Finite Group Databasebeta

← C2×C31×#1661 #21475740

Subgroups — C2×C31×#1661 #21475740

Order 7936.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#562C211yesmaximal, characteristic
2C2×C31×#501C211yesmaximal, characteristic, complemented
3C2×C31×#497C211yesmaximal, characteristic, complemented
4C2×C31×#489C211yesmaximal, characteristic, complemented
5C31×#1661C218yesmaximal, direct product factor
6C2×C31×#330C211yesmaximal, characteristic, complemented
7C2×C31×#329C211yesmaximal, characteristic, complemented
8C2×C31×#329C211yesmaximal, characteristic, complemented
9C22×Q16×C31—21no—
10C22×SD16×C31—21no—
11C31×#562—22no—
12C31×#562—22no—
13C31×#562—22no—
14C31×#562C2214yes—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#125C2211yescharacteristic
17C2×C31×#124C2211yescharacteristic
18C2×C31×#124C2211yescharacteristic
19C2×C31×#123—21no—
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×#121—21no—
24C2×C31×#121—21no—
25C31×#501—22no—
26C31×#501C2214yescomplemented
27C31×#497—22no—
28C31×#497C2214yescomplemented
29C31×#489—22no—
30C31×#489C2214yescomplemented
31C2×C31×#107—21no—
32C2×C31×#106—21no—
33C2×C31×#103—21no—
34C2×C31×#102—21no—
35C2×C31×#97C2211yescharacteristic
36C2×C31×#97C2211yescharacteristic
37C2×C31×#97C2211yescharacteristic
38C31×#330C2214yescomplemented
39C31×#329C2214yescomplemented
40C31×#329C2214yescomplemented
41C22×Q8×C31—21no—
42C22×Q8×C31D411yescharacteristic
43C22×Q8×C31D411yescharacteristic
44C22×D4×C31—21no—
45C22×D4×C31—21no—
46C22×D4×C31D411yescharacteristic
47C22×D4×C31D411yescharacteristic
48C23×C4×C31—21no—
49C23×C4×C31—21no—
50C2×Q16×C31—24no—
51C2×Q16×C31—44no—
52C2×SD16×C31—24no—
53C2×SD16×C31—44no—
54C22×C8×C31—21no—
55C22×C8×C31—21no—
56C22×C8×C31—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8C2312yes—
61C31×#125—21no—
62C31×#125—21no—
63C31×#125D412yes—
64C31×#125C2312yes—
65C31×#124—21no—
66C31×#124—21no—
67C31×#124—21no—
68C31×#124—21no—
69C31×#124D412yes—
70C31×#124D412yes—
71C31×#124C2312yes—
72C31×#124C2312yes—
73C31×#123—24no—
74C31×#123—24no—
75C31×#123—24no—
76C31×#123—24no—
77C31×C22⋊Q8—24no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—24no—
81C31×#121—24no—
82C31×#121—24no—
83C31×#121—24no—
84C31×#121—24no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31D411yescharacteristic
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C42×C31C2311yescharacteristic
100C31×#107—24no—
101C31×#106—24no—
102C31×#103—24no—
103C31×#102—24no—
104C31×#97—21no—
105C31×#97—21no—
106C31×#97—21no—
107C31×#97C2312yes—
108C31×#97C2312yes—
109C31×#97C2312yes—
110C24×C31—21no—
111C24×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—22no—
115C2×Q8×C31—42no—
116C2×Q8×C31—42no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C2×Q8×C31—41no—
120C2×Q8×C31C2×D412yes—
121C2×Q8×C31C2×D412yes—
122C2×D4×C31—24no—
123C2×D4×C31—24no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—42no—
127C2×D4×C31—42no—
128C2×D4×C31—82no—
129C2×D4×C31—82no—
130C2×D4×C31C2×D412yes—
131C2×D4×C31C2×D412yes—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—24no—
137C22×C4×C31—24no—
138C22×C4×C31—41no—
139C22×C4×C31—82no—
140C22×C4×C31—82no—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144Q16×C31—48no—
145SD16×C31—48no—
146C2×C8×C31—24no—
147C2×C8×C31—24no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—42no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—41no—
