Finite Group Databasebeta

← C31×#36577 #21493336

Subgroups — C31×#36577 #21493336

Order 7936.

2312 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#557C211yesmaximal, characteristic, complemented
2C22×C4×Q8×C31C211yesmaximal, characteristic
3C31×#2857C213yesmaximal
4C31×#2582C213yesmaximal, complemented
5C31×#2568C213yesmaximal, complemented
6C31×#2453C216yesmaximal, complemented
7C31×#2442C216yesmaximal
8C31×#2351C212yesmaximal, complemented
9Q8×C4⋊C4×C31C211yesmaximal, characteristic, complemented
10Q8×C22⋊C4×C31C212yesmaximal
11C4×C31×C22⋊Q8C213yesmaximal
12C23×Q8×C31C2211yescharacteristic
13C31×#557—28no—
14C2×C31×C4⋊Q8C2213yes—
15C2×C31×#125C2216yes—
16C2×C31×C22⋊Q8C2213yes—
17C2×C4×Q8×C31—28no—
18C2×C4×Q8×C31C2213yes—
19C2×C4×Q8×C31C2213yes—
20C2×C4×Q8×C31C2212yes—
21C2×C4×Q8×C31C2212yes—
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×Q8×C31C2213yes—
24C2×C4×Q8×C31C2211yescharacteristic
25C22×C4⋊C4×C31C2213yes—
26C22×C42×C31C2213yes—
27C31×#400C22112yes—
28C31×#398C22112yes—
29C31×#391C2213yes—
30C31×#391C2216yes—
31C31×#391C2216yes—
32C31×#391C2216yes—
33C31×#389C2213yes—
34C31×#389C2216yes—
35C31×#389C2216yes—
36C31×#389C2216yes—
37C31×#389C2216yes—
38C31×#387C2216yes—
39C31×#385C2216yes—
40C31×#384C2213yes—
41C31×#382C2216yes—
42C31×#380C2216yes—
43C31×#380C2216yes—
44C31×#380C2211yescharacteristic
45C4×C4⋊C4×C31C2213yes—
46C4×C4⋊C4×C31C2216yes—
47C4×C22⋊C4×C31C2216yes—
48C22×Q8×C31—24no—
49C22×Q8×C31—24no—
50C22×Q8×C31Q818yescomplemented
51C22×Q8×C31C2313yes—
52C22×Q8×C31C2313yes—
53C22×Q8×C31C2311yescharacteristic
54C23×C4×C31C2313yes—
55C23×C4×C31C2313yes—
56C23×C4×C31C2311yescharacteristic
57C31×C4⋊Q8—212no—
58C31×#125—224no—
59C31×C22⋊Q8—212no—
60C4×Q8×C31—24no—
61C4×Q8×C31—28no—
62C4×Q8×C31—28no—
63C4×Q8×C31—212no—
64C4×Q8×C31—212no—
65C4×Q8×C31—212no—
66C4×Q8×C31D418yes—
67C2×C4⋊C4×C31—212no—
68C2×C4⋊C4×C31C2316yes—
69C2×C4⋊C4×C31C2316yes—
70C2×C4⋊C4×C31C2316yes—
71C2×C4⋊C4×C31C2313yes—
72C2×C4⋊C4×C31C2311yescharacteristic
73C2×C4⋊C4×C31C2316yes—
74C2×C4⋊C4×C31C2316yes—
75C2×C4⋊C4×C31C2313yes—
76C2×C4⋊C4×C31C2316yes—
77C2×C4⋊C4×C31C2311yescharacteristic
78C2×C4⋊C4×C31C2316yes—
79C2×C4⋊C4×C31C2316yes—
80C2×C4⋊C4×C31C2313yes—
81C2×C22⋊C4×C31C2316yes—
82C2×C22⋊C4×C31C2312yes—
83C2×C42×C31—212no—
84C2×C42×C31C2313yes—
85C2×C42×C31C2313yes—
86C2×C42×C31C2313yes—
87C2×C42×C31C2316yes—
88C2×C42×C31C2316yes—
89C2×C42×C31C2313yes—
90C31×#95C2316yes—
91C31×#95C23112yes—
92C31×#95C23112yes—
93C31×#95C2316yes—
94C31×#95C2316yes—
95C31×#95C2316yes—
96C31×#95C2312yes—
97C24×C31C2411yescharacteristic
98C2×Q8×C31—232no—
99C2×Q8×C31—26no—
100C2×Q8×C31—26no—
101C2×Q8×C31—26no—
102C2×Q8×C31—26no—
103C2×Q8×C31—26no—
104C2×Q8×C31C2×D414yes—
105C2×Q8×C31C2×Q8112yes—
106C2×Q8×C31C2×Q814yes—
107C2×Q8×C31#4114yes—
108C22×C4×C31—22no—
