Finite Group Databasebeta

← C31×#8219 #21464978

Subgroups — C31×#8219 #21464978

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×C22⋊Q8C211yesmaximal, characteristic
2C31×#1749C211yesmaximal, characteristic
3C2×C31×#328C212yesmaximal
4C4×C31×#95C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C2×C31×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8×C31C411yescharacteristic, complemented
8C22×C42×C31—21no—
9C22×C42×C31C2211yescharacteristic
10C2×C31×C22⋊C8C2212yes—
11C2×C31×#97—21no—
12C31×#391C411yescharacteristic, complemented
13C31×#389C411yescharacteristic, complemented
14C31×#387—21no—
15C31×#385—21no—
16C31×#380C2211yescharacteristic
17C4×C4⋊C4×C31—21no—
18C4×C4⋊C4×C31C411yescharacteristic, complemented
19C4×C22⋊C4×C31—21no—
20C2×C31×#95—21no—
21C2×C31×#95C2212yes—
22C31×#336—21no—
23C31×#328C418yescomplemented
24C22×Q8×C31C2×C411yescharacteristic
25C23×C4×C31—22no—
26C23×C4×C31D411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C23×C4×C31C2311yescharacteristic
29C22×C8×C31—22no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8C2×C414yes—
32C4×Q8×C31—42no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31C2×C411yescharacteristic
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31—24no—
42C2×C42×C31—21no—
43C2×C42×C31—21no—
44C2×C42×C31—42no—
45C2×C42×C31—42no—
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×C414yes—
53C31×#97—42no—
54C31×#95—24no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—22no—
61C31×#95C2×C414yes—
62C24×C31C2×D411yescharacteristic
63C2×Q8×C31—21no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C2×Q8×C31C4212yescomplemented
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—22no—
70C22×C4×C31—22no—
71C22×C4×C31—24no—
72C22×C4×C31—24no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—24no—
76C22×C4×C31—22no—
77C22×C4×C31—22no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—41no—
81C22×C4×C31—41no—
82C22×C4×C31—41no—
83C22×C4×C31—42no—
84C22×C4×C31—42no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31C22⋊C411yescharacteristic
88C22×C4×C31C22⋊C411yescharacteristic
89C22×C4×C31C22⋊C411yescharacteristic
90C22×C4×C31C22⋊C411yescharacteristic
91C22×C4×C31C22×C411yescharacteristic
92C22×C4×C31C22×C412yes—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31#4112yes—
95C2×C8×C31—24no—
96C2×C8×C31—42no—
97C2×C8×C31—42no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—24no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—42no—
103C4⋊C4×C31—42no—
104C4⋊C4×C31C4212yescomplemented
105C22⋊C4×C31—24no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C42×C31—22no—
109C42×C31—22no—
110C42×C31—41no—
111C42×C31—41no—
112C42×C31—48no—
113C42×C31—42no—
114C42×C31—42no—
115#8219C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—21no—
119C23×C31—21no—
120C23×C31C2×C22⋊C411yescharacteristic
121C23×C31C4×D412yes—
122C23×C31AGL(1,ℤ/8)11yescharacteristic
123C23×C31#13711yescharacteristic
124C23×C31#13712yes—
125Q8×C31—42no—
126Q8×C31—42no—
127C2×C4×C31—24no—
128C2×C4×C31—24no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—28no—
140C2×C4×C31—22no—
141C2×C4×C31—22no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—44no—
148C2×C4×C31—44no—
149C2×C4×C31—44no—
150C2×C4×C31—44no—
151C2×C4×C31—44no—
152C2×C4×C31—44no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—44no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31#9912yes—
165C2×C4×C31#10412yes—
166C2×C4×C31#10412yes—
167C2×C4×C31C2×C4211yescharacteristic
168C2×C4×C31#11711yescharacteristic
169C2×C4×C31C4×D412yes—
170C8×C31—44no—
171C4×C22⋊Q8C2×C3111yescharacteristic
172#1749C2×C3111yescharacteristic
173C2×#328C2×C3112yes—
174C4×#95C2×C3111yescharacteristic
175#1294C2×C3112yes—
176C22×C31—22no—
177C22×C31—22no—
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—22no—
181C22×C31—22no—
182C22×C31—21no—
183C22×C31—21no—
184C22×C31C4×C22⋊C411yescharacteristic
185C22×C31C2×#9911yescharacteristic
186C22×C31#41011yescharacteristic
187C22×C31#41711yescharacteristic
188C22×C31#41911yescharacteristic
189C22×C31C2×#10411yescharacteristic
190C22×C31C2×#10411yescharacteristic
191C22×C31#44012yes—
192C22×C31#44112yes—
193C4×C31—22no—
194C4×C31—21no—
195C4×C31—21no—
196C4×C31—22no—
197C4×C31—22no—
198C4×C31—22no—
