Finite Group Databasebeta

← C31×#18442 #21475201

Subgroups — C31×#18442 #21475201

Order 7936.

848 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3563C211yesmaximal, characteristic
2C31×#3238C211yesmaximal, characteristic
3C31×#3229C211yesmaximal, characteristic, complemented
4C31×#2660C211yesmaximal, characteristic, complemented
5C31×#1611C214yesmaximal, complemented
6C31×#1609C214yesmaximal
7C31×#1506C211yesmaximal, characteristic, complemented
8C31×#1500C211yesmaximal, characteristic
9C31×#1470C211yesmaximal, characteristic, complemented
10C31×#564—21no—
11C31×#563—21no—
12C31×#557—21no—
13C31×#554—21no—
14C31×#552—21no—
15C31×#552—22no—
16C31×#552—21no—
17C31×#544—21no—
18C31×#544C2212yes—
19C31×#541—21no—
20C31×#541—22no—
21C31×#539C2212yes—
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#529C2211yescharacteristic
25C2×C31×#125C2211yescharacteristic
26C31×#501—21no—
27C31×#498—21no—
28C31×#489—21no—
29C31×#487—21no—
30C31×#483C2214yes—
31C31×#483C2212yes—
32C31×#480C2212yes—
33C31×#477—21no—
34C31×#476—21no—
35C31×#475—22no—
36C31×#432C2211yescharacteristic
37C31×#431C2211yescharacteristic
38C31×#398C2214yes—
39C31×#395—22no—
40C31×#384—21no—
41C31×#382—21no—
42C31×#376C2211yescharacteristic
43C31×#328C2214yes—
44C31×#328C2212yes—
45C31×#327C2212yes—
46C31×#324C2214yes—
47C22×Q8×C31—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—22no—
50C31×C4⋊Q8—22no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8D411yescharacteristic
56C31×C4⋊Q8D411yescharacteristic
57C31×#126—22no—
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#126—21no—
62C31×#125—22no—
63C31×#125—22no—
64C31×#125—22no—
65C31×#125—22no—
66C31×#125—21no—
67C31×#125—22no—
68C31×#125—22no—
69C31×#125—22no—
70C31×#125—22no—
71C31×#125—21no—
72C31×#125—21no—
73C31×#125—21no—
74C31×#125—21no—
75C31×#125—21no—
76C31×#125D411yescharacteristic
77C31×#124D411yescharacteristic
78C31×#123—22no—
79C31×#123—21no—
80C31×#123—22no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8C2311yescharacteristic
89C31×C22⋊Q8C2312yes—
90C31×#121C2311yescharacteristic
91C4×Q8×C31—21no—
92C4×Q8×C31—22no—
93C4×Q8×C31—21no—
94C4×Q8×C31—21no—
95C4×Q8×C31D412yes—
96C4×Q8×C31D411yescharacteristic
97C4×D4×C31—21no—
98C4×D4×C31—22no—
99C4×D4×C31D411yescharacteristic
100C31×#117—22no—
101C31×#117—21no—
102C31×#117—21no—
103C31×#117—22no—
104C31×#117—21no—
105C2×C4⋊C4×C31—21no—
106C2×C4⋊C4×C31—21no—
107C2×C4⋊C4×C31—24no—
108C2×C4⋊C4×C31D412yes—
109C2×C4⋊C4×C31D411yescharacteristic
110C2×C4⋊C4×C31D411yescharacteristic
111C2×C4⋊C4×C31C2312yes—
112C2×C4⋊C4×C31C2312yes—
113C2×C42×C31—21no—
114C2×C42×C31C2311yescharacteristic
115C31×#107—22no—
116C31×#106—24no—
117C31×#106—22no—
118C31×C4⋊C8—21no—
119C31×C4⋊C8—21no—
120C31×#103—24no—
121C31×#103—22no—
122C31×#102—22no—
123C31×C22⋊C8C2312yes—
124C31×C22⋊C8C2312yes—
125C31×#97—21no—
126C4×C8×C31—21no—
127C31×#95—21no—
128C31×#95—24no—
129C31×#95C2312yes—
130C2×Q8×C31—22no—
131C2×Q8×C31—21no—
132C2×Q8×C31—41no—
133C2×Q8×C31—41no—
134C2×Q8×C31—82no—
135C2×Q8×C31C2×D412yes—
136C2×Q8×C31C2×D411yescharacteristic
137C2×D4×C31—21no—
138C2×D4×C31C2×D411yescharacteristic
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—21no—
