Finite Group Databasebeta

← C31×#62099 #21518858

Subgroups — C31×#62099 #21518858

Order 7936.

1948 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3499C211yesmaximal, characteristic
2Q8×#41×C31C211yesmaximal, characteristic, complemented
3Q8×SD16×C31C212yesmaximal, complemented
4C31×#3359C212yesmaximal, complemented
5C31×#3335C216yesmaximal, complemented
6C31×#3298C216yesmaximal, complemented
7C31×#3246C213yesmaximal, complemented
8C31×#3229C213yesmaximal, complemented
9C31×#3103C213yesmaximal, complemented
10C2×C31×#475C213yesmaximal, complemented
11C31×#2988C211yesmaximal, characteristic, complemented
12C2×C31×2−1+4—21no—
13Q82×C31C2211yescharacteristic
14C31×#557—21no—
15C31×#557C2211yescharacteristic
16C31×#554—23no—
17C31×#554C2213yes—
18D4×Q8×C31—21no—
19D4×Q8×C31C2213yes—
20D4×Q8×C31C2212yes—
21C31×#547—23no—
22C31×#547C2216yes—
23C31×#536—23no—
24C31×#533—23no—
25C31×#533C2213yes—
26C31×#532C2213yes—
27C2×C31×C4⋊Q8C2213yes—
28C4×#41×C31—21no—
29C4×#41×C31C2211yescharacteristic
30C2×C4×Q8×C31C2213yes—
31C31×#498C2216yes—
32C31×#487C2216yes—
33C31×#483C22112yes—
34C31×#481C2216yes—
35C31×#476C2216yes—
36C31×#475C2216yes—
37C31×#475C2213yes—
38C31×#475C2213yes—
39C31×#475C2216yes—
40C31×#475C2216yes—
41C31×#465C22112yes—
42C31×#461C2216yes—
43C31×#460C2211yescharacteristic
44C31×#460C2213yes—
45C31×#448C2216yes—
46C8×Q8×C31C2212yes—
47C4×SD16×C31C2216yes—
48C4×SD16×C31C2216yes—
49C8×D4×C31C2216yes—
50C31×#431C2213yes—
51C2×C31×#106C2216yes—
52C2×C31×C4⋊C8C2213yes—
53C2×C31×#103C2216yes—
54C31×2−1+4—48no—
55C2×#41×C31—26no—
56C2×#41×C31—23no—
57C2×#41×C31D411yescharacteristic
58C22×Q8×C31—22no—
59C22×Q8×C31—23no—
60C22×Q8×C31C2313yes—
61C2×SD16×C31C2312yes—
62C2×SD16×C31C2316yes—
63C22×C8×C31C2316yes—
64C31×C4⋊Q8—23no—
65C31×C4⋊Q8—23no—
66C31×C4⋊Q8—23no—
67C31×C4⋊Q8—23no—
68C31×C4⋊Q8C2313yes—
69C31×C4⋊Q8C2313yes—
70C31×C4⋊Q8C2316yes—
71C31×C4⋊Q8C2313yes—
72C31×#127C2311yescharacteristic
73C31×#125—23no—
74C31×#125—23no—
75C31×#125—23no—
76C31×#124—23no—
77C31×#123—212no—
78C31×C22⋊Q8—23no—
79C31×C22⋊Q8—26no—
80C31×C22⋊Q8—26no—
81C31×C22⋊Q8—26no—
82C31×C22⋊Q8—26no—
83C31×C22⋊Q8C2316yes—
84C31×#121—23no—
85C31×#121C2316yes—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31—23no—
89C4×Q8×C31—23no—
90C4×Q8×C31—23no—
91C4×Q8×C31D411yescharacteristic
92C4×Q8×C31C2313yes—
93C4×Q8×C31C2313yes—
94C4×Q8×C31C2311yescharacteristic
95C4×Q8×C31C2313yes—
96C4×Q8×C31C2311yescharacteristic
97C4×D4×C31—23no—
98C4×D4×C31—23no—
99C4×D4×C31—23no—
100C4×D4×C31D413yes—
101C4×D4×C31C2316yes—
102C4×D4×C31C2313yes—
103C31×#117—23no—
104C31×#117D413yes—
105C2×C4⋊C4×C31—23no—
106C2×C4⋊C4×C31—23no—
107C2×C4⋊C4×C31C2316yes—
108C2×C4⋊C4×C31C2313yes—
109C2×C42×C31—23no—
110C2×C42×C31C2313yes—
111C31×#106C2316yes—
112C31×#106C23112yes—
113C31×#106C2316yes—
114C31×C4⋊C8C2313yes—
115C31×C4⋊C8C2316yes—
116C31×C4⋊C8C2311yescharacteristic
117C31×#103C2316yes—
118C31×#103C23112yes—
119C31×#103C2316yes—
120C31×#102C2312yes—
121C31×#102C2316yes—
