Finite Group Databasebeta

← C31×#8854 #21465613

Subgroups — C31×#8854 #21465613

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2315C211yesmaximal, characteristic
2C31×#1853C212yesmaximal, complemented
3C2×C31×#323C211yesmaximal, characteristic
4C31×#1439C211yesmaximal, characteristic
5C31×#1294C212yesmaximal, complemented
6C22×C4⋊C4×C31C2211yescharacteristic
7C22×C22⋊C4×C31—21no—
8C22×C22⋊C4×C31—21no—
9C22×C22⋊C4×C31C2211yescharacteristic
10C2×C31×#106—22no—
11C2×C31×C22⋊C8C2212yes—
12C31×#385—22no—
13C31×#385—22no—
14C31×#380—21no—
15C31×#380C2212yes—
16C2×C31×#95—21no—
17C2×C31×#95C2211yescharacteristic
18C31×#340—22no—
19C31×#323C414yescomplemented
20C25×C31D411yescharacteristic
21C23×C4×C31—21no—
22C23×C4×C31—21no—
23C23×C4×C31—21no—
24C23×C4×C31—21no—
25C23×C4×C31D411yescharacteristic
26C23×C4×C31C2311yescharacteristic
27C22×C8×C31—22no—
28C2×C4⋊C4×C31—22no—
29C2×C4⋊C4×C31—21no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—42no—
32C2×C4⋊C4×C31D412yes—
33C2×C4⋊C4×C31D412yes—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—42no—
39C2×C22⋊C4×C31—42no—
40C2×C22⋊C4×C31—42no—
41C2×C22⋊C4×C31—42no—
42C2×C22⋊C4×C31—42no—
43C2×C22⋊C4×C31C2×C412yes—
44C2×C22⋊C4×C31Q812yes—
45C31×#106—44no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8C2×C414yescomplemented
48C31×#95—22no—
49C31×#95—22no—
50C31×#95—22no—
51C31×#95—42no—
52C24×C31—21no—
53C24×C31—21no—
54C24×C31—21no—
55C24×C31—21no—
56C24×C31—41no—
57C24×C31—41no—
58C24×C31—41no—
59C24×C31—41no—
60C24×C31C4⋊C412yes—
61C24×C31SD1612yes—
62C24×C31SD1612yes—
63C24×C31C2×D411yescharacteristic
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—41no—
71C22×C4×C31—41no—
72C22×C4×C31—41no—
73C22×C4×C31—41no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31—41no—
77C22×C4×C31—42no—
78C22×C4×C31—42no—
79C22×C4×C31—41no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—41no—
83C22×C4×C31—41no—
84C22×C4×C31—41no—
85C22×C4×C31—41no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31—81no—
89C22×C4×C31—81no—
90C22×C4×C31—81no—
91C22×C4×C31—81no—
92C22×C4×C31C4⋊C412yes—
93C22×C4×C31C22×C411yescharacteristic
94C22×C4×C31C2×D412yes—
95C22×C4×C31C2×Q811yescharacteristic
96C22×C4×C31#4112yes—
97C2×C8×C31—24no—
98C2×C8×C31—42no—
99C2×C8×C31—42no—
100C4⋊C4×C31—24no—
101C4⋊C4×C31—42no—
102C4⋊C4×C31—42no—
103C4⋊C4×C31—42no—
104C4⋊C4×C31—81no—
105C4⋊C4×C31—81no—
106C22⋊C4×C31—24no—
107C22⋊C4×C31—24no—
108C22⋊C4×C31—44no—
109C22⋊C4×C31—48no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112#8854C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—22no—
118C23×C31—22no—
119C23×C31—22no—
120C23×C31—22no—
121C23×C31—22no—
122C23×C31—22no—
123C23×C31—42no—
124C23×C31—42no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—81no—
134C23×C31—81no—
135C23×C31—81no—
136C23×C31—81no—
137C23×C31—81no—
138C23×C31—81no—
139C23×C31#10614yescomplemented
140C23×C31C2×C4⋊C411yescharacteristic
141C23×C31#12011yescharacteristic
142C23×C31#12311yescharacteristic
143C23×C31C2×SD1611yescharacteristic
144C23×C31C2×SD1611yescharacteristic
145C23×C31AGL(1,ℤ/8)11yescharacteristic
146C23×C31#13711yescharacteristic
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—42no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C2×C4×C31—84no—
175C2×C4×C31—84no—
176C2×C4×C31—81no—
177C2×C4×C31—81no—
178C2×C4×C31—81no—
179C2×C4×C31C4×D412yes—
180C2×C4×C31C22⋊Q812yes—
181C8×C31—44no—
182#2315C2×C3111yescharacteristic
183#1853C2×C3112yescomplemented
184C2×#323C2×C3111yescharacteristic
185#1439C2×C3111yescharacteristic
186#1294C2×C3112yescomplemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—24no—
200C22×C31—44no—
201C22×C31—42no—
202C22×C31—42no—
203C22×C31—42no—
204C22×C31—42no—
205C22×C31—42no—
206C22×C31—42no—
207C22×C31—82no—
208C22×C31—82no—
209C22×C31—82no—
210C22×C31#38511yescharacteristic
