Finite Group Databasebeta

← C31×#61319 #21518078

Subgroups — C31×#61319 #21518078

Order 7936.

1836 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3480C211yesmaximal, characteristic
2C2×Q82×C31C211yesmaximal, characteristic, complemented
3C31×#3375C216yesmaximal, complemented
4C31×#3369C216yesmaximal, complemented
5C31×#3363C212yesmaximal, complemented
6C31×#3362C212yesmaximal, complemented
7C31×#3141C213yesmaximal, complemented
8C31×#3079C213yesmaximal, complemented
9C31×#3037C213yesmaximal, complemented
10Q8×M16×C31C211yesmaximal, characteristic, complemented
11C4×C31×#137C213yesmaximal, complemented
12C2×C31×#137C2213yes—
13C31×#559C2213yes—
14Q82×C31—21no—
15Q82×C31—23no—
16Q82×C31C2212yes—
17Q82×C31C2216yes—
18C31×#557C2211yescharacteristic
19C31×#556C2213yes—
20C31×#553—23no—
21C31×#550C2211yescharacteristic
22C31×#548—21no—
23C31×#532C2213yes—
24C2×C31×C4⋊Q8—23no—
25C2×C31×C4⋊Q8C2213yes—
26C31×#529—23no—
27C4×#41×C31—21no—
28C4×#41×C31C2213yes—
29C2×C4×Q8×C31—21no—
30C2×C4×Q8×C31C2213yes—
31C2×C4×Q8×C31C2211yescharacteristic
32C31×#489C22112yes—
33C31×#487C2216yes—
34C31×#486C2216yes—
35C31×#477C2216yes—
36C31×#476C2216yes—
37C31×#475C2216yes—
38C31×#474C2216yes—
39C31×#464C22112yes—
40C31×#462C2216yes—
41C31×#460C2216yes—
42C31×#446C2216yes—
43C8×Q8×C31C2212yes—
44C31×#441C2216yes—
45C31×#440C2216yes—
46C4×Q16×C31C2216yes—
47C4×SD16×C31C2216yes—
48C31×#428C2213yes—
49C31×#423C2213yes—
50C31×#419C2211yescharacteristic
51C31×#417C2216yes—
52C31×#417C2213yes—
53C4×M16×C31C2213yes—
54C2×#41×C31—21no—
55C2×#41×C31C2313yes—
56C22×Q8×C31—23no—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31C2311yescharacteristic
59C31×#137—212no—
60C2×Q16×C31C2316yes—
61C2×SD16×C31C2316yes—
62C2×M16×C31C2311yescharacteristic
63C2×M16×C31C2313yes—
64C31×C4⋊Q8—26no—
65C31×C4⋊Q8—26no—
66C31×C4⋊Q8—212no—
67C31×C4⋊Q8—23no—
68C31×C4⋊Q8—23no—
69C31×C4⋊Q8C2313yes—
70C31×C4⋊Q8C2316yes—
71C31×C4⋊Q8C2316yes—
72C31×#127C2313yes—
73C31×#126—26no—
74C31×#125—23no—
75C31×#124—23no—
76C31×#124C2316yes—
77C31×#123—26no—
78C31×C22⋊Q8—23no—
79C31×#121—23no—
80C31×#121—23no—
81C31×#121—23no—
82C4×Q8×C31—26no—
83C4×Q8×C31—22no—
84C4×Q8×C31—21no—
85C4×Q8×C31—23no—
86C4×Q8×C31—23no—
87C4×Q8×C31D411yescharacteristic
88C4×Q8×C31D413yes—
89C4×Q8×C31C2316yes—
90C4×Q8×C31C2313yes—
91C4×Q8×C31C2316yes—
92C4×Q8×C31C2313yes—
93C4×Q8×C31C2311yescharacteristic
94C4×Q8×C31C2313yes—
95C4×D4×C31—23no—
96C4×D4×C31—23no—
97C4×D4×C31—23no—
98C4×D4×C31—23no—
99C4×D4×C31—23no—
100C4×D4×C31C2313yes—
101C4×D4×C31C2313yes—
102C31×#117—23no—
103C31×#117—23no—
104C31×#117C2311yescharacteristic
105C2×C4⋊C4×C31—23no—
106C2×C4⋊C4×C31—23no—
107C2×C4⋊C4×C31D413yes—
108C2×C4⋊C4×C31C2313yes—
109C2×C42×C31—23no—
110C2×C42×C31—23no—
111C2×C42×C31C2313yes—
112C31×#107C2316yes—
113C31×#106C2316yes—
114C31×C4⋊C8C2316yes—
115C31×C4⋊C8C2316yes—
116C31×#103C2316yes—
117C31×#103C23112yes—
118C31×#103C2312yes—
