Finite Group Databasebeta

← C31×#33096 #21489855

Subgroups — C31×#33096 #21489855

Order 7936.

920 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2854C211yesmaximal, characteristic
2C2×C4×C4⋊C4×C31C211yesmaximal, characteristic, complemented
3C31×#1926C212yesmaximal, complemented
4C31×#1925C212yesmaximal, complemented
5C31×#1921C212yesmaximal, complemented
6C31×#1920C212yesmaximal, complemented
7C31×#1874C211yesmaximal, characteristic
8C31×#1874C211yesmaximal, characteristic
9C31×#1853C211yesmaximal, characteristic, complemented
10C31×#1852C211yesmaximal, characteristic, complemented
11C31×#1817C211yesmaximal, characteristic, complemented
12C2×C31×#123—21no—
13C2×C31×#121C2211yescharacteristic
14C22×C4⋊C4×C31C2211yescharacteristic
15C22×C4⋊C4×C31C2211yescharacteristic
16C22×C42×C31—21no—
17C22×C42×C31C2211yescharacteristic
18C2×C31×#107C2212yes—
19C2×C31×#106C2212yes—
20C2×C31×#102C2212yes—
21C2×C31×#102C2212yes—
22C2×C31×C22⋊C8C2211yescharacteristic
23C2×C31×C22⋊C8C2211yescharacteristic
24C31×#390C2211yescharacteristic
25C31×#390C2211yescharacteristic
26C31×#389C2211yescharacteristic
27C31×#389C2211yescharacteristic
28C31×#388—21no—
29C31×#388—21no—
30C31×#386—21no—
31C31×#380C2211yescharacteristic
32C4×C4⋊C4×C31—21no—
33C4×C4⋊C4×C31—21no—
34C4×C4⋊C4×C31C2212yes—
35C4×C4⋊C4×C31C2212yes—
36C2×C31×#95—21no—
37C31×#340C2212yes—
38C31×#340C2212yes—
39C31×#340C2212yes—
40C31×#340C2212yes—
41C31×#336C2212yes—
42C31×#336C2212yes—
43C22×D4×C31—21no—
44C22×D4×C31C2311yescharacteristic
45C23×C4×C31—21no—
46C23×C4×C31—21no—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31C2311yescharacteristic
50C22×C8×C31C2312yes—
51C22×C8×C31C2312yes—
52C31×#123—44no—
53C31×#121—22no—
54C31×#121—22no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—24no—
58C2×C4⋊C4×C31—24no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31C2311yescharacteristic
62C2×C4⋊C4×C31C2311yescharacteristic
63C2×C4⋊C4×C31C2311yescharacteristic
64C2×C4⋊C4×C31C2312yes—
65C2×C4⋊C4×C31C2312yes—
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C42×C31—22no—
73C2×C42×C31—22no—
74C2×C42×C31—22no—
75C2×C42×C31—22no—
76C2×C42×C31—44no—
77C2×C42×C31D411yescharacteristic
78C2×C42×C31D411yescharacteristic
79C2×C42×C31C2311yescharacteristic
80C2×C42×C31C2311yescharacteristic
81C31×#107—24no—
82C31×#106—24no—
83C31×#102—24no—
84C31×#102—24no—
85C31×C22⋊C8—22no—
86C31×C22⋊C8—22no—
87C31×#95—22no—
88C31×#95—22no—
89C31×#95—21no—
90C31×#95—21no—
91C31×#95—21no—
92C24×C31—21no—
93C24×C31C2×D411yescharacteristic
94C2×D4×C31—22no—
95C2×D4×C31—22no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—42no—
99C2×D4×C31—42no—
100C2×D4×C31—42no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—24no—
107C22×C4×C31—24no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—21no—
114C22×C4×C31—21no—
115C22×C4×C31—21no—
116C22×C4×C31—42no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—42no—
120C22×C4×C31C2×D411yescharacteristic
121C22×C4×C31C2×D411yescharacteristic
122C22×C4×C31C2×D411yescharacteristic
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D411yescharacteristic
125C22×C4×C31#4112yes—
126C22×C4×C31#4112yes—
127C22×C4×C31C2411yescharacteristic
128C2×C8×C31—22no—
129C2×C8×C31—22no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—22no—
133C2×C8×C31—22no—
134C4⋊C4×C31—24no—
135C4⋊C4×C31—24no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—21no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—42no—