211C22×C31C2×#10611yescharacteristic
212C22×C31#42811yescharacteristic
213C22×C31#44811yescharacteristic
214C22×C31#45011yescharacteristic
215C22×C31#45311yescharacteristic
216C22×C31#45511yescharacteristic
217C22×C31B(4,𝔽2)11yescharacteristic
218C22×C31#45811yescharacteristic
219C22×C31#48111yescharacteristic
220C22×C31#48311yescharacteristic
221C4×C31—22no—
222C4×C31—22no—
223C4×C31—42no—
224C4×C31—82no—
225C4×C31—82no—
226C4×C31—82no—
227C4×C31—82no—
228C22×C4⋊C4C22×C3111yescharacteristic
229C22×C22⋊C4—21no—
230C22×C22⋊C4—21no—
231C22×C22⋊C4C22×C3111yescharacteristic
232C2×#106—22no—
233C2×C22⋊C8C22×C3112yes—
234#385—22no—
235#385—22no—
236#380—21no—
237#380C22×C3112yes—
238C2×#95—21no—
239C2×#95C22×C3111yescharacteristic
240#340—22no—
241#323C4×C3114yescomplemented
242C2×C31—21no—
243C2×C31—21no—
244C2×C31—21no—
245C2×C31—21no—
246C2×C31—44no—
247C2×C31#159611yescharacteristic
248C2×C31#160111yescharacteristic
249C2×C31#160811yescharacteristic
250C2×C31#161311yescharacteristic
251C2×C31#185311yescharacteristic
252C2×C31#185611yescharacteristic
253C2×C31#186711yescharacteristic
254C25D4×C3111yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4D4×C3111yescharacteristic
260C23×C4C23×C3111yescharacteristic
261C22×C8—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4D4×C3112yes—
267C2×C4⋊C4D4×C3112yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C2×C4×C3112yes—
278C2×C22⋊C4Q8×C3112yes—
279#106—44no—
280C22⋊C8—22no—
281C22⋊C8C2×C4×C3114yescomplemented
282#95—22no—
283#95—22no—
284#95—22no—
285#95—42no—
286C31#885411yescharacteristic, direct product factor, Sylow
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24C4⋊C4×C3112yes—
296C24SD16×C3112yes—
297C24SD16×C3112yes—
298C24C2×D4×C3111yescharacteristic
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4C4⋊C4×C3112yes—
328C22×C4C22×C4×C3111yescharacteristic
329C22×C4C2×D4×C3112yes—
330C22×C4C2×Q8×C3111yescharacteristic
331C22×C4#41×C3112yes—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—44no—
344C22⋊C4—48no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23C31×#10614yescomplemented
374C23C2×C4⋊C4×C3111yescharacteristic
375C23C31×#12011yescharacteristic
376C23C31×#12311yescharacteristic
377C23C2×SD16×C3111yescharacteristic
378C23C2×SD16×C3111yescharacteristic
379C23C31×AGL(1,ℤ/8)11yescharacteristic
380C23C31×#13711yescharacteristic
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4C4×D4×C3112yes—
414C2×C4C31×C22⋊Q812yes—
415C8—44no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22C31×#38511yescharacteristic
440C22C2×C31×#10611yescharacteristic
441C22C31×#42811yescharacteristic
442C22C31×#44811yescharacteristic
443C22C31×#45011yescharacteristic
444C22C31×#45311yescharacteristic
445C22C31×#45511yescharacteristic
446C22C31×B(4,𝔽2)11yescharacteristic
447C22C31×#45811yescharacteristic
448C22C31×#48111yescharacteristic
449C22C31×#48311yescharacteristic
450C4—22no—
451C4—22no—
452C4—42no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—82no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—44no—
462C2C31×#159611yescharacteristic
463C2C31×#160111yescharacteristic
464C2C31×#160811yescharacteristic
465C2C31×#161311yescharacteristic
466C2C31×#185311yescharacteristic
467C2C31×#185611yescharacteristic
468C2C31×#186711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1853C2semidirect
5. C31×#1294C2semidirect
19. C31×#323C4semidirect
47. C31×C22⋊C8C2×C4semidirect
112. #8854C31direct (N × H)
139. C23×C31#106semidirect
183. #1853C2×C31semidirect
186. #1294C2×C31semidirect
241. #323C4×C31semidirect
281. C22⋊C8C2×C4×C31semidirect
286. C31#8854direct (N × H)
373. C23C31×#106semidirect

Maximal subgroups

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

subgroupindex
C31×#23152
C31×#18532
C2×C31×#3232
C31×#14392
C31×#12942
#885431