Finite Group Databasebeta

← C31×#23547 #21480306

Subgroups — C31×#23547 #21480306

Order 7936.

1458 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#137C211yesmaximal, characteristic
2C2×C31×#423C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic
4C2×C31×#419C211yesmaximal, characteristic, complemented
5C31×#2297C211yesmaximal, characteristic
6C2×C31×#391C211yesmaximal, characteristic, complemented
7C31×#2056C214yesmaximal, complemented
8C31×#2055C214yesmaximal, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C2×C31×#137—24no—
13C2×C31×#137—24no—
14C2×C31×#137—24no—
15C22×Q16×C31C2212yes—
16C22×SD16×C31C2212yes—
17C22×M16×C31C2211yescharacteristic
18C22×M16×C31C2211yescharacteristic
19C2×C31×#117C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C31×#423—24no—
23C2×C31×C4⋊C8C2214yes—
24C31×#419—24no—
25C31×#417—24no—
26C2×C31×#103C2214yes—
27C2×C31×#103C2212yes—
28C2×C31×#102C2212yes—
29C31×#391—21no—
30C31×#391—21no—
31C31×#391C2214yes—
32C31×#391C2212yes—
33C31×#390C2212yes—
34C31×#386—22no—
35C4×C22⋊C4×C31—21no—
36C4×C22⋊C4×C31—21no—
37C2×C31×#95—22no—
38C31×#340C2214yes—
39C2×#41×C31—22no—
40C2×#41×C31—22no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31D414yes—
45C22×Q8×C31—21no—
46C22×Q8×C31—22no—
47C22×Q8×C31—22no—
48C22×Q8×C31—22no—
49C22×Q8×C31—22no—
50C22×Q8×C31—22no—
51C22×Q8×C31D414yes—
52C22×Q8×C31C2312yes—
53C22×Q8×C31C2311yescharacteristic
54C22×D4×C31—21no—
55C22×D4×C31C2311yescharacteristic
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31D412yes—
60C23×C4×C31C2311yescharacteristic
61C31×#137—232no—
62C2×Q16×C31—24no—
63C2×Q16×C31—24no—
64C2×Q16×C31—24no—
65C2×SD16×C31—24no—
66C2×SD16×C31—24no—
67C2×SD16×C31—24no—
68C2×M16×C31—22no—
69C2×M16×C31—22no—
70C2×M16×C31—22no—
71C2×M16×C31—22no—
72C2×M16×C31—22no—
73C2×M16×C31D414yes—
74C22×C8×C31C2312yes—
75C22×C8×C31C2312yes—
76C31×#117—22no—
77C31×#117—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—22no—
85C2×C22⋊C4×C31—22no—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C2×C42×C31—21no—
89C2×C42×C31D411yescharacteristic
90C2×C42×C31D411yescharacteristic
91C2×C42×C31C2312yes—
92C31×C4⋊C8—28no—
93C31×#103—24no—
94C31×#103—28no—
95C31×#102—24no—
96C31×#95—24no—
97C31×#95—24no—
98C31×#95—22no—
99C31×#95—22no—
100C24×C31—21no—
101C24×C31C2×D411yescharacteristic
102#41×C31—28no—
103#41×C31—28no—
104#41×C31—44no—
105#41×C31—44no—
106C2×Q8×C31—24no—
107C2×Q8×C31—22no—
108C2×Q8×C31—21no—
109C2×Q8×C31—28no—
110C2×Q8×C31—28no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—21no—
118C2×Q8×C31—21no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31—44no—
122C2×Q8×C31—44no—
123C2×Q8×C31C2×D414yes—
124C2×Q8×C31C2×D412yes—
125C2×D4×C31—24no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31C2×D412yes—
134C22×C4×C31—24no—
135C22×C4×C31—24no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31—42no—
147C22×C4×C31—44no—
148C22×C4×C31—42no—
149C22×C4×C31C2×D412yes—
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D412yes—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31#4112yes—
156C22×C4×C31#4112yes—
157C22×C4×C31#4112yes—
158C22×C4×C31C2411yescharacteristic
159Q16×C31—216no—
160SD16×C31—216no—
161M16×C31—28no—
162M16×C31—44no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31C2×D414yes—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—42no—
174C22⋊C4×C31—44no—
175C22⋊C4×C31—42no—
176C22⋊C4×C31—44no—
177C42×C31—22no—
178C42×C31—22no—
179C42×C31—22no—
180C42×C31—22no—
181C42×C31—42no—