143C4⋊C4×C31—42no—
144C4⋊C4×C31#4114yes—
145C4⋊C4×C31#4114yes—
146C22⋊C4×C31—42no—
147C22⋊C4×C31—42no—
148C22⋊C4×C31—42no—
149C22⋊C4×C31—42no—
150C22⋊C4×C31—42no—
151C22⋊C4×C31—42no—
152C42×C31—22no—
153C42×C31—22no—
154C42×C31—21no—
155C42×C31—21no—
156C42×C31—21no—
157C42×C31—21no—
158C42×C31—44no—
159C42×C31—44no—
160#33096C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—21no—
164C23×C31—21no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—41no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—41no—
172C23×C31#12312yes—
173C23×C31#13512yes—
174C23×C31AGL(1,ℤ/8)12yes—
175C23×C31C22×D411yescharacteristic
176D4×C31—42no—
177D4×C31—42no—
178D4×C31—84no—
179C2×C4×C31—28no—
180C2×C4×C31—21no—
181C2×C4×C31—21no—
182C2×C4×C31—21no—
183C2×C4×C31—21no—
184C2×C4×C31—21no—
185C2×C4×C31—21no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—21no—
189C2×C4×C31—21no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—48no—
197C2×C4×C31—48no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—44no—
203C2×C4×C31—41no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—41no—
210C2×C4×C31—41no—
211C2×C4×C31—41no—
212C2×C4×C31#12312yes—
213C2×C4×C31AGL(1,ℤ/8)12yes—
214C2×C4×C31#13712yes—
215C2×C4×C31C2×#4111yescharacteristic
216C2×C4×C31C2×#4111yescharacteristic
217C2×C4×C31C2×#4112yes—
218C2×C4×C31C2×#4112yes—
219C8×C31—42no—
220C8×C31—42no—
221#2854C2×C3111yescharacteristic
222C2×C4×C4⋊C4C2×C3111yescharacteristic, complemented
223#1926C2×C3112yescomplemented
224#1925C2×C3112yescomplemented
225#1921C2×C3112yescomplemented
226#1920C2×C3112yescomplemented
227#1874C2×C3111yescharacteristic
228#1874C2×C3111yescharacteristic
229#1853C2×C3111yescharacteristic, complemented
230#1852C2×C3111yescharacteristic, complemented
231#1817C2×C3111yescharacteristic, complemented
232C22×C31—22no—
233C22×C31—22no—
234C22×C31—22no—
235C22×C31—22no—
236C22×C31—22no—
237C22×C31—22no—
238C22×C31—81no—
239C22×C31—81no—
240C22×C31—81no—
241C22×C31—81no—
242C22×C31—81no—
243C22×C31—81no—
244C22×C31—81no—
245C22×C31#48214yes—
246C22×C31C2×#12311yescharacteristic
247C22×C31#52511yescharacteristic
248C22×C31#52511yescharacteristic
249C22×C31C2×#13511yescharacteristic
250C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C31C2×#13711yescharacteristic
253C4×C31—22no—
254C4×C31—22no—
255C4×C31—22no—
256C4×C31—41no—
257C4×C31—41no—
258C4×C31—44no—
259C4×C31—44no—
260C4×C31—81no—
261C4×C31—81no—
262C4×C31—81no—
263C4×C31#52912yes—
264C4×C31#52912yes—
265C2×#123—21no—
266C2×#121C22×C3111yescharacteristic
267C22×C4⋊C4C22×C3111yescharacteristic
268C22×C4⋊C4C22×C3111yescharacteristic
269C22×C42—21no—
270C22×C42C22×C3111yescharacteristic
271C2×#107C22×C3112yes—
272C2×#106C22×C3112yes—
273C2×#102C22×C3112yes—
274C2×#102C22×C3112yes—
275C2×C22⋊C8C22×C3111yescharacteristic
276C2×C22⋊C8C22×C3111yescharacteristic
277#390C22×C3111yescharacteristic
278#390C22×C3111yescharacteristic
279#389C22×C3111yescharacteristic
280#389C22×C3111yescharacteristic
281#388—21no—
282#388—21no—
283#386—21no—
284#380C22×C3111yescharacteristic
285C4×C4⋊C4—21no—
286C4×C4⋊C4—21no—
287C4×C4⋊C4C22×C3112yes—
288C4×C4⋊C4C22×C3112yes—
289C2×#95—21no—
290#340C22×C3112yes—
291#340C22×C3112yes—
292#340C22×C3112yes—
293#340C22×C3112yes—
294#336C22×C3112yes—
295#336C22×C3112yes—
296C2×C31—24no—
297C2×C31—161no—
298C2×C31#285311yescharacteristic