199C4×C31—48no—
200C4×C31—42no—
201C4×C31—42no—
202C4×C31—41no—
203C4×C31—81no—
204C4×C31—81no—
205C2×C22⋊Q8C22×C3111yescharacteristic
206C2×C4×Q8C4×C3111yescharacteristic, complemented
207C22×C42—21no—
208C22×C42C22×C3111yescharacteristic
209C2×C22⋊C8C22×C3112yes—
210C2×#97—21no—
211#391C4×C3111yescharacteristic, complemented
212#389C4×C3111yescharacteristic, complemented
213#387—21no—
214#385—21no—
215#380C22×C3111yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4C4×C3111yescharacteristic, complemented
218C4×C22⋊C4—21no—
219C2×#95—21no—
220C2×#95C22×C3112yes—
221#336—21no—
222#328C4×C3118yescomplemented
223C2×C31—22no—
224C2×C31—22no—
225C2×C31#150411yescharacteristic
226C2×C31#150611yescharacteristic
227C2×C31#150811yescharacteristic
228C2×C31#150911yescharacteristic
229C2×C31C4×#9911yescharacteristic
230C2×C31C4×#10411yescharacteristic
231C2×C31#176311yescharacteristic
232C22×Q8C2×C4×C3111yescharacteristic
233C23×C4—22no—
234C23×C4D4×C3111yescharacteristic
235C23×C4D4×C3111yescharacteristic
236C23×C4C23×C3111yescharacteristic
237C22×C8—22no—
238C22⋊Q8—22no—
239C22⋊Q8C2×C4×C3114yes—
240C4×Q8—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C2×C4×C3111yescharacteristic
245C2×C4⋊C4C2×C4×C3111yescharacteristic
246C2×C4⋊C4C2×C4×C3111yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C2×C4×C3111yescharacteristic
256C2×C42C2×C4×C3111yescharacteristic
257C2×C42D4×C3111yescharacteristic
258C2×C42D4×C3111yescharacteristic
259C22⋊C8—22no—
260C22⋊C8C2×C4×C3114yes—
261#97—42no—
262#95—24no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95C2×C4×C3114yes—
270C31#821911yescharacteristic, direct product factor, Sylow
271C24C2×D4×C3111yescharacteristic
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C42×C3112yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C22⋊C4×C3111yescharacteristic
297C22×C4C22⋊C4×C3111yescharacteristic
298C22×C4C22⋊C4×C3111yescharacteristic
299C22×C4C22⋊C4×C3111yescharacteristic
300C22×C4C22×C4×C3111yescharacteristic
301C22×C4C22×C4×C3112yes—
302C22×C4C2×D4×C3111yescharacteristic
303C22×C4#41×C3112yes—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4C42×C3112yescomplemented
314C22⋊C4—24no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—48no—
322C42—42no—
323C42—42no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23C2×C22⋊C4×C3111yescharacteristic
329C23C4×D4×C3112yes—
330C23C31×AGL(1,ℤ/8)11yescharacteristic
331C23C31×#13711yescharacteristic
332C23C31×#13712yes—
333Q8—42no—
334Q8—42no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—28no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4C31×#9912yes—
373C2×C4C31×#10412yes—
374C2×C4C31×#10412yes—
375C2×C4C2×C42×C3111yescharacteristic
376C2×C4C31×#11711yescharacteristic
377C2×C4C4×D4×C3112yes—
378C8—44no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22C4×C22⋊C4×C3111yescharacteristic
388C22C2×C31×#9911yescharacteristic
389C22C31×#41011yescharacteristic
390C22C31×#41711yescharacteristic
391C22C31×#41911yescharacteristic
392C22C2×C31×#10411yescharacteristic
393C22C2×C31×#10411yescharacteristic
394C22C31×#44012yes—
395C22C31×#44112yes—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—48no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—81no—
407C4—81no—
408C2—22no—
409C2—22no—
410C2C31×#150411yescharacteristic
411C2C31×#150611yescharacteristic
412C2C31×#150811yescharacteristic
413C2C31×#150911yescharacteristic
414C2C4×C31×#9911yescharacteristic
415C2C4×C31×#10411yescharacteristic
416C2C31×#176311yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C2×C42×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
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
7. C2×C4×Q8×C31C4semidirect
12. C31×#391C4semidirect
13. C31×#389C4semidirect
18. C4×C4⋊C4×C31C4semidirect
23. C31×#328C4semidirect
66. C2×Q8×C31C42semidirect
104. C4⋊C4×C31C42semidirect
115. #8219C31direct (N × H)
206. C2×C4×Q8C4×C31semidirect
211. #391C4×C31semidirect
212. #389C4×C31semidirect
217. C4×C4⋊C4C4×C31semidirect
222. #328C4×C31semidirect
270. C31#8219direct (N × H)
275. C2×Q8C42×C31semidirect
313. C4⋊C4C42×C31semidirect

Maximal subgroups

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

subgroupindex
C4×C31×C22⋊Q82
C31×#17492
C2×C31×#3282
C4×C31×#952
C31×#12942
#821931