163C4⋊C4×C31—41no—
164C4⋊C4×C31C2×D412yes—
165C4⋊C4×C31C2×D412yes—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C42×C31C2×D411yescharacteristic
177C42×C31C2×D411yescharacteristic
178C42×C31C2×D411yescharacteristic
179C42×C31C2411yescharacteristic
180C2×#1661C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—22no—
182C23×C31—22no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31#12011yescharacteristic
190Q8×C31—42no—
191Q8×C31—42no—
192Q8×C31—42no—
193Q8×C31—42no—
194Q8×C31—81no—
195Q8×C31—81no—
196Q8×C31—82no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—81no—
200D4×C31—81no—
201D4×C31—84no—
202D4×C31—84no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—22no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—84no—
216C2×C4×C31—84no—
217C2×C4×C31—82no—
218C2×C4×C31#12011yescharacteristic
219C2×C4×C31#12011yescharacteristic
220C2×C4×C31#12011yescharacteristic
221C2×C4×C31C22×D411yescharacteristic
222C2×C4×C31C22×D411yescharacteristic
223C2×C4×C31C22×D411yescharacteristic
224C8×C31—24no—
225C8×C31—24no—
226C8×C31—42no—
227C2×#562C2×C3111yescharacteristic
228C2×#501C2×C3111yescharacteristic, complemented
229C2×#497C2×C3111yescharacteristic, complemented
230C2×#489C2×C3111yescharacteristic, complemented
231#1661C2×C3118yesdirect product factor
232C2×#330C2×C3111yescharacteristic, complemented
233C2×#329C2×C3111yescharacteristic, complemented
234C2×#329C2×C3111yescharacteristic, complemented
235C22×C31—42no—
236C22×C31—42no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31—42no—
241C22×C31—84no—
242C22×C31—84no—
243C22×C31#45312yes—
244C22×C31#45512yes—
245C22×C31#45612yes—
246C22×C31C2×#12011yescharacteristic
247C4×C31—21no—
248C4×C31—21no—
249C4×C31—21no—
250C4×C31—21no—
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—42no—
254C4×C31—42no—
255C4×C31—42no—
256C4×C31—42no—
257C4×C31—82no—
258C22×Q16—21no—
259C22×SD16—21no—
260#562—22no—
261#562—22no—
262#562—22no—
263#562C22×C3114yes—
264C2×C4⋊Q8C22×C3111yescharacteristic
265C2×#125C22×C3111yescharacteristic
266C2×#124C22×C3111yescharacteristic
267C2×#124C22×C3111yescharacteristic
268C2×#123—21no—
269C2×#123—21no—
270C2×C22⋊Q8—21no—
271C2×C22⋊Q8—21no—
272C2×#121—21no—
273C2×#121—21no—
274#501—22no—
275#501C22×C3114yescomplemented
276#497—22no—
277#497C22×C3114yescomplemented
278#489—22no—
279#489C22×C3114yescomplemented
280C2×#107—21no—
281C2×#106—21no—
282C2×#103—21no—
283C2×#102—21no—
284C2×#97C22×C3111yescharacteristic
285C2×#97C22×C3111yescharacteristic
286C2×#97C22×C3111yescharacteristic
287#330C22×C3114yescomplemented
288#329C22×C3114yescomplemented
289#329C22×C3114yescomplemented
290C2×C31—82no—
291C2×C31—82no—
292C2×C31#166114yesdirect product factor
293C2×C31C2×#45311yescharacteristic
294C2×C31C2×#45511yescharacteristic
295C2×C31C2×#45611yescharacteristic
296C22×Q8—21no—
297C22×Q8D4×C3111yescharacteristic
298C22×Q8D4×C3111yescharacteristic
299C22×D4—21no—
300C22×D4—21no—
301C22×D4D4×C3111yescharacteristic
302C22×D4D4×C3111yescharacteristic
303C23×C4—21no—
304C23×C4—21no—
305C2×Q16—24no—
306C2×Q16—44no—
307C2×SD16—24no—
308C2×SD16—44no—
309C22×C8—21no—
310C22×C8—21no—
311C22×C8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8C23×C3112yes—
316#125—21no—
317#125—21no—
318#125D4×C3112yes—
319#125C23×C3112yes—
320#124—21no—
