Finite Group Databasebeta

← C31×#18650 #21475409

Subgroups — C31×#18650 #21475409

Order 7936.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3499C211yesmaximal, characteristic
2C31×#3226C211yesmaximal, characteristic, complemented
3C2×C31×#500C211yesmaximal, characteristic, complemented
4C31×#3039C211yesmaximal, characteristic, complemented
5C31×#1649C214yesmaximal, complemented
6C31×#1647C214yesmaximal, complemented
7C31×#1546C211yesmaximal, characteristic, complemented
8C31×#1492C211yesmaximal, characteristic, complemented
9C31×#1476C211yesmaximal, characteristic, complemented
10C2×C31×2−1+4—21no—
11C2×C31×#137—21no—
12D4×Q8×C31—21no—
13D4×Q8×C31C2212yes—
14C31×#547—22no—
15C31×#547—21no—
16C31×#547—22no—
17C31×#547C2212yes—
18C31×#536—22no—
19C31×#536—21no—
20C31×#532—21no—
21C31×#532C2211yescharacteristic
22C2×C31×C4⋊Q8—21no—
23C2×C31×C4⋊Q8C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C4×#41×C31C2211yescharacteristic
26C31×#500—22no—
27C31×#500—21no—
28C31×#500C2214yes—
29C31×#486—21no—
30C31×#480—22no—
31C31×#474C2214yes—
32C31×#462—21no—
33C31×#462C2212yes—
34C31×#461—21no—
35C31×#461C2212yes—
36C31×#431C2211yescharacteristic
37C2×C31×#107—22no—
38C31×#423C2211yescharacteristic
39C31×#419—21no—
40C2×C4×C8×C31C2211yescharacteristic
41C31×#332C2214yes—
42C31×Q8⋊C8C2212yes—
43C31×D4⋊C8C2212yes—
44C31×D4⋊C8C2214yes—
45C31×2−1+4—44no—
46C31×2−1+4—44no—
47C2×#41×C31—22no—
48C2×#41×C31—24no—
49C2×#41×C31—22no—
50C2×#41×C31—21no—
51C2×#41×C31D411yescharacteristic
52C22×Q8×C31—22no—
53C22×Q8×C31—22no—
54C22×Q8×C31—21no—
55C22×Q8×C31—21no—
56C31×#137—44no—
57C2×Q16×C31—22no—
58C2×SD16×C31—22no—
59C2×M16×C31—21no—
60C22×C8×C31—21no—
61C31×C4⋊Q8—22no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8—21no—
69C31×C4⋊Q8D411yescharacteristic
70C31×C4⋊Q8D412yes—
71C31×C4⋊Q8C2312yes—
72C31×C4⋊Q8C2312yes—
73C31×#127D411yescharacteristic
74C31×#124—22no—
75C31×#124—21no—
76C31×#123—24no—
77C31×#123—24no—
78C31×#123—24no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—24no—
82C31×C22⋊Q8—22no—
83C31×#121—22no—
84C31×#121—22no—
85C31×#121—21no—
86C31×#121D412yes—
87C4×Q8×C31D411yescharacteristic
88C4×Q8×C31C2311yescharacteristic
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—21no—
94C4×D4×C31D411yescharacteristic
95C4×D4×C31C2311yescharacteristic
96C4×D4×C31C2312yes—
97C31×#117—21no—
98C31×#117D411yescharacteristic
99C2×C4⋊C4×C31—22no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31—21no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31D412yes—
105C2×C42×C31—21no—
106C2×C42×C31C2311yescharacteristic
107C31×#107—28no—
108C31×#107—24no—
109C31×C4⋊C8—21no—
110C31×C4⋊C8C2312yes—
111C31×C4⋊C8C2312yes—
112C31×#103—22no—
113C31×#103—22no—
114C31×#102—24no—
115C31×C22⋊C8—21no—
116C4×C8×C31—21no—
117C4×C8×C31—21no—
118C4×C8×C31C2312yes—
119#41×C31—22no—
120#41×C31—48no—
121#41×C31—416no—
122#41×C31—44no—
123#41×C31—44no—
124#41×C31—42no—
125#41×C31—42no—
126#41×C31D814yes—
127C2×Q8×C31—24no—
128C2×Q8×C31—22no—
129C2×Q8×C31—21no—
130C2×Q8×C31—22no—
131C2×Q8×C31—22no—
132C2×Q8×C31—48no—
133C2×Q8×C31—44no—
134C2×Q8×C31—44no—
135C2×Q8×C31—41no—
136C2×Q8×C31—42no—
137C2×Q8×C31—42no—
138C2×Q8×C31—42no—
139C2×Q8×C31—42no—
140C2×Q8×C31C2×D411yescharacteristic
141C2×D4×C31—21no—
142C2×D4×C31—24no—
143C2×D4×C31—22no—
144C2×D4×C31—21no—
145C2×D4×C31—21no—
146C2×D4×C31C2×D412yes—
147C2×D4×C31C2×D411yescharacteristic
148C22×C4×C31—21no—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—22no—
152C22×C4×C31—21no—
153C22×C4×C31—21no—
154C22×C4×C31—22no—
155C22×C4×C31—41no—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159Q16×C31—44no—
160SD16×C31—44no—