122C31×C22⋊C8C2316yes—
123C4×C8×C31C2316yes—
124#41×C31—424no—
125#41×C31—412no—
126#41×C31SD1618yescomplemented
127C2×Q8×C31—23no—
128C2×Q8×C31—26no—
129C2×Q8×C31—23no—
130C2×Q8×C31—23no—
131C2×Q8×C31—26no—
132C2×Q8×C31—43no—
133C2×Q8×C31—48no—
134C2×Q8×C31—412no—
135C2×Q8×C31C2×D411yescharacteristic
136C2×Q8×C31C2413yes—
137C2×Q8×C31C2411yescharacteristic
138C2×D4×C31—26no—
139C2×D4×C31—21no—
140C2×D4×C31—21no—
141C2×D4×C31C2×D413yes—
142C2×D4×C31C2412yes—
143C22×C4×C31—23no—
144C22×C4×C31—26no—
145C22×C4×C31—23no—
146C22×C4×C31—26no—
147C22×C4×C31C2×D413yes—
148C22×C4×C31C2×D413yes—
149C22×C4×C31C2413yes—
150SD16×C31—24no—
151SD16×C31—46no—
152C2×C8×C31—46no—
153C2×C8×C31C2412yes—
154C2×C8×C31C2416yes—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—23no—
157C4⋊C4×C31—23no—
158C4⋊C4×C31—23no—
159C4⋊C4×C31—23no—
160C4⋊C4×C31—26no—
161C4⋊C4×C31—23no—
162C4⋊C4×C31—26no—
163C4⋊C4×C31—23no—
164C4⋊C4×C31—23no—
165C4⋊C4×C31—23no—
166C4⋊C4×C31—26no—
167C4⋊C4×C31C2×D413yes—
168C4⋊C4×C31C2×D413yes—
169C4⋊C4×C31C2411yescharacteristic
170C4⋊C4×C31C2413yes—
171C4⋊C4×C31C2416yes—
172C22⋊C4×C31—26no—
173C22⋊C4×C31—26no—
174C22⋊C4×C31—26no—
175C22⋊C4×C31C2×D416yes—
176C42×C31—21no—
177C42×C31—23no—
178C42×C31—23no—
179C42×C31—23no—
180C42×C31C2×D413yes—
181C42×C31C2×D413yes—
182C42×C31C2413yes—
183C42×C31C2411yescharacteristic
184#62099C3111yesmaximal, characteristic, direct product factor, Sylow
185C23×C31—22no—
186C23×C31C22×D413yes—
187Q8×C31—21no—
188Q8×C31—23no—
189Q8×C31—23no—
190Q8×C31—46no—
191Q8×C31—412no—
192Q8×C31—46no—
193Q8×C31—83no—
194Q8×C31C2×SD1614yes—
195Q8×C312−1+412yescomplemented
196D4×C31—22no—
197D4×C31—22no—
198D4×C31—42no—
199D4×C31—412no—
200D4×C31—42no—
201D4×C31C2×SD16112yes—
202C2×C4×C31—21no—
203C2×C4×C31—23no—
204C2×C4×C31—23no—
205C2×C4×C31—21no—
206C2×C4×C31—23no—
207C2×C4×C31—23no—
208C2×C4×C31—26no—
209C2×C4×C31—43no—
210C2×C4×C31—46no—
211C2×C4×C31—412no—
212C2×C4×C31—412no—
213C2×C4×C31—83no—
214C2×C4×C31C2×SD16112yes—
215C2×C4×C31C22×D411yescharacteristic
216C2×C4×C31C22×D413yes—
217C2×C4×C31C22×D413yes—
218C2×C4×C31C22×D411yescharacteristic
219C2×C4×C31C22×D413yes—
220C2×C4×C31C2511yescharacteristic
221C8×C31—22no—
222C8×C31—26no—
223#3499C2×C3111yescharacteristic
224Q8×#41C2×C3111yescharacteristic, complemented
225Q8×SD16C2×C3112yescomplemented
226#3359C2×C3112yescomplemented
227#3335C2×C3116yescomplemented
228#3298C2×C3116yescomplemented
229#3246C2×C3113yescomplemented
230#3229C2×C3113yescomplemented
231#3103C2×C3113yescomplemented
232C2×#475C2×C3113yescomplemented
233#2988C2×C3111yescharacteristic, complemented
234C22×C31—23no—
235C22×C31—23no—
236C22×C31—42no—
237C22×C31—42no—
238C22×C31—42no—
239C22×C31C22×SD1616yes—
240C22×C31C23×D411yescharacteristic
241C4×C31—21no—
242C4×C31—23no—
243C4×C31—23no—
244C4×C31—22no—
245C4×C31—41no—
246C4×C31—43no—
247C4×C31—43no—
248C4×C31—46no—
249C4×C31C22×SD1616yes—
250C4×C31C22×SD1612yes—
251C4×C31C2×2−1+411yescharacteristic
