Finite Group Databasebeta

← C31×#10912 #21467671

Subgroups — C31×#10912 #21467671

Order 7936.

1266 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#385C211yesmaximal, characteristic
2C2×C4×C22⋊C4×C31C211yesmaximal, characteristic
3C2×C31×#342C212yesmaximal
4C2×C31×#342C212yesmaximal
5C31×C22⋊#95C211yesmaximal, characteristic
6C24×C4×C31C2211yescharacteristic
7C22×C4⋊C4×C31C412yescomplemented
8C22×C22⋊C4×C31C2211yescharacteristic
9C22×C22⋊C4×C31C2212yes—
10C22×C22⋊C4×C31C2211yescharacteristic
11C22×C22⋊C4×C31C2212yes—
12C22×C42×C31—21no—
13C31×#385—21no—
14C31×#385—22no—
15C31×#385—21no—
16C4×C22⋊C4×C31—24no—
17C2×C31×#95—21no—
18C2×C31×#95—22no—
19C2×C31×#95C412yescomplemented
20C31×#342—24no—
21C31×#342C418yescomplemented
22C25×C31C2311yescharacteristic
23C23×C4×C31—22no—
24C23×C4×C31—22no—
25C23×C4×C31—22no—
26C23×C4×C31—22no—
27C23×C4×C31—44no—
28C23×C4×C31C2×C412yes—
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31C2×C411yescharacteristic
31C23×C4×C31C2×C412yes—
32C23×C4×C31D411yescharacteristic
33C23×C4×C31D412yes—
34C23×C4×C31D411yescharacteristic
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31—44no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—22no—
52C2×C22⋊C4×C31—22no—
53C2×C22⋊C4×C31C2×C414yes—
54C2×C22⋊C4×C31C2×C414yes—
55C2×C22⋊C4×C31C2×C414yes—
56C2×C22⋊C4×C31D414yes—
57C2×C22⋊C4×C31Q814yes—
58C2×C42×C31—24no—
59C2×C42×C31—44no—
60C31×#95—24no—
61C31×#95—22no—
62C31×#95—22no—
63C31×#95—44no—
64C24×C31—42no—
65C24×C31—42no—
66C24×C31C22×C411yescharacteristic
67C24×C31C22×C412yes—
68C24×C31C2×D411yescharacteristic
69C24×C31C2×D411yescharacteristic
70C24×C31C2×D412yes—
71C24×C31C2×Q812yes—
72C24×C31#4112yes—
73C24×C31#4112yes—
74C24×C31#4112yes—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—24no—
78C22×C4×C31—22no—
79C22×C4×C31—22no—
80C22×C4×C31—24no—
81C22×C4×C31—24no—
82C22×C4×C31—21no—
83C22×C4×C31—24no—
84C22×C4×C31—24no—
85C22×C4×C31—24no—
86C22×C4×C31—21no—
87C22×C4×C31—24no—
88C22×C4×C31—24no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—24no—
92C22×C4×C31—22no—
93C22×C4×C31—22no—
94C22×C4×C31—42no—
95C22×C4×C31—42no—
96C22×C4×C31—42no—
97C22×C4×C31—48no—
98C22×C4×C31—48no—
99C22×C4×C31—44no—
100C22×C4×C31—44no—
101C22×C4×C31—42no—
102C22×C4×C31—42no—
103C22×C4×C31—44no—
104C22×C4×C31—44no—
105C22×C4×C31—42no—
106C22×C4×C31—42no—
107C22×C4×C31—42no—
108C22×C4×C31—42no—
109C22×C4×C31C4214yescomplemented
110C22×C4×C31C22⋊C412yes—
111C22×C4×C31C22⋊C412yes—
112C4⋊C4×C31—44no—
113C4⋊C4×C31—44no—
114C22⋊C4×C31—24no—
115C22⋊C4×C31—24no—
116C22⋊C4×C31—24no—
117C22⋊C4×C31—24no—
118C22⋊C4×C31—24no—
119C22⋊C4×C31—24no—
120C22⋊C4×C31—24no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—42no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—42no—
125C22⋊C4×C31—42no—
126C22⋊C4×C31C4⋊C418yescomplemented
127C42×C31—48no—
128#10912C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—22no—
130C23×C31—22no—
131C23×C31—22no—
132C23×C31—22no—
133C23×C31—22no—
134C23×C31—22no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—44no—