161M16×C31—42no—
162C2×C8×C31—24no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C4⋊C4×C31—24no—
167C4⋊C4×C31—24no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—41no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31C2×D411yescharacteristic
179C4⋊C4×C31C2×D411yescharacteristic
180C4⋊C4×C31C2×D412yes—
181C4⋊C4×C31C2×D414yes—
182C22⋊C4×C31—24no—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—21no—
185C22⋊C4×C31—21no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—22no—
189C42×C31—21no—
190C42×C31—21no—
191C42×C31—21no—
192C42×C31C2×D411yescharacteristic
193C42×C31C2×D411yescharacteristic
194C42×C31C2×D411yescharacteristic
195C42×C31C2411yescharacteristic
196#18650C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—21no—
198C23×C31—22no—
199C23×C31#12011yescharacteristic
200Q8×C31—21no—
201Q8×C31—48no—
202Q8×C31—42no—
203Q8×C31—42no—
204Q8×C31—42no—
205Q8×C31—44no—
206Q8×C31—42no—
207Q8×C31—42no—
208Q8×C31—82no—
209Q8×C31C2×D812yes—
210D4×C31—24no—
211D4×C31—21no—
212D4×C31—24no—
213D4×C31—48no—
214D4×C31—44no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31C2×D812yes—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—24no—
225C2×C4×C31—22no—
226C2×C4×C31—48no—
227C2×C4×C31—48no—
228C2×C4×C31—44no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—42no—
234C2×C4×C31—82no—
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31C2×D812yes—
239C2×C4×C31AGL(1,ℤ/8)12yes—
240C2×C4×C31C22×D411yescharacteristic
241C2×C4×C31C22×D411yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C8×C31—24no—
244C8×C31—42no—
245C8×C31—42no—
246#3499C2×C3111yescharacteristic
247#3226C2×C3111yescharacteristic, complemented
248C2×#500C2×C3111yescharacteristic, complemented
249#3039C2×C3111yescharacteristic, complemented
250#1649C2×C3114yescomplemented
251#1647C2×C3114yescomplemented
252#1546C2×C3111yescharacteristic, complemented
253#1492C2×C3111yescharacteristic, complemented
254#1476C2×C3111yescharacteristic, complemented
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—22no—
258C22×C31—41no—
259C22×C31—41no—
260C22×C31—42no—
261C22×C31—42no—
262C22×C31—42no—
263C22×C31#44712yes—
264C22×C31C2×#12011yescharacteristic
265C4×C31—22no—
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—44no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—41no—
272C4×C31—82no—
273C4×C31#44712yes—
274C4×C31#45612yes—
275C4×C31C22×D811yescharacteristic
276C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C31#57812yes—
278C2×2−1+4—21no—
279C2×#137—21no—
280D4×Q8—21no—
281D4×Q8C22×C3112yes—
282#547—22no—
283#547—21no—
284#547—22no—
285#547C22×C3112yes—
286#536—22no—
287#536—21no—
288#532—21no—
289#532C22×C3111yescharacteristic
290C2×C4⋊Q8—21no—
291C2×C4⋊Q8C22×C3111yescharacteristic
292C2×C4⋊Q8C22×C3111yescharacteristic
293C4×#41C22×C3111yescharacteristic
294#500—22no—
295#500—21no—
296#500C22×C3114yes—
297#486—21no—
298#480—22no—
299#474C22×C3114yes—
300#462—21no—
301#462C22×C3112yes—
302#461—21no—
303#461C22×C3112yes—
304#431C22×C3111yescharacteristic
305C2×#107—22no—
306#423C22×C3111yescharacteristic
307#419—21no—
308C2×C4×C8C22×C3111yescharacteristic
309#332C22×C3114yes—
310Q8⋊C8C22×C3112yes—
311D4⋊C8C22×C3112yes—
312D4⋊C8C22×C3114yes—
313C2×C31—22no—
314C2×C31—42no—
315C2×C31—82no—
316C2×C31C2×#44711yescharacteristic
317C2×C31#301111yescharacteristic
318C2×C31C2×#45611yescharacteristic
3192−1+4—44no—
3202−1+4—44no—
321C2×#41—22no—
322C2×#41—24no—
323C2×#41—22no—
324C2×#41—21no—
325C2×#41D4×C3111yescharacteristic
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—21no—
330#137—44no—
331C2×Q16—22no—
332C2×SD16—22no—
333C2×M16—21no—
