Finite Group Databasebeta

← C31×#8985 #21465744

Subgroups — C31×#8985 #21465744

Order 7936.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#461C211yesmaximal, characteristic
2D4×C4⋊C4×C31C211yesmaximal, characteristic
3C31×#1990C211yesmaximal, characteristic, complemented
4C31×#1988C211yesmaximal, characteristic, complemented
5C2×C31×D4⋊C8C211yesmaximal, characteristic
6C31×#1296C211yesmaximal, characteristic, complemented
7C31×#1279C211yesmaximal, characteristic, complemented
8C22×SD16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C4×D4×C31—21no—
11C2×C4×D4×C31C2211yescharacteristic
12C22×C4⋊C4×C31—21no—
13C31×#461—22no—
14C31×#461C414yescomplemented
15C2×C31×C4⋊C8C2211yescharacteristic
16C2×C31×C4⋊C8C2211yescharacteristic
17C2×C31×#103—21no—
18C2×C4×C8×C31C2211yescharacteristic
19C31×#389—21no—
20C31×#385—22no—
21C31×#384C2211yescharacteristic
22C31×#380—21no—
23C4×C4⋊C4×C31C2211yescharacteristic
24C31×#340—21no—
25C31×#340—21no—
26C31×#340—22no—
27C31×D4⋊C8—22no—
28C22×Q8×C31—21no—
29C22×D4×C31D411yescharacteristic
30C23×C4×C31—21no—
31C23×C4×C31—22no—
32C2×SD16×C31—24no—
33C2×SD16×C31—42no—
34C2×SD16×C31—42no—
35C22×C8×C31—21no—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8C2×C412yescomplemented
42C4×D4×C31—21no—
43C4×D4×C31—21no—
44C4×D4×C31—42no—
45C4×D4×C31—42no—
46C4×D4×C31C2×C412yes—
47C4×D4×C31D412yes—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—22no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—42no—
54C2×C4⋊C4×C31—42no—
55C2×C4⋊C4×C31D411yescharacteristic
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D412yes—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—22no—
61C2×C42×C31—21no—
62C2×C42×C31C2311yescharacteristic
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8C2×C412yescomplemented
67C31×#103—24no—
68C4×C8×C31—21no—
69C4×C8×C31—21no—
70C31×#95—22no—
71C31×#95—21no—
72C24×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31C22⋊C412yes—
83C2×D4×C31D812yes—
84C2×D4×C31SD1612yes—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—22no—
88C22×C4×C31—22no—
89C22×C4×C31—21no—
90C22×C4×C31—22no—
91C22×C4×C31—44no—
92C22×C4×C31—41no—
93C22×C4×C31—41no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—44no—
97C22×C4×C31—41no—
98C22×C4×C31—82no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31C2×D411yescharacteristic
102SD16×C31—48no—
103C2×C8×C31—22no—
104C2×C8×C31—22no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—44no—
119C4⋊C4×C31—42no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C4⋊C4×C31—41no—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31C22⋊C412yes—
127C4⋊C4×C31SD1612yescomplemented
128C22⋊C4×C31—22no—
129C22⋊C4×C31—22no—
130C22⋊C4×C31—44no—
131C42×C31—41no—
132C42×C31—41no—
133C42×C31C22×C411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31#4111yescharacteristic
136C42×C31#4111yescharacteristic
137#8985C3111yesmaximal, characteristic, direct product factor, Sylow
138C23×C31—22no—
139C23×C31—22no—
140C23×C31—22no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31#12011yescharacteristic
146Q8×C31—42no—
147Q8×C31—42no—
148Q8×C31—81no—
149Q8×C31—81no—
150D4×C31—22no—
151D4×C31—22no—
152D4×C31—22no—
153D4×C31#10214yes—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—22no—
157C2×C4×C31—22no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—88no—
174C2×C4×C31—81no—
175C2×C4×C31—81no—
176C2×C4×C31C2×C22⋊C411yescharacteristic
177C2×C4×C31C4×D411yescharacteristic
178C2×C4×C31C4×D411yescharacteristic
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31#12311yescharacteristic
182C2×C4×C31C2×D811yescharacteristic
183C2×C4×C31C2×SD1611yescharacteristic
184C2×C4×C31C2×SD1611yescharacteristic
185C2×C4×C31#13511yescharacteristic
186C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C31#13711yescharacteristic
189C2×C4×C31#13711yescharacteristic
190C8×C31—42no—
191C8×C31—42no—
192C8×C31—81no—
193C2×#461C2×C3111yescharacteristic
194D4×C4⋊C4C2×C3111yescharacteristic
195#1990C2×C3111yescharacteristic, complemented
196#1988C2×C3111yescharacteristic, complemented
197C2×D4⋊C8C2×C3111yescharacteristic
198#1296C2×C3111yescharacteristic, complemented
199#1279C2×C3111yescharacteristic, complemented
200C22×C31—24no—
201C22×C31—42no—
202C22×C31—42no—
203C22×C31—42no—
204C22×C31—42no—
205C22×C31—42no—
206C22×C31—42no—
207C22×C31#38611yescharacteristic
208C22×C31#44711yescharacteristic
209C22×C31#44811yescharacteristic
210C22×C31#45011yescharacteristic
211C22×C31#45211yescharacteristic
212C22×C31#45511yescharacteristic
213C22×C31#45611yescharacteristic
214C4×C31—21no—
215C4×C31—21no—
216C4×C31—42no—
217C4×C31—42no—
218C4×C31—82no—
219C4×C31—82no—
220C4×C31—81no—
221C4×C31C2×#10211yescharacteristic
222C4×C31#41711yescharacteristic