142C22×C4×C31—22no—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147C22×C4×C31C2411yescharacteristic
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—41no—
178C4⋊C4×C31—41no—
179C4⋊C4×C31—84no—
180C4⋊C4×C31C2×D411yescharacteristic
181C4⋊C4×C31C2×D412yes—
182C4⋊C4×C31C2×D411yescharacteristic
183C22⋊C4×C31—22no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C22⋊C4×C31—22no—
187C22⋊C4×C31—21no—
188C22⋊C4×C31—21no—
189C22⋊C4×C31—21no—
190C22⋊C4×C31—21no—
191C22⋊C4×C31—21no—
192C42×C31—24no—
193C42×C31—22no—
194C42×C31—22no—
195C42×C31—22no—
196C42×C31—22no—
197C42×C31—21no—
198C42×C31C2×D411yescharacteristic
199C42×C31C2×D411yescharacteristic
200C42×C31C2×D411yescharacteristic
201C42×C31C2×D411yescharacteristic
202#18442C3111yesmaximal, characteristic, direct product factor, Sylow
203C23×C31—21no—
204C23×C31C22×D411yescharacteristic
205Q8×C31—41no—
206Q8×C31—42no—
207Q8×C31—42no—
208Q8×C31—82no—
209Q8×C31—82no—
210Q8×C31—82no—
211D4×C31—41no—
212D4×C31—42no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—44no—
231C2×C4×C31—82no—
232C2×C4×C31#12011yescharacteristic
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31#12011yescharacteristic
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31C22×D411yescharacteristic
237C2×C4×C31C22×D411yescharacteristic
238C8×C31—42no—
239C8×C31—42no—
240#3563C2×C3111yescharacteristic
241#3238C2×C3111yescharacteristic
242#3229C2×C3111yescharacteristic, complemented
243#2660C2×C3111yescharacteristic, complemented
244#1611C2×C3114yescomplemented
245#1609C2×C3114yes—
246#1506C2×C3111yescharacteristic, complemented
247#1500C2×C3111yescharacteristic
248#1470C2×C3111yescharacteristic, complemented
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—41no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31C2×#12011yescharacteristic
255C22×C31#57712yes—
256C4×C31—21no—
257C4×C31—21no—
258C4×C31—21no—
259C4×C31—21no—
260C4×C31—21no—
261C4×C31—44no—
262C4×C31—42no—
263C4×C31—41no—
264C4×C31—41no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—42no—
268C4×C31—41no—
269C4×C31—82no—
270C4×C31#45612yes—
271#564—21no—
272#563—21no—
273#557—21no—
274#554—21no—
275#552—21no—
276#552—21no—
277#552—22no—
278#544—21no—
279#544C22×C3112yes—
280#541—21no—
281#541—22no—
282#539C22×C3112yes—
283#533C22×C3111yescharacteristic
284C2×C4⋊Q8C22×C3111yescharacteristic
285#529C22×C3111yescharacteristic
286C2×#125C22×C3111yescharacteristic
287#501—21no—
288#498—21no—
289#489—21no—
290#487—21no—
291#483C22×C3114yes—
292#483C22×C3112yes—
293#480C22×C3112yes—
294#477—21no—
295#476—21no—
296#475—22no—
297#432C22×C3111yescharacteristic
298#431C22×C3111yescharacteristic
299#398C22×C3114yes—
300#395—22no—
301#384—21no—
302#382—21no—
303#376C22×C3111yescharacteristic
304#328C22×C3114yes—
305#328C22×C3112yes—
306#327C22×C3112yes—
307#324C22×C3114yes—
308C2×C31—22no—
309C2×C31—81no—
310C2×C31#301011yescharacteristic
311C2×C31C2×#45611yescharacteristic
312C2×C31#302811yescharacteristic
313C22×Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8D4×C3111yescharacteristic
322C4⋊Q8D4×C3111yescharacteristic
323#126—22no—
324#126—22no—
325#126—22no—
326#126—22no—
327#126—21no—
328#125—22no—
329#125—22no—