252C4×C31#58511yescharacteristic
253C2×2−1+4—21no—
254Q82C22×C3111yescharacteristic
255#557—21no—
256#557C22×C3111yescharacteristic
257#554—23no—
258#554C22×C3113yes—
259D4×Q8—21no—
260D4×Q8C22×C3113yes—
261D4×Q8C22×C3112yes—
262#547—23no—
263#547C22×C3116yes—
264#536—23no—
265#533—23no—
266#533C22×C3113yes—
267#532C22×C3113yes—
268C2×C4⋊Q8C22×C3113yes—
269C4×#41—21no—
270C4×#41C22×C3111yescharacteristic
271C2×C4×Q8C22×C3113yes—
272#498C22×C3116yes—
273#487C22×C3116yes—
274#483C22×C31112yes—
275#481C22×C3116yes—
276#476C22×C3116yes—
277#475C22×C3113yes—
278#475C22×C3116yes—
279#475C22×C3116yes—
280#475C22×C3116yes—
281#475C22×C3113yes—
282#465C22×C31112yes—
283#461C22×C3116yes—
284#460C22×C3111yescharacteristic
285#460C22×C3113yes—
286#448C22×C3116yes—
287C8×Q8C22×C3112yes—
288C4×SD16C22×C3116yes—
289C4×SD16C22×C3116yes—
290C8×D4C22×C3116yes—
291#431C22×C3113yes—
292C2×#106C22×C3116yes—
293C2×C4⋊C8C22×C3113yes—
294C2×#103C22×C3116yes—
295C2×C31—26no—
296C2×C31—82no—
297C2×C31#348911yescharacteristic
298C2×C31C23×SD1611yescharacteristic
299C2×C31#358611yescharacteristic
3002−1+4—48no—
301C2×#41—26no—
302C2×#41—23no—
303C2×#41D4×C3111yescharacteristic
304C22×Q8—22no—
305C22×Q8—23no—
306C22×Q8C23×C3113yes—
307C2×SD16C23×C3112yes—
308C2×SD16C23×C3116yes—
309C22×C8C23×C3116yes—
310C4⋊Q8—23no—
311C4⋊Q8—23no—
312C4⋊Q8—23no—
313C4⋊Q8—23no—
314C4⋊Q8C23×C3113yes—
315C4⋊Q8C23×C3113yes—
316C4⋊Q8C23×C3116yes—
317C4⋊Q8C23×C3113yes—
318#127C23×C3111yescharacteristic
319#125—23no—
320#125—23no—
321#125—23no—
322#124—23no—
323#123—212no—
324C22⋊Q8—23no—
325C22⋊Q8—26no—
326C22⋊Q8—26no—
327C22⋊Q8—26no—
328C22⋊Q8—26no—
329C22⋊Q8C23×C3116yes—
330#121—23no—
331#121C23×C3116yes—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—23no—
335C4×Q8—23no—
336C4×Q8—23no—
337C4×Q8D4×C3111yescharacteristic
338C4×Q8C23×C3111yescharacteristic
339C4×Q8C23×C3113yes—
340C4×Q8C23×C3113yes—
341C4×Q8C23×C3113yes—
342C4×Q8C23×C3111yescharacteristic
343C4×D4—23no—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4D4×C3113yes—
347C4×D4C23×C3116yes—
348C4×D4C23×C3113yes—
349#117—23no—
350#117D4×C3113yes—
351C2×C4⋊C4—23no—
352C2×C4⋊C4—23no—
353C2×C4⋊C4C23×C3113yes—
354C2×C4⋊C4C23×C3116yes—
355C2×C42—23no—
356C2×C42C23×C3113yes—
357#106C23×C3116yes—
358#106C23×C31112yes—
359#106C23×C3116yes—
360C4⋊C8C23×C3113yes—
361C4⋊C8C23×C3116yes—
362C4⋊C8C23×C3111yescharacteristic
363#103C23×C3116yes—
364#103C23×C31112yes—
365#103C23×C3116yes—
366#102C23×C3112yes—
367#102C23×C3116yes—
368C22⋊C8C23×C3116yes—
369C4×C8C23×C3116yes—
370C31#6209911yescharacteristic, direct product factor, Sylow
371#41—424no—
372#41—412no—
373#41SD16×C3118yescomplemented
374C2×Q8—23no—
375C2×Q8—26no—
376C2×Q8—23no—
377C2×Q8—23no—
378C2×Q8—26no—
379C2×Q8—43no—
380C2×Q8—48no—
381C2×Q8—412no—
382C2×Q8C2×D4×C3111yescharacteristic
383C2×Q8C24×C3113yes—
384C2×Q8C24×C3111yescharacteristic
385C2×D4—26no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4C2×D4×C3113yes—
389C2×D4C24×C3112yes—
390C22×C4—23no—