138C23×C31—42no—
139C23×C31—44no—
140C23×C31—42no—
141C23×C31—44no—
142C23×C31—44no—
143C23×C31C2×C4211yescharacteristic
144C23×C31C2×C22⋊C411yescharacteristic
145C23×C31C2×C4⋊C412yes—
146C23×C31#11711yescharacteristic
147C23×C31#11712yes—
148C23×C31C4×D412yes—
149C23×C31C4×D412yes—
150C23×C31C4×D412yes—
151C23×C31C4×D412yes—
152C23×C31C4×Q812yes—
153C23×C31C4×Q812yes—
154C23×C31#12011yescharacteristic
155C23×C31#12112yes—
156C23×C31C22⋊Q812yes—
157C23×C31C22⋊Q812yes—
158C23×C31#12311yescharacteristic
159C23×C31#12312yes—
160C23×C31#12411yescharacteristic
161C23×C31#12512yes—
162C23×C31#12612yes—
163C23×C31C4⋊Q811yescharacteristic
164C2×C4×C31—24no—
165C2×C4×C31—24no—
166C2×C4×C31—22no—
167C2×C4×C31—22no—
168C2×C4×C31—24no—
169C2×C4×C31—28no—
170C2×C4×C31—22no—
171C2×C4×C31—22no—
172C2×C4×C31—22no—
173C2×C4×C31—48no—
174C2×C4×C31—48no—
175C2×C4×C31—44no—
176C2×C4×C31—44no—
177C2×C4×C31—48no—
178C2×C4×C31—48no—
179C2×C4×C31—44no—
180C2×C4×C31—42no—
181C2×C4×C31—42no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—44no—
185C2×C4×C31—48no—
186C2×C4×C31—41no—
187C2×C4×C31—41no—
188C2×C4×C31—44no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—82no—
192C2×C4×C31—82no—
193C2×C4×C31—84no—
194C2×C4×C31—82no—
195C2×C4×C31—82no—
196C2×C4×C31#9914yes—
197C2×#385C2×C3111yescharacteristic
198C2×C4×C22⋊C4C2×C3111yescharacteristic
199C2×#342C2×C3112yes—
200C2×#342C2×C3112yes—
201C22⋊#95C2×C3111yescharacteristic
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—22no—
205C22×C31—22no—
206C22×C31—21no—
207C22×C31—22no—
208C22×C31—22no—
209C22×C31—21no—
210C22×C31—24no—
211C22×C31—22no—
212C22×C31—24no—
213C22×C31—22no—
214C22×C31—22no—
215C22×C31—22no—
216C22×C31—24no—
217C22×C31—24no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—42no—
222C22×C31—42no—
223C22×C31—42no—
224C22×C31—42no—
225C22×C31—42no—
226C22×C31C4×C22⋊C411yescharacteristic
227C22×C31C4×C4⋊C412yes—
228C22×C31#38512yes—
229C22×C31#38611yescharacteristic
230C22×C31#38712yes—
231C22×C31#38712yes—
232C22×C31#38812yes—
233C22×C31#38912yes—
234C22×C31#39111yescharacteristic
235C22×C31C2×#9912yes—
236C22×C31#41012yes—
237C4×C31—24no—
238C4×C31—48no—
239C4×C31—42no—
240C4×C31—44no—
241C4×C31—84no—
242C4×C31—81no—
243C4×C31—81no—
244C24×C4C22×C3111yescharacteristic
245C22×C4⋊C4C4×C3112yescomplemented
246C22×C22⋊C4C22×C3111yescharacteristic
247C22×C22⋊C4C22×C3112yes—
248C22×C22⋊C4C22×C3111yescharacteristic
249C22×C22⋊C4C22×C3112yes—
250C22×C42—21no—
251#385—21no—
252#385—22no—
253#385—21no—
254C4×C22⋊C4—24no—
255C2×#95—21no—
256C2×#95—22no—
257C2×#95C4×C3112yescomplemented
258#342—24no—
259#342C4×C3118yescomplemented
260C2×C31—22no—
261C2×C31—22no—
262C2×C31—22no—
263C2×C31—22no—
264C2×C31—22no—
265C2×C31—22no—
266C2×C31#143611yescharacteristic
267C2×C31C4×#9912yes—
268C2×C31#189811yescharacteristic
269C2×C31#189911yescharacteristic
270C2×C31#199612yes—
271C25C23×C3111yescharacteristic
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—44no—
277C23×C4C2×C4×C3112yes—
278C23×C4C2×C4×C3111yescharacteristic