119C31×#103C2312yes—
120C31×#102C2316yes—
121C31×#102C23112yes—
122C31×#97C2316yes—
123C4×C8×C31C2316yes—
124#41×C31—26no—
125#41×C31—26no—
126#41×C31—44no—
127C2×Q8×C31—24no—
128C2×Q8×C31—21no—
129C2×Q8×C31—26no—
130C2×Q8×C31—26no—
131C2×Q8×C31—22no—
132C2×Q8×C31—86no—
133C2×Q8×C31C2×D413yes—
134C2×Q8×C31C2×D411yescharacteristic
135C2×Q8×C31#4114yescomplemented
136C2×Q8×C31C2413yes—
137C2×Q8×C31C2412yes—
138C2×D4×C31—23no—
139C2×D4×C31—23no—
140C2×D4×C31—23no—
141C2×D4×C31C2413yes—
142C22×C4×C31—21no—
143C22×C4×C31—23no—
144C22×C4×C31—23no—
145C22×C4×C31—23no—
146C22×C4×C31—23no—
147C22×C4×C31C2×D413yes—
148C22×C4×C31C2×D413yes—
149C22×C4×C31C2411yescharacteristic
150Q16×C31—212no—
151SD16×C31—212no—
152M16×C31—26no—
153M16×C31—22no—
154C2×C8×C31C2412yes—
155C2×C8×C31C2416yes—
156C4⋊C4×C31—26no—
157C4⋊C4×C31—26no—
158C4⋊C4×C31—26no—
159C4⋊C4×C31—26no—
160C4⋊C4×C31—23no—
161C4⋊C4×C31—23no—
162C4⋊C4×C31—23no—
163C4⋊C4×C31—23no—
164C4⋊C4×C31—23no—
165C4⋊C4×C31C2×D413yes—
166C4⋊C4×C31C2×D413yes—
167C4⋊C4×C31C2×D416yes—
168C4⋊C4×C31C2416yes—
169C4⋊C4×C31C2411yescharacteristic
170C4⋊C4×C31C2411yescharacteristic
171C22⋊C4×C31—21no—
172C22⋊C4×C31—26no—
173C22⋊C4×C31—23no—
174C22⋊C4×C31—23no—
175C22⋊C4×C31—26no—
176C42×C31—21no—
177C42×C31—23no—
178C42×C31—23no—
179C42×C31—26no—
180C42×C31—26no—
181C42×C31C2×D413yes—
182C42×C31C2×D413yes—
183C42×C31C2413yes—
184C42×C31C2413yes—
185#61319C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—23no—
187C23×C31C22×D411yescharacteristic
188Q8×C31—24no—
189Q8×C31—26no—
190Q8×C31—22no—
191Q8×C31—23no—
192Q8×C31—23no—
193Q8×C31—86no—
194Q8×C31—86no—
195Q8×C31#13714yescomplemented
196Q8×C31C2×#4114yes—
197D4×C31—23no—
198D4×C31—23no—
199D4×C31—46no—
200D4×C31—46no—
201D4×C31—83no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—26no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—23no—
208C2×C4×C31—26no—
209C2×C4×C31—43no—
210C2×C4×C31—46no—
211C2×C4×C31—46no—
212C2×C4×C31—46no—
213C2×C4×C31—83no—
214C2×C4×C31C22×D413yes—
215C2×C4×C31C22×D413yes—
216C2×C4×C31C22×D413yes—
217C2×C4×C31C22×D411yescharacteristic
218C2×C4×C31C22×D413yes—
219C2×C4×C31C2×#4112yes—
220C2×C4×C312−1+412yes—
221C2×C4×C31C2511yescharacteristic
222C8×C31—26no—
223C8×C31—22no—
224#3480C2×C3111yescharacteristic
225C2×Q82C2×C3111yescharacteristic, complemented
226#3375C2×C3116yescomplemented
227#3369C2×C3116yescomplemented
228#3363C2×C3112yescomplemented
229#3362C2×C3112yescomplemented
230#3141C2×C3113yescomplemented
231#3079C2×C3113yescomplemented
232#3037C2×C3113yescomplemented
233Q8×M16C2×C3111yescharacteristic, complemented
234C4×#137C2×C3113yescomplemented
235C22×C31—21no—
236C22×C31—21no—
237C22×C31—43no—
238C22×C31—43no—
239C22×C31—43no—
240C22×C31#54812yes—
241C22×C31C23×D411yescharacteristic
242C4×C31—23no—
243C4×C31—23no—
244C4×C31—24no—
245C4×C31—23no—
246C4×C31—46no—
247C4×C31—41no—
248C4×C31—43no—
249C4×C31—41no—