391C22×C4—26no—
392C22×C4—23no—
393C22×C4—26no—
394C22×C4C2×D4×C3113yes—
395C22×C4C2×D4×C3113yes—
396C22×C4C24×C3113yes—
397SD16—24no—
398SD16—46no—
399C2×C8—46no—
400C2×C8C24×C3112yes—
401C2×C8C24×C3116yes—
402C4⋊C4—21no—
403C4⋊C4—23no—
404C4⋊C4—23no—
405C4⋊C4—23no—
406C4⋊C4—23no—
407C4⋊C4—26no—
408C4⋊C4—23no—
409C4⋊C4—26no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4—26no—
414C4⋊C4C2×D4×C3113yes—
415C4⋊C4C2×D4×C3113yes—
416C4⋊C4C24×C3111yescharacteristic
417C4⋊C4C24×C3113yes—
418C4⋊C4C24×C3116yes—
419C22⋊C4—26no—
420C22⋊C4—26no—
421C22⋊C4—26no—
422C22⋊C4C2×D4×C3116yes—
423C42—21no—
424C42—23no—
425C42—23no—
426C42—23no—
427C42C2×D4×C3113yes—
428C42C2×D4×C3113yes—
429C42C24×C3113yes—
430C42C24×C3111yescharacteristic
431C23—22no—
432C23C22×D4×C3113yes—
433Q8—21no—
434Q8—23no—
435Q8—23no—
436Q8—46no—
437Q8—412no—
438Q8—46no—
439Q8—83no—
440Q8C2×SD16×C3114yes—
441Q8C31×2−1+412yescomplemented
442D4—22no—
443D4—22no—
444D4—42no—
445D4—412no—
446D4—42no—
447D4C2×SD16×C31112yes—
448C2×C4—21no—
449C2×C4—23no—
450C2×C4—23no—
451C2×C4—21no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—26no—
455C2×C4—43no—
456C2×C4—46no—
457C2×C4—412no—
458C2×C4—412no—
459C2×C4—83no—
460C2×C4C2×SD16×C31112yes—
461C2×C4C22×D4×C3111yescharacteristic
462C2×C4C22×D4×C3113yes—
463C2×C4C22×D4×C3113yes—
464C2×C4C22×D4×C3111yescharacteristic
465C2×C4C22×D4×C3113yes—
466C2×C4C25×C3111yescharacteristic
467C8—22no—
468C8—26no—
469C22—23no—
470C22—23no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22C22×SD16×C3116yes—
475C22C23×D4×C3111yescharacteristic
476C4—23no—
477C4—22no—
478C4—21no—
479C4—23no—
480C4—43no—
481C4—46no—
482C4—41no—
483C4—43no—
484C4C22×SD16×C3116yes—
485C4C22×SD16×C3112yes—
486C4C2×C31×2−1+411yescharacteristic
487C4C31×#58511yescharacteristic
488C2—26no—
489C2—82no—
490C2C31×#348911yescharacteristic
491C2C23×SD16×C3111yescharacteristic
492C2C31×#358611yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC31×#3499C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C4×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×#41×C31C2semidirect
3. Q8×SD16×C31C2semidirect
4. C31×#3359C2semidirect
5. C31×#3335C2semidirect
6. C31×#3298C2semidirect
7. C31×#3246C2semidirect
8. C31×#3229C2semidirect
9. C31×#3103C2semidirect
10. C2×C31×#475C2semidirect
11. C31×#2988C2semidirect
126. #41×C31SD16semidirect
184. #62099C31direct (N × H)
195. Q8×C312−1+4semidirect
224. Q8×#41C2×C31semidirect
225. Q8×SD16C2×C31semidirect
226. #3359C2×C31semidirect
227. #3335C2×C31semidirect
228. #3298C2×C31semidirect
229. #3246C2×C31semidirect
230. #3229C2×C31semidirect
231. #3103C2×C31semidirect
232. C2×#475C2×C31semidirect
233. #2988C2×C31semidirect
370. C31#62099direct (N × H)
373. #41SD16×C31semidirect
441. Q8C31×2−1+4semidirect

Maximal subgroups

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

subgroupindex
C31×#34992
Q8×#41×C312
Q8×SD16×C312
C31×#33592
C31×#33352
C31×#32982
C31×#32462
C31×#32292
C31×#31032
C2×C31×#4752
C31×#29882
#6209931