182#23547C3111yesmaximal, characteristic, direct product factor, Sylow
183C23×C31—22no—
184C23×C31—22no—
185C23×C31—22no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31#12112yes—
191C23×C31#12112yes—
192C23×C31#12112yes—
193C23×C31C22×D411yescharacteristic
194Q8×C31—24no—
195Q8×C31—22no—
196Q8×C31—24no—
197Q8×C31—22no—
198Q8×C31—42no—
199Q8×C31—41no—
200Q8×C31—42no—
201Q8×C31—41no—
202Q8×C31—48no—
203D4×C31—22no—
204D4×C31—22no—
205D4×C31—48no—
206D4×C31—41no—
207D4×C31—41no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—24no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—48no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—48no—
221C2×C4×C31—48no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—44no—
227C2×C4×C31—84no—
228C2×C4×C31#12014yes—
229C2×C4×C31#12112yes—
230C2×C4×C31#12112yes—
231C2×C4×C31#12112yes—
232C2×C4×C31#12314yes—
233C2×C4×C31#12314yes—
234C2×C4×C31#12414yes—
235C2×C4×C31#13714yescomplemented
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C2×C4×C31C22×D411yescharacteristic
239C2×C4×C31C2×#4111yescharacteristic
240C2×C4×C31C2×#4111yescharacteristic
241C2×C4×C31C2×#4111yescharacteristic
242C8×C31—24no—
243C8×C31—42no—
244C22×#137C2×C3111yescharacteristic
245C2×#423C2×C3111yescharacteristic, complemented
246C2×#417C2×C3111yescharacteristic
247C2×#419C2×C3111yescharacteristic, complemented
248#2297C2×C3111yescharacteristic
249C2×#391C2×C3111yescharacteristic, complemented
250#2056C2×C3114yescomplemented
251#2055C2×C3114yescomplemented
252#1736C2×C3111yescharacteristic, complemented
253C22×C31—24no—
254C22×C31—22no—
255C22×C31—22no—
256C22×C31—24no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31#39414yes—
263C22×C31C2×#12111yescharacteristic
264C22×C31C2×#12111yescharacteristic
265C22×C31C2×#12111yescharacteristic
266C22×C31C2×#13712yes—
267C22×C31#57512yes—
268C4×C31—24no—
269C4×C31—42no—
270C4×C31—44no—
271C4×C31—42no—
272C4×C31—82no—
273C4×C31—82no—
274C4×C31#39414yes—
275C4×C31C2×#12011yescharacteristic
276C4×C31C2×#12311yescharacteristic
277C4×C31C2×#12311yescharacteristic
278C4×C31C2×#12411yescharacteristic
279C22×#41C22×C3111yescharacteristic
280C23×Q8C22×C3111yescharacteristic
281C2×#137—24no—
282C2×#137—24no—
283C2×#137—24no—
284C22×Q16C22×C3112yes—
285C22×SD16C22×C3112yes—
286C22×M16C22×C3111yescharacteristic
287C22×M16C22×C3111yescharacteristic
288C2×#117C22×C3111yescharacteristic
289C22×C4⋊C4C22×C3111yescharacteristic
290C22×C42C22×C3111yescharacteristic
291#423—24no—
292C2×C4⋊C8C22×C3114yes—
293#419—24no—
294#417—24no—
295C2×#103C22×C3114yes—
296C2×#103C22×C3112yes—
297C2×#102C22×C3112yes—
298#391—21no—
299#391—21no—
300#391C22×C3114yes—
301#391C22×C3112yes—
302#390C22×C3112yes—
303#386—22no—
304C4×C22⋊C4—21no—
305C4×C22⋊C4—21no—
306C2×#95—22no—
307#340C22×C3114yes—
308C2×C31—24no—
309C2×C31—82no—
310C2×C31C2×#39411yescharacteristic
311C2×C31#303712yes—
312C2×C31#303911yescharacteristic
313C2×C31#304011yescharacteristic
314C2×C31#305612yes—
315C2×#41—22no—
316C2×#41—22no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41D4×C3114yes—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—22no—
327C22×Q8D4×C3114yes—
328C22×Q8C23×C3112yes—
329C22×Q8C23×C3111yescharacteristic
330C22×D4—21no—
331C22×D4C23×C3111yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4D4×C3112yes—
336C23×C4C23×C3111yescharacteristic
337#137—232no—
338C2×Q16—24no—
339C2×Q16—24no—
340C2×Q16—24no—
341C2×SD16—24no—
342C2×SD16—24no—
343C2×SD16—24no—