330#125—22no—
331#125—22no—
332#125—21no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—21no—
338#125—21no—
339#125—21no—
340#125—21no—
341#125—21no—
342#125D4×C3111yescharacteristic
343#124D4×C3111yescharacteristic
344#123—22no—
345#123—21no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8C23×C3111yescharacteristic
355C22⋊Q8C23×C3112yes—
356#121C23×C3111yescharacteristic
357C4×Q8—21no—
358C4×Q8—22no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8D4×C3111yescharacteristic
362C4×Q8D4×C3112yes—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4D4×C3111yescharacteristic
366#117—22no—
367#117—21no—
368#117—21no—
369#117—22no—
370#117—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4D4×C3111yescharacteristic
375C2×C4⋊C4D4×C3112yes—
376C2×C4⋊C4D4×C3111yescharacteristic
377C2×C4⋊C4C23×C3112yes—
378C2×C4⋊C4C23×C3112yes—
379C2×C42—21no—
380C2×C42C23×C3111yescharacteristic
381#107—22no—
382#106—24no—
383#106—22no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386#103—24no—
387#103—22no—
388#102—22no—
389C22⋊C8C23×C3112yes—
390C22⋊C8C23×C3112yes—
391#97—21no—
392C4×C8—21no—
393#95—21no—
394#95—24no—
395#95C23×C3112yes—
396C31#1844211yescharacteristic, direct product factor, Sylow
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—82no—
402C2×Q8C2×D4×C3112yes—
403C2×Q8C2×D4×C3111yescharacteristic
404C2×D4—21no—
405C2×D4C2×D4×C3111yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4C2×D4×C3112yes—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3112yes—
414C22×C4C24×C3111yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—84no—
447C4⋊C4C2×D4×C3111yescharacteristic
448C4⋊C4C2×D4×C3112yes—
449C4⋊C4C2×D4×C3111yescharacteristic
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42C2×D4×C3111yescharacteristic
466C42C2×D4×C3111yescharacteristic
467C42C2×D4×C3111yescharacteristic
468C42C2×D4×C3111yescharacteristic
469C23—21no—
470C23C22×D4×C3111yescharacteristic
471Q8—41no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475Q8—82no—
476Q8—82no—
477D4—41no—
478D4—42no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—82no—
498C2×C4C31×#12011yescharacteristic
499C2×C4C31×#12011yescharacteristic
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#12011yescharacteristic
502C2×C4C22×D4×C3111yescharacteristic
503C2×C4C22×D4×C3111yescharacteristic
504C8—42no—
505C8—42no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22C2×C31×#12011yescharacteristic
512C22C31×#57712yes—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—44no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—82no—
527C4C31×#45612yes—
528C2—22no—
529C2—81no—
530C2C31×#301011yescharacteristic
531C2C2×C31×#45611yescharacteristic
532C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#3229C2semidirect
4. C31×#2660C2semidirect
5. C31×#1611C2semidirect
7. C31×#1506C2semidirect
9. C31×#1470C2semidirect
202. #18442C31direct (N × H)
242. #3229C2×C31semidirect
243. #2660C2×C31semidirect
244. #1611C2×C31semidirect
246. #1506C2×C31semidirect
248. #1470C2×C31semidirect
396. C31#18442direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35632
C31×#32382
C31×#32292
C31×#26602
C31×#16112
C31×#16092
C31×#15062
C31×#15002
C31×#14702
#1844231