Finite Group Databasebeta

← C31×#8216 #21464975

Subgroups — C31×#8216 #21464975

Order 7936.

712 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×#121C211yesmaximal, characteristic
2C31×#1749C211yesmaximal, characteristic, complemented
3C2×C31×#327C212yesmaximal
4C31×#1434C211yesmaximal, characteristic, complemented
5C31×#1294C212yesmaximal, complemented
6C2×C31×#121C2211yescharacteristic
7C2×C4×D4×C31—21no—
8C2×C4×D4×C31C411yescharacteristic, complemented
9C22×C42×C31C2211yescharacteristic
10C2×C31×C22⋊C8C2212yes—
11C2×C31×#97—21no—
12C31×#390C411yescharacteristic, complemented
13C31×#389C411yescharacteristic, complemented
14C31×#388—21no—
15C31×#386—21no—
16C31×#380C2211yescharacteristic
17C4×C4⋊C4×C31C411yescharacteristic, complemented
18C4×C22⋊C4×C31—21no—
19C2×C31×#95—21no—
20C2×C31×#95—21no—
21C2×C31×#95C2212yes—
22C31×#336—21no—
23C31×#327C418yescomplemented
24C22×D4×C31—21no—
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31—22no—
27C23×C4×C31—21no—
28C23×C4×C31D411yescharacteristic
29C23×C4×C31D411yescharacteristic
30C23×C4×C31C2311yescharacteristic
31C22×C8×C31—22no—
32C31×#121—22no—
33C31×#121C2×C414yes—
34C4×D4×C31—42no—
35C4×D4×C31—44no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C4⋊C4×C31C2×C411yescharacteristic
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C42×C31—21no—
46C2×C42×C31—42no—
47C2×C42×C31C2×C411yescharacteristic
48C2×C42×C31C2×C411yescharacteristic
49C2×C42×C31D411yescharacteristic
50C2×C42×C31D411yescharacteristic
51C31×C22⋊C8—22no—
52C31×C22⋊C8C2×C414yescomplemented
53C31×#97—42no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—22no—
58C31×#95—42no—
59C31×#95—42no—
60C31×#95C2×C414yescomplemented
61C24×C31—21no—
62C24×C31C2×D411yescharacteristic
63C2×D4×C31—24no—
64C2×D4×C31—21no—
65C2×D4×C31—41no—
66C2×D4×C31—41no—
67C2×D4×C31—42no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31C4212yes—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—22no—
74C22×C4×C31—21no—
75C22×C4×C31—22no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—24no—
79C22×C4×C31—41no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—41no—
83C22×C4×C31—41no—
84C22×C4×C31—42no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31—42no—
89C22×C4×C31—42no—
90C22×C4×C31—42no—
91C22×C4×C31—42no—
92C22×C4×C31—42no—
93C22×C4×C31C22⋊C411yescharacteristic
94C22×C4×C31C22⋊C411yescharacteristic
95C22×C4×C31C22⋊C411yescharacteristic
96C22×C4×C31C22⋊C411yescharacteristic
97C22×C4×C31C22×C411yescharacteristic
98C22×C4×C31C22×C412yes—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31#4112yes—
101C2×C8×C31—24no—
102C2×C8×C31—42no—
103C2×C8×C31—42no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—42no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31C4212yes—
109C22⋊C4×C31—24no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—42no—
114C42×C31—22no—
115C42×C31—22no—
116C42×C31—41no—
117C42×C31—41no—
118C42×C31—42no—
119#8216C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—21no—
121C23×C31—21no—
122C23×C31—21no—
123C23×C31—21no—
124C23×C31—22no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31C2×C22⋊C411yescharacteristic
132C23×C31C4×D412yes—
133C23×C31AGL(1,ℤ/8)11yescharacteristic
134C23×C31AGL(1,ℤ/8)12yes—
135C23×C31#13711yescharacteristic
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—44no—
139D4×C31—44no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—22no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—44no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—44no—
164C2×C4×C31—44no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—44no—
172C2×C4×C31—84no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31—82no—
176C2×C4×C31—82no—
177C2×C4×C31#9912yes—
178C2×C4×C31C2×C4211yescharacteristic
179C2×C4×C31#11711yescharacteristic
180C2×C4×C31C4×D412yes—
181C8×C31—44no—
182C4×#121C2×C3111yescharacteristic
183#1749C2×C3111yescharacteristic, complemented
184C2×#327C2×C3112yes—
185#1434C2×C3111yescharacteristic, complemented
186#1294C2×C3112yescomplemented
187C22×C31—22no—
188C22×C31—22no—
189C22×C31—22no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—22no—
195C22×C31—42no—
196C22×C31—42no—
197C22×C31—42no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31C4×C22⋊C411yescharacteristic
203C22×C31C2×#9911yescharacteristic
204C22×C31#41011yescharacteristic
205C22×C31#41711yescharacteristic
206C22×C31#41811yescharacteristic
207C22×C31#42111yescharacteristic