344C2×M16—22no—
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—22no—
349C2×M16D4×C3114yes—
350C22×C8C23×C3112yes—
351C22×C8C23×C3112yes—
352#117—22no—
353#117—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4C23×C3112yes—
358C2×C4⋊C4C23×C3112yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42D4×C3111yescharacteristic
366C2×C42D4×C3111yescharacteristic
367C2×C42C23×C3112yes—
368C4⋊C8—28no—
369#103—24no—
370#103—28no—
371#102—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376C31#2354711yescharacteristic, direct product factor, Sylow
377C24—21no—
378C24C2×D4×C3111yescharacteristic
379#41—28no—
380#41—28no—
381#41—44no—
382#41—44no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8C2×D4×C3114yes—
401C2×Q8C2×D4×C3112yes—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×D4×C3112yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4C2×D4×C3112yes—
427C22×C4C2×D4×C3112yes—
428C22×C4C2×D4×C3112yes—
429C22×C4C2×D4×C3112yes—
430C22×C4C2×D4×C3112yes—
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4#41×C3112yes—
433C22×C4#41×C3112yes—
434C22×C4#41×C3112yes—
435C22×C4C24×C3111yescharacteristic
436Q16—216no—
437SD16—216no—
438M16—28no—
439M16—44no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8C2×D4×C3114yes—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—42no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23C31×#12112yes—
467C23C31×#12112yes—
468C23C31×#12112yes—
469C23C22×D4×C3111yescharacteristic
470Q8—24no—
471Q8—22no—
472Q8—24no—
473Q8—22no—
474Q8—42no—
475Q8—41no—
476Q8—42no—
477Q8—41no—
478Q8—48no—
479D4—22no—
480D4—22no—
481D4—48no—
482D4—41no—
483D4—41no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—84no—
504C2×C4C31×#12014yes—
505C2×C4C31×#12112yes—
506C2×C4C31×#12112yes—
507C2×C4C31×#12112yes—
508C2×C4C31×#12314yes—
509C2×C4C31×#12314yes—
510C2×C4C31×#12414yes—
511C2×C4C31×#13714yescomplemented
512C2×C4C22×D4×C3111yescharacteristic
513C2×C4C22×D4×C3111yescharacteristic
514C2×C4C22×D4×C3111yescharacteristic
515C2×C4C2×#41×C3111yescharacteristic
516C2×C4C2×#41×C3111yescharacteristic
517C2×C4C2×#41×C3111yescharacteristic
518C8—24no—
519C8—42no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22C31×#39414yes—
530C22C2×C31×#12111yescharacteristic
531C22C2×C31×#12111yescharacteristic
532C22C2×C31×#12111yescharacteristic
533C22C2×C31×#13712yes—
534C22C31×#57512yes—
535C4—24no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—82no—
540C4—82no—
541C4C31×#39414yes—
542C4C2×C31×#12011yescharacteristic
543C4C2×C31×#12311yescharacteristic
544C4C2×C31×#12311yescharacteristic
545C4C2×C31×#12411yescharacteristic
546C2—24no—
547C2—82no—
548C2C2×C31×#39411yescharacteristic
549C2C31×#303712yes—
550C2C31×#303911yescharacteristic
551C2C31×#304011yescharacteristic
552C2C31×#305612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
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×C31×#423C2semidirect
4. C2×C31×#419C2semidirect
6. C2×C31×#391C2semidirect
7. C31×#2056C2semidirect
8. C31×#2055C2semidirect
9. C31×#1736C2semidirect
182. #23547C31direct (N × H)
235. C2×C4×C31#137semidirect
245. C2×#423C2×C31semidirect
247. C2×#419C2×C31semidirect
249. C2×#391C2×C31semidirect
250. #2056C2×C31semidirect
251. #2055C2×C31semidirect
252. #1736C2×C31semidirect
376. C31#23547direct (N × H)
511. C2×C4C31×#137semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1372
C2×C31×#4232
C2×C31×#4172
C2×C31×#4192
C31×#22972
C2×C31×#3912
C31×#20562
C31×#20552
C31×#17362
#2354731