250C4×C31#54812yes—
251C4×C31C2×#13716yes—
252C4×C31C22×#4111yescharacteristic
253C4×C31C2×2−1+411yescharacteristic
254C2×#137C22×C3113yes—
255#559C22×C3113yes—
256Q82—21no—
257Q82—23no—
258Q82C22×C3112yes—
259Q82C22×C3116yes—
260#557C22×C3111yescharacteristic
261#556C22×C3113yes—
262#553—23no—
263#550C22×C3111yescharacteristic
264#548—21no—
265#532C22×C3113yes—
266C2×C4⋊Q8—23no—
267C2×C4⋊Q8C22×C3113yes—
268#529—23no—
269C4×#41—21no—
270C4×#41C22×C3113yes—
271C2×C4×Q8—21no—
272C2×C4×Q8C22×C3113yes—
273C2×C4×Q8C22×C3111yescharacteristic
274#489C22×C31112yes—
275#487C22×C3116yes—
276#486C22×C3116yes—
277#477C22×C3116yes—
278#476C22×C3116yes—
279#475C22×C3116yes—
280#474C22×C3116yes—
281#464C22×C31112yes—
282#462C22×C3116yes—
283#460C22×C3116yes—
284#446C22×C3116yes—
285C8×Q8C22×C3112yes—
286#441C22×C3116yes—
287#440C22×C3116yes—
288C4×Q16C22×C3116yes—
289C4×SD16C22×C3116yes—
290#428C22×C3113yes—
291#423C22×C3113yes—
292#419C22×C3111yescharacteristic
293#417C22×C3116yes—
294#417C22×C3113yes—
295C4×M16C22×C3113yes—
296C2×C31—22no—
297C2×C31—83no—
298C2×C31C2×#54811yescharacteristic
299C2×C31C22×#13711yescharacteristic
300C2×C31#358511yescharacteristic
301C2×#41—21no—
302C2×#41C23×C3113yes—
303C22×Q8—23no—
304C22×Q8D4×C3111yescharacteristic
305C22×Q8C23×C3111yescharacteristic
306#137—212no—
307C2×Q16C23×C3116yes—
308C2×SD16C23×C3116yes—
309C2×M16C23×C3113yes—
310C2×M16C23×C3111yescharacteristic
311C4⋊Q8—26no—
312C4⋊Q8—26no—
313C4⋊Q8—212no—
314C4⋊Q8—23no—
315C4⋊Q8—23no—
316C4⋊Q8C23×C3113yes—
317C4⋊Q8C23×C3116yes—
318C4⋊Q8C23×C3116yes—
319#127C23×C3113yes—
320#126—26no—
321#125—23no—
322#124—23no—
323#124C23×C3116yes—
324#123—26no—
325C22⋊Q8—23no—
326#121—23no—
327#121—23no—
328#121—23no—
329C4×Q8—26no—
330C4×Q8—22no—
331C4×Q8—21no—
332C4×Q8—23no—
333C4×Q8—23no—
334C4×Q8D4×C3111yescharacteristic
335C4×Q8D4×C3113yes—
336C4×Q8C23×C3116yes—
337C4×Q8C23×C3116yes—
338C4×Q8C23×C3113yes—
339C4×Q8C23×C3113yes—
340C4×Q8C23×C3111yescharacteristic
341C4×Q8C23×C3113yes—
342C4×D4—23no—
343C4×D4—23no—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4C23×C3113yes—
348C4×D4C23×C3113yes—
349#117—23no—
350#117—23no—
351#117C23×C3111yescharacteristic
352C2×C4⋊C4—23no—
353C2×C4⋊C4—23no—
354C2×C4⋊C4D4×C3113yes—
355C2×C4⋊C4C23×C3113yes—
356C2×C42—23no—
357C2×C42—23no—
358C2×C42C23×C3113yes—
359#107C23×C3116yes—
360#106C23×C3116yes—
361C4⋊C8C23×C3116yes—
362C4⋊C8C23×C3116yes—
363#103C23×C3116yes—
364#103C23×C31112yes—
365#103C23×C3112yes—
366#103C23×C3112yes—
367#102C23×C3116yes—
368#102C23×C31112yes—
369#97C23×C3116yes—
370C4×C8C23×C3116yes—
371C31#6131911yescharacteristic, direct product factor, Sylow
372#41—26no—
373#41—26no—
374#41—44no—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—26no—
378C2×Q8—26no—
379C2×Q8—22no—
380C2×Q8—86no—
381C2×Q8C2×D4×C3113yes—
382C2×Q8C2×D4×C3111yescharacteristic
383C2×Q8#41×C3114yescomplemented
384C2×Q8C24×C3113yes—
385C2×Q8C24×C3112yes—