279C23×C4C2×C4×C3111yescharacteristic
280C23×C4C2×C4×C3112yes—
281C23×C4D4×C3111yescharacteristic
282C23×C4D4×C3112yes—
283C23×C4D4×C3111yescharacteristic
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4C2×C4×C3114yes—
303C2×C22⋊C4C2×C4×C3114yes—
304C2×C22⋊C4C2×C4×C3114yes—
305C2×C22⋊C4D4×C3114yes—
306C2×C22⋊C4Q8×C3114yes—
307C2×C42—24no—
308C2×C42—44no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—44no—
313C31#1091211yescharacteristic, direct product factor, Sylow
314C24—42no—
315C24—42no—
316C24C22×C4×C3111yescharacteristic
317C24C22×C4×C3112yes—
318C24C2×D4×C3111yescharacteristic
319C24C2×D4×C3111yescharacteristic
320C24C2×D4×C3112yes—
321C24C2×Q8×C3112yes—
322C24#41×C3112yes—
323C24#41×C3112yes—
324C24#41×C3112yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4C42×C3114yescomplemented
360C22×C4C22⋊C4×C3112yes—
361C22×C4C22⋊C4×C3112yes—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4C4⋊C4×C3118yescomplemented
377C42—48no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—44no—
391C23—44no—
392C23C2×C42×C3111yescharacteristic
393C23C2×C22⋊C4×C3111yescharacteristic
394C23C2×C4⋊C4×C3112yes—
395C23C31×#11711yescharacteristic
396C23C31×#11712yes—
397C23C4×D4×C3112yes—
398C23C4×D4×C3112yes—
399C23C4×D4×C3112yes—
400C23C4×D4×C3112yes—
401C23C4×Q8×C3112yes—
402C23C4×Q8×C3112yes—
403C23C31×#12011yescharacteristic
404C23C31×#12112yes—
405C23C31×C22⋊Q812yes—
406C23C31×C22⋊Q812yes—
407C23C31×#12311yescharacteristic
408C23C31×#12312yes—
409C23C31×#12411yescharacteristic
410C23C31×#12512yes—
411C23C31×#12612yes—
412C23C31×C4⋊Q811yescharacteristic
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4C31×#9914yes—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—21no—
454C22—24no—
455C22—22no—
456C22—24no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—24no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22C4×C22⋊C4×C3111yescharacteristic
471C22C4×C4⋊C4×C3112yes—
472C22C31×#38512yes—
473C22C31×#38611yescharacteristic
474C22C31×#38712yes—
475C22C31×#38712yes—
476C22C31×#38812yes—
477C22C31×#38912yes—
478C22C31×#39111yescharacteristic
479C22C2×C31×#9912yes—
480C22C31×#41012yes—
481C4—24no—
482C4—48no—
483C4—42no—
484C4—44no—
485C4—84no—
486C4—81no—
487C4—81no—
488C2—22no—
489C2—22no—
490C2—22no—
491C2—22no—
492C2—22no—
493C2—22no—
494C2C31×#143611yescharacteristic
495C2C4×C31×#9912yes—
496C2C31×#189811yescharacteristic
497C2C31×#189911yescharacteristic
498C2C31×#199612yes—
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC23C2×C42×C31
Frattini, cosocleC25C23×C31
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
7. C22×C4⋊C4×C31C4semidirect
19. C2×C31×#95C4semidirect
21. C31×#342C4semidirect
109. C22×C4×C31C42semidirect
126. C22⋊C4×C31C4⋊C4semidirect
128. #10912C31direct (N × H)
245. C22×C4⋊C4C4×C31semidirect
257. C2×#95C4×C31semidirect
259. #342C4×C31semidirect
313. C31#10912direct (N × H)
359. C22×C4C42×C31semidirect
376. C22⋊C4C4⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#3852
C2×C4×C22⋊C4×C312
C2×C31×#3422
C31×C22⋊#952
#1091231