Finite Group Databasebeta

← C31×#8310 #21465069

Subgroups — C31×#8310 #21465069

Order 7936.

1520 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C31C211yesmaximal, characteristic
2C31×#1797C211yesmaximal, characteristic, complemented
3C4×C31×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×C31×D4⋊C8C214yesmaximal, complemented
5C23×D4×C31C411yescharacteristic
6C24×C4×C31—21no—
7C2×C4×D4×C31—28no—
8C2×C4×D4×C31—21no—
9C2×C4×D4×C31—21no—
10C2×C4×D4×C31C412yes—
11C2×C4×D4×C31C2212yes—
12C22×C4⋊C4×C31C411yescharacteristic
13C22×C22⋊C4×C31—21no—
14C22×C42×C31C2211yescharacteristic
15C2×C31×C4⋊C8C2212yescomplemented
16C2×C31×C22⋊C8—21no—
17C2×C31×C22⋊C8—21no—
18C2×C4×C8×C31C2212yescomplemented
19C31×#336—21no—
20C31×#336—21no—
21C31×D4⋊C8—28no—
22C25×C31—21no—
23C22×D4×C31—24no—
24C22×D4×C31—24no—
25C22×D4×C31—21no—
26C22×D4×C31—22no—
27C22×D4×C31C814yescomplemented
28C22×D4×C31C2×C412yes—
29C23×C4×C31—28no—
30C23×C4×C31—21no—
31C23×C4×C31—22no—
32C23×C4×C31—22no—
33C23×C4×C31—44no—
34C23×C4×C31—42no—
35C23×C4×C31—42no—
36C23×C4×C31C2×C411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31D411yescharacteristic
39C22×C8×C31—22no—
40C22×C8×C31—22no—
41C4×D4×C31—28no—
42C4×D4×C31—28no—
43C4×D4×C31—24no—
44C4×D4×C31—24no—
45C4×D4×C31—24no—
46C4×D4×C31D418yes—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31—24no—
49C2×C4⋊C4×C31C2×C412yes—
50C2×C22⋊C4×C31—28no—
51C2×C22⋊C4×C31—22no—
52C2×C22⋊C4×C31—22no—
53C2×C42×C31—24no—
54C2×C42×C31C2×C411yescharacteristic
55C2×C42×C31D411yescharacteristic
56C2×C42×C31D411yescharacteristic
57C2×C42×C31C2311yescharacteristic
58C31×C4⋊C8—24no—
59C31×C22⋊C8—24no—
60C31×C22⋊C8—42no—
61C4×C8×C31—24no—
62C24×C31—24no—
63C24×C31—24no—
64C24×C31—24no—
65C24×C31—22no—
66C24×C31—42no—
67C24×C31—42no—
68C24×C31—42no—
69C24×C31—42no—
70C24×C31C22⋊C411yescharacteristic
71C2×D4×C31—24no—
72C2×D4×C31—24no—
73C2×D4×C31—216no—
74C2×D4×C31—216no—
75C2×D4×C31—24no—
76C2×D4×C31—22no—
77C2×D4×C31—22no—
78C2×D4×C31—22no—
79C2×D4×C31C22⋊C414yes—
80C2×D4×C31C2×C814yes—
81C2×D4×C31M1614yes—
82C22×C4×C31—22no—
83C22×C4×C31—22no—
84C22×C4×C31—22no—
85C22×C4×C31—22no—
86C22×C4×C31—24no—
87C22×C4×C31—28no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—28no—
92C22×C4×C31—41no—
93C22×C4×C31—44no—
94C22×C4×C31—416no—
95C22×C4×C31—416no—
96C22×C4×C31—42no—
97C22×C4×C31—42no—
98C22×C4×C31—44no—
99C22×C4×C31—42no—
100C22×C4×C31—42no—
101C22×C4×C31—44no—
102C22×C4×C31C22⋊C411yescharacteristic
103C22×C4×C31C22⋊C411yescharacteristic
104C22×C4×C31C22⋊C411yescharacteristic
105C22×C4×C31C2×C812yes—
106C22×C4×C31M1612yes—
107C22×C4×C31D812yes—
108C22×C4×C31SD1612yes—
109C22×C4×C31C22×C411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2×D411yescharacteristic
112C2×C8×C31—24no—
113C2×C8×C31—42no—
114C2×C8×C31—42no—
115C2×C8×C31—42no—
116C2×C8×C31—42no—
117C2×C8×C31—42no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31C22⋊C414yes—
121C22⋊C4×C31—28no—
122C22⋊C4×C31—28no—
123C42×C31—22no—
124C42×C31—22no—
125C42×C31—22no—
126C42×C31C2×D414yes—
127#8310C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—21no—
129C23×C31—21no—
130C23×C31—21no—
131C23×C31—21no—
132C23×C31—216no—
133C23×C31—24no—
134C23×C31—24no—
135C23×C31—24no—
136C23×C31—48no—
137C23×C31—48no—
138C23×C31—48no—
139C23×C31—48no—
140C23×C31—48no—
141C23×C31—48no—
142C23×C31—42no—
143C23×C31—42no—
144C23×C31—42no—