386C2×D4—23no—
387C2×D4—23no—
388C2×D4—23no—
389C2×D4C24×C3113yes—
390C22×C4—21no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4C2×D4×C3113yes—
396C22×C4C2×D4×C3113yes—
397C22×C4C24×C3111yescharacteristic
398Q16—212no—
399SD16—212no—
400M16—26no—
401M16—22no—
402C2×C8C24×C3112yes—
403C2×C8C24×C3116yes—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—23no—
409C4⋊C4—23no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4C2×D4×C3113yes—
414C4⋊C4C2×D4×C3113yes—
415C4⋊C4C2×D4×C3116yes—
416C4⋊C4C24×C3116yes—
417C4⋊C4C24×C3111yescharacteristic
418C4⋊C4C24×C3111yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—26no—
421C22⋊C4—23no—
422C22⋊C4—23no—
423C22⋊C4—26no—
424C42—21no—
425C42—23no—
426C42—23no—
427C42—26no—
428C42—26no—
429C42C2×D4×C3113yes—
430C42C2×D4×C3113yes—
431C42C24×C3113yes—
432C42C24×C3113yes—
433C23—23no—
434C23C22×D4×C3111yescharacteristic
435Q8—24no—
436Q8—26no—
437Q8—22no—
438Q8—23no—
439Q8—23no—
440Q8—86no—
441Q8—86no—
442Q8C31×#13714yescomplemented
443Q8C2×#41×C3114yes—
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—26no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—26no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C3113yes—
462C2×C4C22×D4×C3113yes—
463C2×C4C22×D4×C3113yes—
464C2×C4C22×D4×C3111yescharacteristic
465C2×C4C22×D4×C3113yes—
466C2×C4C2×#41×C3112yes—
467C2×C4C31×2−1+412yes—
468C2×C4C25×C3111yescharacteristic
469C8—26no—
470C8—22no—
471C22—21no—
472C22—21no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22C31×#54812yes—
477C22C23×D4×C3111yescharacteristic
478C4—24no—
479C4—23no—
480C4—23no—
481C4—23no—
482C4—43no—
483C4—41no—
484C4—46no—
485C4—41no—
486C4C31×#54812yes—
487C4C2×C31×#13716yes—
488C4C22×#41×C3111yescharacteristic
489C4C2×C31×2−1+411yescharacteristic
490C2—22no—
491C2—83no—
492C2C2×C31×#54811yescharacteristic
493C2C22×C31×#13711yescharacteristic
494C2C31×#358511yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC31×#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×Q82×C31C2semidirect
3. C31×#3375C2semidirect
4. C31×#3369C2semidirect
5. C31×#3363C2semidirect
6. C31×#3362C2semidirect
7. C31×#3141C2semidirect
8. C31×#3079C2semidirect
9. C31×#3037C2semidirect
10. Q8×M16×C31C2semidirect
11. C4×C31×#137C2semidirect
135. C2×Q8×C31#41semidirect
185. #61319C31direct (N × H)
195. Q8×C31#137semidirect
225. C2×Q82C2×C31semidirect
226. #3375C2×C31semidirect
227. #3369C2×C31semidirect
228. #3363C2×C31semidirect
229. #3362C2×C31semidirect
230. #3141C2×C31semidirect
231. #3079C2×C31semidirect
232. #3037C2×C31semidirect
233. Q8×M16C2×C31semidirect
234. C4×#137C2×C31semidirect
371. C31#61319direct (N × H)
383. C2×Q8#41×C31semidirect
442. Q8C31×#137semidirect

Maximal subgroups

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

subgroupindex
C31×#34802
C2×Q82×C312
C31×#33752
C31×#33692
C31×#33632
C31×#33622
C31×#31412
C31×#30792
C31×#30372
Q8×M16×C312
C4×C31×#1372
#6131931