299C2×C31C2×#48211yescharacteristic
300C2×C31#309211yescharacteristic
301C2×C31#310511yescharacteristic
302C2×C31#310811yescharacteristic
303C2×C31#310911yescharacteristic
304C2×C31#311211yescharacteristic
305C22×D4—21no—
306C22×D4C23×C3111yescharacteristic
307C23×C4—21no—
308C23×C4—21no—
309C23×C4D4×C3111yescharacteristic
310C23×C4D4×C3111yescharacteristic
311C23×C4C23×C3111yescharacteristic
312C22×C8C23×C3112yes—
313C22×C8C23×C3112yes—
314#123—44no—
315#121—22no—
316#121—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C23×C3111yescharacteristic
324C2×C4⋊C4C23×C3111yescharacteristic
325C2×C4⋊C4C23×C3111yescharacteristic
326C2×C4⋊C4C23×C3112yes—
327C2×C4⋊C4C23×C3112yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42D4×C3111yescharacteristic
340C2×C42D4×C3111yescharacteristic
341C2×C42C23×C3111yescharacteristic
342C2×C42C23×C3111yescharacteristic
343#107—24no—
344#106—24no—
345#102—24no—
346#102—24no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—21no—
354C31#3309611yescharacteristic, direct product factor, Sylow
355C24—21no—
356C24C2×D4×C3111yescharacteristic
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×D4×C3111yescharacteristic
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387C22×C4C2×D4×C3111yescharacteristic
388C22×C4#41×C3112yes—
389C22×C4#41×C3112yes—
390C22×C4C24×C3111yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4#41×C3114yes—
408C4⋊C4#41×C3114yes—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—44no—
422C42—44no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23C31×#12312yes—
435C23C31×#13512yes—
436C23C31×AGL(1,ℤ/8)12yes—
437C23C22×D4×C3111yescharacteristic
438D4—42no—
439D4—42no—
440D4—84no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4C31×#12312yes—
475C2×C4C31×AGL(1,ℤ/8)12yes—
476C2×C4C31×#13712yes—
477C2×C4C2×#41×C3111yescharacteristic
478C2×C4C2×#41×C3111yescharacteristic
479C2×C4C2×#41×C3112yes—
480C2×C4C2×#41×C3112yes—
481C8—42no—
482C8—42no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22C31×#48214yes—
497C22C2×C31×#12311yescharacteristic
498C22C31×#52511yescharacteristic
499C22C31×#52511yescharacteristic
500C22C2×C31×#13511yescharacteristic
501C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
502C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
503C22C2×C31×#13711yescharacteristic
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—44no—
510C4—44no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4C31×#52912yes—
515C4C31×#52912yes—
516C2—24no—
517C2—161no—
518C2C31×#285311yescharacteristic
519C2C2×C31×#48211yescharacteristic
520C2C31×#309211yescharacteristic
521C2C31×#310511yescharacteristic
522C2C31×#310811yescharacteristic
523C2C31×#310911yescharacteristic
524C2C31×#311211yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊C4×C31C2semidirect
3. C31×#1926C2semidirect
4. C31×#1925C2semidirect
5. C31×#1921C2semidirect
6. C31×#1920C2semidirect
9. C31×#1853C2semidirect
10. C31×#1852C2semidirect
11. C31×#1817C2semidirect
160. #33096C31direct (N × H)
222. C2×C4×C4⋊C4C2×C31semidirect
223. #1926C2×C31semidirect
224. #1925C2×C31semidirect
225. #1921C2×C31semidirect
226. #1920C2×C31semidirect
229. #1853C2×C31semidirect
230. #1852C2×C31semidirect
231. #1817C2×C31semidirect
354. C31#33096direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28542
C2×C4×C4⋊C4×C312
C31×#19262
C31×#19252
C31×#19212
C31×#19202
C31×#18742
C31×#18532
C31×#18522
C31×#18172
#3309631