145C23×C31—42no—
146C23×C31C22⋊C812yes—
147C23×C31#10212yes—
148C23×C31#10412yes—
149C23×C31C2×C22⋊C411yescharacteristic
150D4×C31—216no—
151D4×C31—24no—
152D4×C31—24no—
153D4×C31—24no—
154D4×C31C22⋊C818yescomplemented
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—24no—
162C2×C4×C31—24no—
163C2×C4×C31—24no—
164C2×C4×C31—24no—
165C2×C4×C31—21no—
166C2×C4×C31—24no—
167C2×C4×C31—21no—
168C2×C4×C31—21no—
169C2×C4×C31—44no—
170C2×C4×C31—416no—
171C2×C4×C31—48no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—48no—
175C2×C4×C31—48no—
176C2×C4×C31C22⋊C812yes—
177C2×C4×C31#10212yes—
178C2×C4×C31C2×C22⋊C412yes—
179C2×C4×C31#12012yes—
180C2×C4×C31#12012yes—
181C2×C4×C31C22×C811yescharacteristic
182C2×C4×C31C2×M1612yes—
183C2×C4×C31C2×M1611yescharacteristic
184C2×C4×C31C2×D811yescharacteristic
185C2×C4×C31C2×SD1611yescharacteristic
186C2×C4×C31AGL(1,ℤ/8)12yes—
187C8×C31—44no—
188C8×C31—82no—
189C22×C4×D4C2×C3111yescharacteristic
190#1797C2×C3111yescharacteristic, complemented
191C4×C22⋊C8C2×C3111yescharacteristic, complemented
192C2×D4⋊C8C2×C3114yescomplemented
193C22×C31—22no—
194C22×C31—22no—
195C22×C31—22no—
196C22×C31—22no—
197C22×C31—22no—
198C22×C31—22no—
199C22×C31—28no—
200C22×C31—432no—
201C22×C31—44no—
202C22×C31—44no—
203C22×C31—44no—
204C22×C31—44no—
205C22×C31—44no—
206C22×C31—44no—
207C22×C31D4⋊C814yescomplemented
208C22×C31#37911yescharacteristic
209C22×C31C2×C22⋊C811yescharacteristic
210C22×C31#40711yescharacteristic
211C22×C31C2×#10211yescharacteristic
212C22×C31#41811yescharacteristic
213C22×C31C2×#10411yescharacteristic
214C22×C31#42111yescharacteristic
215C4×C31—22no—
216C4×C31—22no—
217C4×C31—22no—
218C4×C31—44no—
219C4×C31—41no—
220C4×C31C2×C22⋊C812yes—
221C4×C31#40712yes—
222C4×C31#44712yes—
223C4×C31#45012yes—
224C23×D4C4×C3111yescharacteristic
225C24×C4—21no—
226C2×C4×D4—28no—
227C2×C4×D4—21no—
228C2×C4×D4—21no—
229C2×C4×D4C4×C3112yes—
230C2×C4×D4C22×C3112yes—
231C22×C4⋊C4C4×C3111yescharacteristic
232C22×C22⋊C4—21no—
233C22×C42C22×C3111yescharacteristic
234C2×C4⋊C8C22×C3112yescomplemented
235C2×C22⋊C8—21no—
236C2×C22⋊C8—21no—
237C2×C4×C8C22×C3112yescomplemented
238#336—21no—
239#336—21no—
240D4⋊C8—28no—
241C2×C31—24no—
242C2×C31—48no—
243C2×C31C2×D4⋊C811yescharacteristic
244C2×C31#147811yescharacteristic
245C2×C31#148611yescharacteristic
246C2×C31#149011yescharacteristic
247C2×C31#177911yescharacteristic
248C2×C31#178611yescharacteristic
249C2×C31#178911yescharacteristic
250C25—21no—
251C22×D4—24no—
252C22×D4—24no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4C8×C3114yescomplemented
256C22×D4C2×C4×C3112yes—
257C23×C4—28no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—44no—
262C23×C4—42no—
263C23×C4—42no—
264C23×C4C2×C4×C3111yescharacteristic
265C23×C4D4×C3111yescharacteristic
266C23×C4D4×C3111yescharacteristic
267C22×C8—22no—
268C22×C8—22no—
269C4×D4—28no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4D4×C3118yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4C2×C4×C3112yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C42—24no—
282C2×C42C2×C4×C3111yescharacteristic
283C2×C42D4×C3111yescharacteristic
284C2×C42D4×C3111yescharacteristic
285C2×C42C23×C3111yescharacteristic
286C4⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—42no—
289C4×C8—24no—
290C31#831011yescharacteristic, direct product factor, Sylow
291C24—24no—
292C24—24no—