321#124—21no—
322#124—21no—
323#124—21no—
324#124D4×C3112yes—
325#124D4×C3112yes—
326#124C23×C3112yes—
327#124C23×C3112yes—
328#123—24no—
329#123—24no—
330#123—24no—
331#123—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336#121—24no—
337#121—24no—
338#121—24no—
339#121—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4D4×C3111yescharacteristic
347C2×C4⋊C4D4×C3111yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C42C23×C3111yescharacteristic
355#107—24no—
356#106—24no—
357#103—24no—
358#102—24no—
359#97—21no—
360#97—21no—
361#97—21no—
362#97C23×C3112yes—
363#97C23×C3112yes—
364#97C23×C3112yes—
365C31C2×#166111yescharacteristic, direct product factor, Sylow
366C24—21no—
367C24—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—41no—
376C2×Q8C2×D4×C3112yes—
377C2×Q8C2×D4×C3112yes—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4C2×D4×C3112yes—
387C2×D4C2×D4×C3112yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—41no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4C2×D4×C3111yescharacteristic
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3111yescharacteristic
400Q16—48no—
401SD16—48no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—42no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4C2×D4×C3112yes—
421C4⋊C4C2×D4×C3112yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42C2×D4×C3111yescharacteristic
433C42C2×D4×C3111yescharacteristic
434C42C2×D4×C3111yescharacteristic
435C42C24×C3111yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23C31×#12011yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—81no—
450Q8—81no—
451Q8—82no—
452D4—42no—
453D4—42no—
454D4—81no—
455D4—81no—
456D4—84no—
457D4—84no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—82no—
473C2×C4C31×#12011yescharacteristic
474C2×C4C31×#12011yescharacteristic
475C2×C4C31×#12011yescharacteristic
476C2×C4C22×D4×C3111yescharacteristic
477C2×C4C22×D4×C3111yescharacteristic
478C2×C4C22×D4×C3111yescharacteristic
479C8—24no—
480C8—24no—
481C8—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—84no—
490C22C31×#45312yes—
491C22C31×#45512yes—
492C22C31×#45612yes—
493C22C2×C31×#12011yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C2—82no—
506C2—82no—
507C2C31×#166114yesdirect product factor
508C2C2×C31×#45311yescharacteristic
509C2C2×C31×#45511yescharacteristic
510C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#501C2semidirect
3. C2×C31×#497C2semidirect
4. C2×C31×#489C2semidirect
5. C31×#1661C2direct (N × H)
6. C2×C31×#330C2semidirect
7. C2×C31×#329C2semidirect
8. C2×C31×#329C2semidirect
26. C31×#501C22semidirect
28. C31×#497C22semidirect
30. C31×#489C22semidirect
38. C31×#330C22semidirect
39. C31×#329C22semidirect
40. C31×#329C22semidirect
180. C2×#1661C31direct (N × H)
228. C2×#501C2×C31semidirect
229. C2×#497C2×C31semidirect
230. C2×#489C2×C31semidirect
231. #1661C2×C31direct (N × H)
232. C2×#330C2×C31semidirect
233. C2×#329C2×C31semidirect
234. C2×#329C2×C31semidirect
275. #501C22×C31semidirect
277. #497C22×C31semidirect
279. #489C22×C31semidirect
287. #330C22×C31semidirect
288. #329C22×C31semidirect
289. #329C22×C31semidirect
292. C2×C31#1661direct (N × H)
365. C31C2×#1661direct (N × H)
507. C2C31×#1661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5622
C2×C31×#5012
C2×C31×#4972
C2×C31×#4892
C31×#16612
C2×C31×#3302
C2×C31×#3292
C2×#166131