109C22×C4×C31—26no—
110C22×C4×C31—26no—
111C22×C4×C31—22no—
112C22×C4×C31—26no—
113C22×C4×C31—26no—
114C22×C4×C31—43no—
115C22×C4×C31C2×Q8112yes—
116C22×C4×C31#4114yes—
117C22×C4×C31C2412yes—
118C22×C4×C31C2413yes—
119C22×C4×C31C2416yes—
120C22×C4×C31C2416yes—
121C22×C4×C31C2412yes—
122C22×C4×C31C2413yes—
123C22×C4×C31C2413yes—
124C22×C4×C31C2411yescharacteristic
125C22×C4×C31C2413yes—
126C22×C4×C31C2411yescharacteristic
127C4⋊C4×C31—212no—
128C4⋊C4×C31—26no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—212no—
131C4⋊C4×C31—212no—
132C4⋊C4×C31—212no—
133C4⋊C4×C31—212no—
134C4⋊C4×C31—26no—
135C4⋊C4×C31—212no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—46no—
138C4⋊C4×C31—46no—
139C4⋊C4×C31C2×D4112yes—
140C22⋊C4×C31—212no—
141C22⋊C4×C31—24no—
142C42×C31—26no—
143C42×C31—26no—
144C42×C31—26no—
145C42×C31—212no—
146C42×C31—212no—
147C42×C31C2×D4112yes—
148#36577C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—21no—
150C23×C31—21no—
151C23×C31—21no—
152C23×C31—21no—
153C23×C31C22×Q812yes—
154C23×C31C2×#4112yes—
155C23×C312−1+412yes—
156C23×C31C2511yescharacteristic
157Q8×C31—224no—
158Q8×C31—412no—
159Q8×C31C22⋊Q8116yescomplemented
160C2×C4×C31—21no—
161C2×C4×C31—23no—
162C2×C4×C31—23no—
163C2×C4×C31—21no—
164C2×C4×C31—23no—
165C2×C4×C31—23no—
166C2×C4×C31—224no—
167C2×C4×C31—28no—
168C2×C4×C31—22no—
169C2×C4×C31—23no—
170C2×C4×C31—26no—
171C2×C4×C31—22no—
172C2×C4×C31—26no—
173C2×C4×C31—26no—
174C2×C4×C31—26no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—23no—
179C2×C4×C31—26no—
180C2×C4×C31—26no—
181C2×C4×C31—22no—
182C2×C4×C31—23no—
183C2×C4×C31—412no—
184C2×C4×C31C22×D416yes—
185C2×C4×C31C22×D416yes—
186C2×C4×C31C22×D412yes—
187C2×C4×C31C22×Q816yes—
188C2×C4×C31C22×Q816yes—
189C2×C4×C31C2×#4112yes—
190C2×C4×C31C2×#4116yes—
191C2×C4×C31C2×#4112yes—
192C2×C4×C312−1+412yes—
193C2×#557C2×C3111yescharacteristic, complemented
194C22×C4×Q8C2×C3111yescharacteristic
195#2857C2×C3113yes—
196#2582C2×C3113yescomplemented
197#2568C2×C3113yescomplemented
198#2453C2×C3116yescomplemented
199#2442C2×C3116yes—
200#2351C2×C3112yescomplemented
201Q8×C4⋊C4C2×C3111yescharacteristic, complemented
202Q8×C22⋊C4C2×C3112yes—
203C4×C22⋊Q8C2×C3113yes—
204C22×C31—22no—
205C22×C31—22no—
206C22×C31—22no—
207C22×C31—22no—
208C22×C31—22no—
209C22×C31—22no—
210C22×C31#55714yescomplemented
211C22×C31C23×D411yescharacteristic
212C22×C31C23×Q811yescharacteristic
213C22×C31C22×#4111yescharacteristic
214C22×C31C22×#4111yescharacteristic
215C22×C31C2×2−1+411yescharacteristic
216C22×C31C2×2−1+411yescharacteristic
217C22×C31#58511yescharacteristic
218C4×C31—26no—
219C4×C31—22no—
220C4×C31—26no—
221C4×C31—46no—
222C4×C31—46no—
223C4×C31—42no—
224C4×C31—43no—
225C4×C31—42no—
226C4×C31C2×C22⋊Q8112yes—
227C4×C31#54814yes—
228C23×Q8C22×C3111yescharacteristic
229#557—28no—