208C22×C31#42111yescharacteristic
209C22×C31#44012yes—
210C22×C31#44212yes—
211C4×C31—22no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—22no—
215C4×C31—42no—
216C4×C31—41no—
217C4×C31—41no—
218C4×C31—41no—
219C4×C31—84no—
220C4×C31—81no—
221C4×C31—81no—
222C2×#121C22×C3111yescharacteristic
223C2×C4×D4—21no—
224C2×C4×D4C4×C3111yescharacteristic, complemented
225C22×C42C22×C3111yescharacteristic
226C2×C22⋊C8C22×C3112yes—
227C2×#97—21no—
228#390C4×C3111yescharacteristic, complemented
229#389C4×C3111yescharacteristic, complemented
230#388—21no—
231#386—21no—
232#380C22×C3111yescharacteristic
233C4×C4⋊C4C4×C3111yescharacteristic, complemented
234C4×C22⋊C4—21no—
235C2×#95—21no—
236C2×#95—21no—
237C2×#95C22×C3112yes—
238#336—21no—
239#327C4×C3118yescomplemented
240C2×C31—22no—
241C2×C31—22no—
242C2×C31—82no—
243C2×C31#151011yescharacteristic
244C2×C31#151211yescharacteristic
245C2×C31#151311yescharacteristic
246C2×C31#151411yescharacteristic
247C2×C31C4×#9911yescharacteristic
248C2×C31#175911yescharacteristic
249C2×C31#176211yescharacteristic
250C22×D4—21no—
251C22×D4C2×C4×C3111yescharacteristic
252C23×C4—22no—
253C23×C4—21no—
254C23×C4D4×C3111yescharacteristic
255C23×C4D4×C3111yescharacteristic
256C23×C4C23×C3111yescharacteristic
257C22×C8—22no—
258#121—22no—
259#121C2×C4×C3114yes—
260C4×D4—42no—
261C4×D4—44no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×C4×C3111yescharacteristic
264C2×C4⋊C4C2×C4×C3111yescharacteristic
265C2×C4⋊C4C2×C4×C3111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—21no—
272C2×C42—42no—
273C2×C42C2×C4×C3111yescharacteristic
274C2×C42C2×C4×C3111yescharacteristic
275C2×C42D4×C3111yescharacteristic
276C2×C42D4×C3111yescharacteristic
277C22⋊C8—22no—
278C22⋊C8C2×C4×C3114yescomplemented
279#97—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—42no—
285#95—42no—
286#95C2×C4×C3114yescomplemented
287C31#821611yescharacteristic, direct product factor, Sylow
288C24—21no—
289C24C2×D4×C3111yescharacteristic
290C2×D4—24no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4C42×C3112yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4C22⋊C4×C3111yescharacteristic
321C22×C4C22⋊C4×C3111yescharacteristic
322C22×C4C22⋊C4×C3111yescharacteristic
323C22×C4C22⋊C4×C3111yescharacteristic
324C22×C4C22×C4×C3111yescharacteristic
325C22×C4C22×C4×C3112yes—
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4#41×C3112yes—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4C42×C3112yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—42no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C2×C22⋊C4×C3111yescharacteristic
358C23C4×D4×C3112yes—
359C23C31×AGL(1,ℤ/8)11yescharacteristic
360C23C31×AGL(1,ℤ/8)12yes—
361C23C31×#13711yescharacteristic
362D4—42no—
363D4—42no—
364D4—44no—
365D4—44no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4C31×#9912yes—
404C2×C4C2×C42×C3111yescharacteristic
405C2×C4C31×#11711yescharacteristic
406C2×C4C4×D4×C3112yes—
407C8—44no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22C4×C22⋊C4×C3111yescharacteristic
424C22C2×C31×#9911yescharacteristic
425C22C31×#41011yescharacteristic
426C22C31×#41711yescharacteristic
427C22C31×#41811yescharacteristic
428C22C31×#42111yescharacteristic
429C22C31×#42111yescharacteristic
430C22C31×#44012yes—
431C22C31×#44212yes—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—84no—
441C4—81no—
442C4—81no—
443C2—22no—
444C2—22no—
445C2—82no—
446C2C31×#151011yescharacteristic
447C2C31×#151211yescharacteristic
448C2C31×#151311yescharacteristic
449C2C31×#151411yescharacteristic
450C2C4×C31×#9911yescharacteristic
451C2C31×#175911yescharacteristic
452C2C31×#176211yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C2×C42×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1749C2semidirect
4. C31×#1434C2semidirect
5. C31×#1294C2semidirect
8. C2×C4×D4×C31C4semidirect
12. C31×#390C4semidirect
13. C31×#389C4semidirect
17. C4×C4⋊C4×C31C4semidirect
23. C31×#327C4semidirect
52. C31×C22⋊C8C2×C4semidirect
60. C31×#95C2×C4semidirect
119. #8216C31direct (N × H)
183. #1749C2×C31semidirect
185. #1434C2×C31semidirect
186. #1294C2×C31semidirect
224. C2×C4×D4C4×C31semidirect
228. #390C4×C31semidirect
229. #389C4×C31semidirect
233. C4×C4⋊C4C4×C31semidirect
239. #327C4×C31semidirect
278. C22⋊C8C2×C4×C31semidirect
286. #95C2×C4×C31semidirect
287. C31#8216direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×#1212
C31×#17492
C2×C31×#3272
C31×#14342
C31×#12942
#821631