteristic
51C31×#124D412yes—
52C31×#123—21no—
53C31×#123—22no—
54C31×#123—22no—
55C31×#123—22no—
56C31×#123—21no—
57C31×#123—22no—
58C31×#123—22no—
59C31×C22⋊Q8—21no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×#121—21no—
65C31×#121—22no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#120—21no—
70C31×#120—22no—
71C31×#120—21no—
72C31×#120—22no—
73C4×Q8×C31—21no—
74C4×Q8×C31—42no—
75C4×D4×C31—21no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C4×D4×C31—22no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117—22no—
82C31×#117—21no—
83C31×#117D412yes—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—22no—
87C2×C42×C31D411yescharacteristic
88C31×#104—44no—
89C31×#102—42no—
90C31×#101—22no—
91C31×#101—22no—
92C31×#101—22no—
93C24×C31—21no—
94C24×C31—21no—
95#41×C31—42no—
96#41×C31—42no—
97C2×Q8×C31—21no—
98C2×Q8×C31—41no—
99C2×Q8×C31—41no—
100C2×Q8×C31C22⋊C412yes—
101C2×Q8×C31C22×C411yescharacteristic
102C2×Q8×C31C2×D411yescharacteristic
103C2×Q8×C31C2×D412yes—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—21no—
108C2×D4×C31—22no—
109C2×D4×C31—21no—
110C2×D4×C31—41no—
111C2×D4×C31—41no—
112C2×D4×C31—81no—
113C2×D4×C31—81no—
114C2×D4×C31C22⋊C412yes—
115C2×D4×C31C22⋊C411yescharacteristic
116C2×D4×C31C22⋊C411yescharacteristic
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—22no—
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D412yes—
124M16×C31—42no—
125M16×C31—41no—
126M16×C31—41no—
127M16×C31—42no—
128C2×C8×C31—41no—
129C2×C8×C31—42no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
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—42no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—21no—
144C22⋊C4×C31—21no—
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—
154C22⋊C4×C31—22no—
155C22⋊C4×C31—22no—
156C22⋊C4×C31—41no—
157C42×C31—22no—
158C42×C31—21no—
159C42×C31—24no—
160C42×C31—41no—
161C42×C31C22⋊C411yescharacteristic
162C42×C31C22⋊C411yescharacteristic
163C42×C31C22⋊C412yes—
164C42×C31C22⋊C412yes—
165#12781C3111yesmaximal, characteristic, direct product factor, Sylow
166C23×C31—21no—
167C23×C31—21no—
168C23×C31—22no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—81no—
172C23×C31—82no—
173C23×C31—82no—
174C23×C31#12011yescharacteristic
175Q8×C31—41no—
176Q8×C31—41no—
177Q8×C31—42no—
178Q8×C31—42no—
179Q8×C31—81no—
180D4×C31—41no—
181D4×C31—41no—
182D4×C31—42no—
183D4×C31—42no—
184D4×C31—42no—
185D4×C31—81no—
186D4×C31—82no—
187D4×C31—82no—
188D4×C31—81no—
189D4×C31—82no—
190C2×C4×C31—22no—
191C2×C4×C31—21no—
192C2×C4×C31—22no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—22no—
196C2×C4×C31—21no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—42no—
200C2×C4×C31—82no—
201C2×C4×C31—82no—
202C2×C4×C31—81no—
203C2×C4×C31—82no—
204C2×C4×C31—81no—
205C2×C4×C31C2×C22⋊C412yes—
206C2×C4×C31C2×C22⋊C411yescharacteristic
207C2×C4×C31#12011yescharacteristic
208C2×C4×C31#12012yes—
209C8×C31—82no—
210C8×C31—81no—
211#3545C2×C3111yescharacteristic
212#2133C2×C3112yescomplemented
213C2×#355C2×C3111yescharacteristic, complemented
214#1912C2×C3112yescomplemented
215#1888C2×C3111yescharacteristic, complemented
216C22×C31—41no—
217C22×C31—42no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—42no—
222C22×C31—41no—
223C22×C31—41no—
224C22×C31—161no—
225C22×C31—161no—
226C22×C31—162no—
227C22×C31#37911yescharacteristic
228C22×C31B(4,𝔽2)11yescharacteristic
229C22×C31#45811yescharacteristic
230C4×C31—22no—
231C4×C31—21no—
232C4×C31—21no—
233C4×C31—22no—
234C4×C31—41no—
235C4×C31—42no—
236C4×C31—42no—
237C4×C31—41no—
238C4×C31—42no—
239C4×C31—82no—
240#563C4×C3111yescharacteristic, complemented
241#562—21no—
242H(3,𝔽4)C4×C3111yescharacteristic, complemented
243#555C4×C3112yescomplemented
244#551—21no—
245#551—22no—
246#546—22no—
247#542—22no—
248#538—22no—
249#536C22×C3112yes—
250#534—21no—
251#529—21no—