223C4×C31C4×SD1611yescharacteristic
224C4×C31#44011yescharacteristic
225C4×C31#46111yescharacteristic
226C4×C31#46311yescharacteristic
227C4×C31#48011yescharacteristic
228C4×C31#48111yescharacteristic
229C22×SD16—21no—
230C2×C4⋊Q8C22×C3111yescharacteristic
231C2×C4×D4—21no—
232C2×C4×D4C22×C3111yescharacteristic
233C22×C4⋊C4—21no—
234#461—22no—
235#461C4×C3114yescomplemented
236C2×C4⋊C8C22×C3111yescharacteristic
237C2×C4⋊C8C22×C3111yescharacteristic
238C2×#103—21no—
239C2×C4×C8C22×C3111yescharacteristic
240#389—21no—
241#385—22no—
242#384C22×C3111yescharacteristic
243#380—21no—
244C4×C4⋊C4C22×C3111yescharacteristic
245#340—21no—
246#340—21no—
247#340—22no—
248D4⋊C8—22no—
249C2×C31—44no—
250C2×C31#163911yescharacteristic
251C2×C31#164311yescharacteristic
252C2×C31#164711yescharacteristic
253C2×C31#165111yescharacteristic
254C2×C31#187411yescharacteristic
255C2×C31#187711yescharacteristic
256C2×C31#188211yescharacteristic
257C22×Q8—21no—
258C22×D4D4×C3111yescharacteristic
259C23×C4—21no—
260C23×C4—22no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×SD16—42no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8C2×C4×C3112yescomplemented
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4C2×C4×C3112yes—
276C4×D4D4×C3112yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4D4×C3111yescharacteristic
285C2×C4⋊C4D4×C3111yescharacteristic
286C2×C4⋊C4D4×C3111yescharacteristic
287C2×C4⋊C4D4×C3112yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C42—21no—
291C2×C42C23×C3111yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8C2×C4×C3112yescomplemented
296#103—24no—
297C4×C8—21no—
298C4×C8—21no—
299#95—22no—
300#95—21no—
301C31#898511yescharacteristic, direct product factor, Sylow
302C24—21no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4C22⋊C4×C3112yes—
313C2×D4D8×C3112yes—
314C2×D4SD16×C3112yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4C2×D4×C3111yescharacteristic
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3111yescharacteristic
332SD16—48no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4C22⋊C4×C3112yes—
357C4⋊C4SD16×C3112yescomplemented
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—44no—
361C42—41no—
362C42—41no—
363C42C22×C4×C3111yescharacteristic
364C42C2×D4×C3111yescharacteristic
365C42#41×C3111yescharacteristic
366C42#41×C3111yescharacteristic
367C23—22no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23C31×#12011yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—81no—
378Q8—81no—
379D4—22no—
380D4—22no—
381D4—22no—
382D4C31×#10214yes—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—88no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊C4×C3111yescharacteristic
406C2×C4C4×D4×C3111yescharacteristic
407C2×C4C4×D4×C3111yescharacteristic
408C2×C4C31×#12111yescharacteristic
409C2×C4C31×#12111yescharacteristic
410C2×C4C31×#12311yescharacteristic
411C2×C4C2×D8×C3111yescharacteristic
412C2×C4C2×SD16×C3111yescharacteristic
413C2×C4C2×SD16×C3111yescharacteristic
414C2×C4C31×#13511yescharacteristic
415C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
417C2×C4C31×#13711yescharacteristic
418C2×C4C31×#13711yescharacteristic
419C8—42no—
420C8—42no—
421C8—81no—
422C22—24no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22C31×#38611yescharacteristic
430C22C31×#44711yescharacteristic
431C22C31×#44811yescharacteristic
432C22C31×#45011yescharacteristic
433C22C31×#45211yescharacteristic
434C22C31×#45511yescharacteristic
435C22C31×#45611yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4C2×C31×#10211yescharacteristic
444C4C31×#41711yescharacteristic
445C4C4×SD16×C3111yescharacteristic
446C4C31×#44011yescharacteristic
447C4C31×#46111yescharacteristic
448C4C31×#46311yescharacteristic
449C4C31×#48011yescharacteristic
450C4C31×#48111yescharacteristic
451C2—44no—
452C2C31×#163911yescharacteristic
453C2C31×#164311yescharacteristic
454C2C31×#164711yescharacteristic
455C2C31×#165111yescharacteristic
456C2C31×#187411yescharacteristic
457C2C31×#187711yescharacteristic
458C2C31×#188211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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
3. C31×#1990C2semidirect
4. C31×#1988C2semidirect
6. C31×#1296C2semidirect
7. C31×#1279C2semidirect
14. C31×#461C4semidirect
41. C31×C4⋊Q8C2×C4semidirect
66. C31×C4⋊C8C2×C4semidirect
127. C4⋊C4×C31SD16semidirect
137. #8985C31direct (N × H)
195. #1990C2×C31semidirect
196. #1988C2×C31semidirect
198. #1296C2×C31semidirect
199. #1279C2×C31semidirect
235. #461C4×C31semidirect
270. C4⋊Q8C2×C4×C31semidirect
295. C4⋊C8C2×C4×C31semidirect
301. C31#8985direct (N × H)
357. C4⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4612
D4×C4⋊C4×C312
C31×#19902
C31×#19882
C2×C31×D4⋊C82
C31×#12962
C31×#12792
#898531