293C24—24no—
294C24—22no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24C22⋊C4×C3111yescharacteristic
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—216no—
303C2×D4—216no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4C22⋊C4×C3114yes—
309C2×D4C2×C8×C3114yes—
310C2×D4M16×C3114yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—416no—
324C22×C4—416no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4C22⋊C4×C3111yescharacteristic
332C22×C4C22⋊C4×C3111yescharacteristic
333C22×C4C22⋊C4×C3111yescharacteristic
334C22×C4C2×C8×C3112yes—
335C22×C4M16×C3112yes—
336C22×C4D8×C3112yes—
337C22×C4SD16×C3112yes—
338C22×C4C22×C4×C3111yescharacteristic
339C22×C4C2×D4×C3111yescharacteristic
340C22×C4C2×D4×C3111yescharacteristic
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4C22⋊C4×C3114yes—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42C2×D4×C3114yes—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—216no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—48no—
365C23—48no—
366C23—48no—
367C23—48no—
368C23—48no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—48no—
374C23C31×C22⋊C812yes—
375C23C31×#10212yes—
376C23C31×#10412yes—
377C23C2×C22⋊C4×C3111yescharacteristic
378D4—216no—
379D4—24no—
380D4—24no—
381D4—24no—
382D4C31×C22⋊C818yescomplemented
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—44no—
398C2×C4—416no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4C31×C22⋊C812yes—
405C2×C4C31×#10212yes—
406C2×C4C2×C22⋊C4×C3112yes—
407C2×C4C31×#12012yes—
408C2×C4C31×#12012yes—
409C2×C4C22×C8×C3111yescharacteristic
410C2×C4C2×M16×C3112yes—
411C2×C4C2×M16×C3111yescharacteristic
412C2×C4C2×D8×C3111yescharacteristic
413C2×C4C2×SD16×C3111yescharacteristic
414C2×C4C31×AGL(1,ℤ/8)12yes—
415C8—44no—
416C8—82no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—28no—
424C22—432no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22C31×D4⋊C814yescomplemented
432C22C31×#37911yescharacteristic
433C22C2×C31×C22⋊C811yescharacteristic
434C22C31×#40711yescharacteristic
435C22C2×C31×#10211yescharacteristic
436C22C31×#41811yescharacteristic
437C22C2×C31×#10411yescharacteristic
438C22C31×#42111yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—44no—
443C4—41no—
444C4C2×C31×C22⋊C812yes—
445C4C31×#40712yes—
446C4C31×#44712yes—
447C4C31×#45012yes—
448C2—24no—
449C2—48no—
450C2C2×C31×D4⋊C811yescharacteristic
451C2C31×#147811yescharacteristic
452C2C31×#148611yescharacteristic
453C2C31×#149011yescharacteristic
454C2C31×#177911yescharacteristic
455C2C31×#178611yescharacteristic
456C2C31×#178911yescharacteristic
subgroupG/H
centerC22×C4×C31C2×D4
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
socleC23×C31C2×C22⋊C4
prime radicalC23×D4×C31C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1797C2semidirect
3. C4×C31×C22⋊C8C2semidirect
4. C2×C31×D4⋊C8C2semidirect
15. C2×C31×C4⋊C8C22semidirect
18. C2×C4×C8×C31C22semidirect
27. C22×D4×C31C8semidirect
127. #8310C31direct (N × H)
154. D4×C31C22⋊C8semidirect
190. #1797C2×C31semidirect
191. C4×C22⋊C8C2×C31semidirect
192. C2×D4⋊C8C2×C31semidirect
207. C22×C31D4⋊C8semidirect
234. C2×C4⋊C8C22×C31semidirect
237. C2×C4×C8C22×C31semidirect
255. C22×D4C8×C31semidirect
290. C31#8310direct (N × H)
382. D4C31×C22⋊C8semidirect
431. C22C31×D4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D4×C312
C31×#17972
C4×C31×C22⋊C82
C2×C31×D4⋊C82
#831031