252C2×#124C22×C3111yescharacteristic
253#519C22×C3111yescharacteristic, complemented
254#421—22no—
255#418—21no—
256C2×#101C22×C3112yescomplemented
257C2×#101C22×C3111yescharacteristic, complemented
258#356—22no—
259#355—21no—
260#355—21no—
261#355—22no—
262#339—21no—
263H(3,ℤ/4)—22no—
264C2×C31—21no—
265C2×C31—21no—
266C2×C31—21no—
267C2×C31—81no—
268C2×C31—82no—
269C2×C31—81no—
270C2×C31#179011yescharacteristic
271C2×#41D4×C3112yes—
272C22×Q8C23×C3111yescharacteristic
273C22×D4—21no—
274C22×D4D4×C3111yescharacteristic
275C2×M16—22no—
276C2×M16—21no—
277C4⋊Q8C2×C4×C3112yes—
278C4⋊Q8D4×C3112yes—
279#126—22no—
280#126—22no—
281#126—22no—
282#125—21no—
283#124—22no—
284#124—22no—
285#124—21no—
286#124—22no—
287#124C2×C4×C3111yescharacteristic
288#124C2×C4×C3111yescharacteristic
289#124C2×C4×C3112yes—
290#124D4×C3111yescharacteristic
291#124D4×C3111yescharacteristic
292#124D4×C3112yes—
293#123—21no—
294#123—22no—
295#123—22no—
296#123—22no—
297#123—21no—
298#123—22no—
299#123—22no—
300C22⋊Q8—21no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305#121—21no—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—22no—
310#120—21no—
311#120—22no—
312#120—21no—
313#120—22no—
314C4×Q8—21no—
315C4×Q8—42no—
316C4×D4—21no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—21no—
324#117D4×C3112yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42D4×C3111yescharacteristic
329#104—44no—
330#102—42no—
331#101—22no—
332#101—22no—
333#101—22no—
334C31#1278111yescharacteristic, direct product factor, Sylow
335C24—21no—
336C24—21no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C22⋊C4×C3112yes—
343C2×Q8C22×C4×C3111yescharacteristic
344C2×Q8C2×D4×C3111yescharacteristic
345C2×Q8C2×D4×C3112yes—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D4C22⋊C4×C3112yes—
357C2×D4C22⋊C4×C3111yescharacteristic
358C2×D4C22⋊C4×C3111yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4C2×D4×C3111yescharacteristic
365C22×C4C2×D4×C3112yes—
366M16—42no—
367M16—41no—
368M16—41no—
369M16—42no—
370C2×C8—41no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—41no—
399C42—22no—
400C42—21no—
401C42—24no—
402C42—41no—
403C42C22⋊C4×C3111yescharacteristic
404C42C22⋊C4×C3111yescharacteristic
405C42C22⋊C4×C3112yes—
406C42C22⋊C4×C3112yes—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23C31×#12011yescharacteristic
416Q8—41no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—81no—
421D4—41no—
422D4—41no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—81no—
427D4—82no—
428D4—82no—
429D4—81no—
430D4—82no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4C2×C22⋊C4×C3112yes—
447C2×C4C2×C22⋊C4×C3111yescharacteristic
448C2×C4C31×#12011yescharacteristic
449C2×C4C31×#12012yes—
450C8—82no—
451C8—81no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—161no—
461C22—161no—
462C22—162no—
463C22C31×#37911yescharacteristic
464C22C31×B(4,𝔽2)11yescharacteristic
465C22C31×#45811yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—81no—
480C2—81no—
481C2—82no—
482C2C31×#179011yescharacteristic
subgroupG/H
center, socleC2×C31#1790
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC31×#3545C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2133C2semidirect
3. C2×C31×#355C2semidirect
4. C31×#1912C2semidirect
5. C31×#1888C2semidirect
6. C31×#563C4semidirect
8. C31×H(3,𝔽4)C4semidirect
9. C31×#555C4semidirect
19. C31×#519C22semidirect
22. C2×C31×#101C22semidirect
23. C2×C31×#101C22semidirect
165. #12781C31direct (N × H)
212. #2133C2×C31semidirect
213. C2×#355C2×C31semidirect
214. #1912C2×C31semidirect
215. #1888C2×C31semidirect
240. #563C4×C31semidirect
242. H(3,𝔽4)C4×C31semidirect
243. #555C4×C31semidirect
253. #519C22×C31semidirect
256. C2×#101C22×C31semidirect
257. C2×#101C22×C31semidirect
334. C31#12781direct (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×#21332
C2×C31×#3552
C31×#19122
C31×#18882
#1278131