230C2×C4⋊Q8C22×C3113yes—
231C2×#125C22×C3116yes—
232C2×C22⋊Q8C22×C3113yes—
233C2×C4×Q8—28no—
234C2×C4×Q8C22×C3113yes—
235C2×C4×Q8C22×C3113yes—
236C2×C4×Q8C22×C3112yes—
237C2×C4×Q8C22×C3112yes—
238C2×C4×Q8C22×C3111yescharacteristic
239C2×C4×Q8C22×C3113yes—
240C2×C4×Q8C22×C3111yescharacteristic
241C22×C4⋊C4C22×C3113yes—
242C22×C42C22×C3113yes—
243#400C22×C31112yes—
244#398C22×C31112yes—
245#391C22×C3113yes—
246#391C22×C3116yes—
247#391C22×C3116yes—
248#391C22×C3116yes—
249#389C22×C3113yes—
250#389C22×C3116yes—
251#389C22×C3116yes—
252#389C22×C3116yes—
253#389C22×C3116yes—
254#387C22×C3116yes—
255#385C22×C3116yes—
256#384C22×C3113yes—
257#382C22×C3116yes—
258#380C22×C3116yes—
259#380C22×C3116yes—
260#380C22×C3111yescharacteristic
261C4×C4⋊C4C22×C3113yes—
262C4×C4⋊C4C22×C3116yes—
263C4×C22⋊C4C22×C3116yes—
264C2×C31—24no—
265C2×C31C22×C22⋊Q811yescharacteristic
266C2×C31C2×#54811yescharacteristic
267C2×C31C2×#55711yescharacteristic
268C2×C31#348011yescharacteristic
269C2×C31#348911yescharacteristic
270C2×C31#350211yescharacteristic
271C2×C31#352311yescharacteristic
272C22×Q8—24no—
273C22×Q8—24no—
274C22×Q8Q8×C3118yescomplemented
275C22×Q8C23×C3113yes—
276C22×Q8C23×C3113yes—
277C22×Q8C23×C3111yescharacteristic
278C23×C4C23×C3113yes—
279C23×C4C23×C3113yes—
280C23×C4C23×C3111yescharacteristic
281C4⋊Q8—212no—
282#125—224no—
283C22⋊Q8—212no—
284C4×Q8—24no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—212no—
288C4×Q8—212no—
289C4×Q8—212no—
290C4×Q8D4×C3118yes—
291C2×C4⋊C4—212no—
292C2×C4⋊C4C23×C3113yes—
293C2×C4⋊C4C23×C3116yes—
294C2×C4⋊C4C23×C3111yescharacteristic
295C2×C4⋊C4C23×C3116yes—
296C2×C4⋊C4C23×C3116yes—
297C2×C4⋊C4C23×C3116yes—
298C2×C4⋊C4C23×C3113yes—
299C2×C4⋊C4C23×C3116yes—
300C2×C4⋊C4C23×C3116yes—
301C2×C4⋊C4C23×C3111yescharacteristic
302C2×C4⋊C4C23×C3116yes—
303C2×C4⋊C4C23×C3116yes—
304C2×C4⋊C4C23×C3113yes—
305C2×C22⋊C4C23×C3116yes—
306C2×C22⋊C4C23×C3112yes—
307C2×C42—212no—
308C2×C42C23×C3116yes—
309C2×C42C23×C3113yes—
310C2×C42C23×C3113yes—
311C2×C42C23×C3113yes—
312C2×C42C23×C3116yes—
313C2×C42C23×C3113yes—
314#95C23×C3116yes—
315#95C23×C3116yes—
316#95C23×C3112yes—
317#95C23×C3116yes—
318#95C23×C3116yes—
319#95C23×C31112yes—
320#95C23×C31112yes—
321C31#3657711yescharacteristic, direct product factor, Sylow
322C24C24×C3111yescharacteristic
323C2×Q8—232no—
324C2×Q8—26no—
325C2×Q8—26no—
326C2×Q8—26no—
327C2×Q8—26no—
328C2×Q8—26no—
329C2×Q8C2×D4×C3114yes—
330C2×Q8C2×Q8×C31112yes—
331C2×Q8C2×Q8×C3114yes—
332C2×Q8#41×C3114yes—
333C22×C4—22no—
334C22×C4—26no—
335C22×C4—26no—
336C22×C4—22no—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—43no—
340C22×C4C2×Q8×C31112yes—
341C22×C4#41×C3114yes—
342C22×C4C24×C3112yes—
343C22×C4C24×C3113yes—
344C22×C4C24×C3116yes—
345C22×C4C24×C3116yes—
346C22×C4C24×C3112yes—
347C22×C4C24×C3113yes—
348C22×C4C24×C3113yes—