334C22×C8—21no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8D4×C3111yescharacteristic
344C4⋊Q8D4×C3112yes—
345C4⋊Q8C23×C3112yes—
346C4⋊Q8C23×C3112yes—
347#127D4×C3111yescharacteristic
348#124—22no—
349#124—21no—
350#123—24no—
351#123—24no—
352#123—24no—
353C22⋊Q8—24no—
354C22⋊Q8—22no—
355C22⋊Q8—24no—
356C22⋊Q8—22no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121D4×C3112yes—
361C4×Q8D4×C3111yescharacteristic
362C4×Q8C23×C3111yescharacteristic
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4D4×C3111yescharacteristic
369C4×D4C23×C3111yescharacteristic
370C4×D4C23×C3112yes—
371#117—21no—
372#117D4×C3111yescharacteristic
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C3112yes—
379C2×C42—21no—
380C2×C42C23×C3111yescharacteristic
381#107—28no—
382#107—24no—
383C4⋊C8—21no—
384C4⋊C8C23×C3112yes—
385C4⋊C8C23×C3112yes—
386#103—22no—
387#103—22no—
388#102—24no—
389C22⋊C8—21no—
390C4×C8—21no—
391C4×C8—21no—
392C4×C8C23×C3112yes—
393C31#1865011yescharacteristic, direct product factor, Sylow
394#41—22no—
395#41—48no—
396#41—416no—
397#41—44no—
398#41—44no—
399#41—42no—
400#41—42no—
401#41D8×C3114yes—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—41no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8C2×D4×C3111yescharacteristic
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×D4×C3112yes—
422C2×D4C2×D4×C3111yescharacteristic
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434Q16—44no—
435SD16—44no—
436M16—42no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4C2×D4×C3111yescharacteristic
454C4⋊C4C2×D4×C3111yescharacteristic
455C4⋊C4C2×D4×C3112yes—
456C4⋊C4C2×D4×C3114yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—21no—
463C22⋊C4—22no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42C2×D4×C3111yescharacteristic
468C42C2×D4×C3111yescharacteristic
469C42C2×D4×C3111yescharacteristic
470C42C24×C3111yescharacteristic
471C23—21no—
472C23—22no—
473C23C31×#12011yescharacteristic
474Q8—21no—
475Q8—48no—
476Q8—42no—
477Q8—44no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—82no—
483Q8C2×D8×C3112yes—
484D4—24no—
485D4—21no—
486D4—24no—
487D4—48no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4C2×D8×C3112yes—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12011yescharacteristic
512C2×C4C2×D8×C3112yes—
513C2×C4C31×AGL(1,ℤ/8)12yes—
514C2×C4C22×D4×C3111yescharacteristic
515C2×C4C22×D4×C3111yescharacteristic
516C2×C4C22×D4×C3111yescharacteristic
517C8—24no—
518C8—42no—
519C8—42no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22C31×#44712yes—
529C22C2×C31×#12011yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—44no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—82no—
538C4C31×#44712yes—
539C4C31×#45612yes—
540C4C22×D8×C3111yescharacteristic
541C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
542C4C31×#57812yes—
543C2—22no—
544C2—42no—
545C2—82no—
546C2C2×C31×#44711yescharacteristic
547C2C31×#301111yescharacteristic
548C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3499C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3226C2semidirect
3. C2×C31×#500C2semidirect
4. C31×#3039C2semidirect
5. C31×#1649C2semidirect
6. C31×#1647C2semidirect
7. C31×#1546C2semidirect
8. C31×#1492C2semidirect
9. C31×#1476C2semidirect
196. #18650C31direct (N × H)
247. #3226C2×C31semidirect
248. C2×#500C2×C31semidirect
249. #3039C2×C31semidirect
250. #1649C2×C31semidirect
251. #1647C2×C31semidirect
252. #1546C2×C31semidirect
253. #1492C2×C31semidirect
254. #1476C2×C31semidirect
393. C31#18650direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34992
C31×#32262
C2×C31×#5002
C31×#30392
C31×#16492
C31×#16472
C31×#15462
C31×#14922
C31×#14762
#1865031