349C22×C4C24×C3111yescharacteristic
350C22×C4C24×C3113yes—
351C22×C4C24×C3111yescharacteristic
352C4⋊C4—212no—
353C4⋊C4—26no—
354C4⋊C4—22no—
355C4⋊C4—212no—
356C4⋊C4—212no—
357C4⋊C4—212no—
358C4⋊C4—212no—
359C4⋊C4—26no—
360C4⋊C4—212no—
361C4⋊C4—22no—
362C4⋊C4—46no—
363C4⋊C4—46no—
364C4⋊C4C2×D4×C31112yes—
365C22⋊C4—212no—
366C22⋊C4—24no—
367C42—26no—
368C42—26no—
369C42—26no—
370C42—212no—
371C42—212no—
372C42C2×D4×C31112yes—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23C22×Q8×C3112yes—
378C23C2×#41×C3112yes—
379C23C31×2−1+412yes—
380C23C25×C3111yescharacteristic
381Q8—224no—
382Q8—412no—
383Q8C31×C22⋊Q8116yescomplemented
384C2×C4—21no—
385C2×C4—23no—
386C2×C4—23no—
387C2×C4—21no—
388C2×C4—23no—
389C2×C4—23no—
390C2×C4—224no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—23no—
394C2×C4—26no—
395C2×C4—22no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—23no—
403C2×C4—26no—
404C2×C4—26no—
405C2×C4—22no—
406C2×C4—23no—
407C2×C4—412no—
408C2×C4C22×D4×C3116yes—
409C2×C4C22×D4×C3116yes—
410C2×C4C22×D4×C3112yes—
411C2×C4C22×Q8×C3116yes—
412C2×C4C22×Q8×C3116yes—
413C2×C4C2×#41×C3112yes—
414C2×C4C2×#41×C3116yes—
415C2×C4C2×#41×C3112yes—
416C2×C4C31×2−1+412yes—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22C31×#55714yescomplemented
424C22C23×D4×C3111yescharacteristic
425C22C23×Q8×C3111yescharacteristic
426C22C22×#41×C3111yescharacteristic
427C22C22×#41×C3111yescharacteristic
428C22C2×C31×2−1+411yescharacteristic
429C22C2×C31×2−1+411yescharacteristic
430C22C31×#58511yescharacteristic
431C4—26no—
432C4—22no—
433C4—26no—
434C4—46no—
435C4—46no—
436C4—43no—
437C4—42no—
438C4—42no—
439C4C2×C31×C22⋊Q8112yes—
440C4C31×#54814yes—
441C2—24no—
442C2C22×C31×C22⋊Q811yescharacteristic
443C2C2×C31×#54811yescharacteristic
444C2C2×C31×#55711yescharacteristic
445C2C31×#348011yescharacteristic
446C2C31×#348911yescharacteristic
447C2C31×#350211yescharacteristic
448C2C31×#352311yescharacteristic
subgroupG/H
center, socleC23×C31C25
commutator, Frattini, cosocleC23C25×C31
prime radicalC24×C31C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#557C2semidirect
4. C31×#2582C2semidirect
5. C31×#2568C2semidirect
6. C31×#2453C2semidirect
8. C31×#2351C2semidirect
9. Q8×C4⋊C4×C31C2semidirect
50. C22×Q8×C31Q8semidirect
148. #36577C31direct (N × H)
159. Q8×C31C22⋊Q8semidirect
193. C2×#557C2×C31semidirect
196. #2582C2×C31semidirect
197. #2568C2×C31semidirect
198. #2453C2×C31semidirect
200. #2351C2×C31semidirect
201. Q8×C4⋊C4C2×C31semidirect
210. C22×C31#557semidirect
274. C22×Q8Q8×C31semidirect
321. C31#36577direct (N × H)
383. Q8C31×C22⋊Q8semidirect
423. C22C31×#557semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5572
C22×C4×Q8×C312
C31×#28572
C31×#25822
C31×#25682
C31×#24532
C31×#24422
C31×#23512
Q8×C4⋊C4×C312
Q8×C22⋊C4×C312
C4×C31×C22⋊Q82
#3657731