Finite Group Databasebeta

← C31×#8318 #21465077

Subgroups — C31×#8318 #21465077

Order 7936.

1058 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×H(5,𝔽2)C211yesmaximal, characteristic
2C31×#1954C211yesmaximal, characteristic, complemented
3C31×#1477C212yesmaximal, complemented
4C31×#1476C212yesmaximal, complemented
5C31×#1460C211yesmaximal, characteristic, complemented
6C2×C31×H(5,𝔽2)C411yescharacteristic
7C31×#520—22no—
8C31×#520C412yes—
9C31×#518—22no—
10C31×#518—22no—
11C31×#518C411yescharacteristic
12C4×#41×C31—22no—
13C4×#41×C31C2212yes—
14C2×C4×D4×C31—22no—
15C2×C4×D4×C31—22no—
16C2×C4×D4×C31C2211yescharacteristic
17C31×#431C2212yescomplemented
18C2×C31×C4⋊C8C2211yescharacteristic, complemented
19C2×C31×C22⋊C8—22no—
20C2×C4×C8×C31C2211yescharacteristic, complemented
21C31×Q8⋊C8—22no—
22C31×Q8⋊C8—22no—
23C31×D4⋊C8—22no—
24C31×D4⋊C8—22no—
25C31×#323—22no—
26C31×H(5,𝔽2)—24no—
27C31×H(5,𝔽2)—22no—
28C31×H(5,𝔽2)C814yescomplemented
29C2×#41×C31—22no—
30C2×#41×C31C2×C412yes—
31C22×D4×C31—22no—
32C22×D4×C31—22no—
33C22×D4×C31C2×C411yescharacteristic
34C23×C4×C31—22no—
35C23×C4×C31D412yes—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C4×Q8×C31—22no—
39C4×Q8×C31D412yes—
40C4×Q8×C31D412yes—
41C4×D4×C31—24no—
42C4×D4×C31—24no—
43C4×D4×C31—24no—
44C4×D4×C31—24no—
45C4×D4×C31—22no—
46C4×D4×C31—24no—
47C4×D4×C31—22no—
48C4×D4×C31—22no—
49C4×D4×C31—22no—
50C4×D4×C31—22no—
51C4×D4×C31—22no—
52C4×D4×C31—22no—
53C4×D4×C31D412yes—
54C4×D4×C31D412yes—
55C31×#117—24no—
56C31×#117—22no—
57C31×#117—22no—
58C31×#117C2×C412yes—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31C2×C411yescharacteristic
62C2×C22⋊C4×C31—22no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31D412yes—
66C2×C42×C31—22no—
67C2×C42×C31—22no—
68C2×C42×C31C2311yescharacteristic
69C31×C4⋊C8—22no—
70C31×C4⋊C8—22no—
71C31×C22⋊C8—22no—
72C31×C22⋊C8—44no—
73C4×C8×C31—22no—
74C4×C8×C31—22no—
75C24×C31—22no—
76C24×C31C22⋊C412yes—
77#41×C31—28no—
78#41×C31—28no—
79#41×C31—22no—
80#41×C31—22no—
81#41×C31C2×C814yes—
82#41×C31M1614yes—
83C2×Q8×C31C22⋊C412yes—
84C2×D4×C31—24no—
85C2×D4×C31—28no—
86C2×D4×C31—28no—
87C2×D4×C31—28no—
88C2×D4×C31—22no—
89C2×D4×C31—28no—
90C2×D4×C31—22no—
91C2×D4×C31—22no—
92C2×D4×C31C22⋊C412yes—
93C2×D4×C31C2×C812yes—
94C2×D4×C31M1612yes—
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—24no—
99C22×C4×C31—22no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—42no—
105C22×C4×C31—42no—
106C22×C4×C31—82no—
107C22×C4×C31—82no—
108C22×C4×C31C22⋊C412yes—
109C22×C4×C31C22×C411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2×D411yescharacteristic
112C2×C8×C31—22no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C2×C8×C31—42no—
116C2×C8×C31—42no—
117C2×C8×C31—41no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31C22⋊C412yes—
123C4⋊C4×C31C22⋊C412yes—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—24no—
126C22⋊C4×C31—22no—
127C22⋊C4×C31—22no—
128C22⋊C4×C31—22no—
129C22⋊C4×C31—24no—
130C42×C31—24no—
131C42×C31—22no—
132C42×C31—22no—
133C42×C31—22no—
134C42×C31C2×D412yes—
135C42×C31C2×D412yes—
136#8318C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—28no—
138C23×C31—22no—
139C23×C31—22no—
140C23×C31—22no—
141C23×C31—42no—
142C23×C31—42no—
143C23×C31—82no—
144C23×C31—82no—
145C23×C31C22⋊C814yes—
146C23×C31C2×C22⋊C411yescharacteristic
147Q8×C31—22no—
148Q8×C31C22⋊C814yescomplemented
149D4×C31—216no—
150D4×C31—24no—
151D4×C31—24no—
152D4×C31—24no—
153D4×C31—22no—
154D4×C31—24no—
155D4×C31C22⋊C814yescomplemented
156C2×C4×C31—28no—
157C2×C4×C31—22no—
158C2×C4×C31—24no—
159C2×C4×C31—24no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—22no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—82no—
172C2×C4×C31—82no—
173C2×C4×C31—82no—
174C2×C4×C31—84no—
175C2×C4×C31—82no—
176C2×C4×C31C2×C22⋊C412yes—
177C2×C4×C31#12012yes—
178C2×C4×C31#12011yescharacteristic
179C2×C4×C31#12011yescharacteristic
180C2×C4×C31C22×C811yescharacteristic
181C2×C4×C31C2×M1612yes—
182C2×C4×C31C2×M1611yescharacteristic
183C8×C31—42no—
184C8×C31—82no—
185C8×C31—81no—
186C4×H(5,𝔽2)C2×C3111yescharacteristic
187#1954C2×C3111yescharacteristic, complemented
188#1477C2×C3112yescomplemented
189#1476C2×C3112yescomplemented
190#1460C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—24no—
194C22×C31—24no—
195C22×C31—44no—
196C22×C31—42no—
197C22×C31—42no—
198C22×C31—42no—
199C22×C31—42no—
200C22×C31—82no—
201C22×C31—88no—
202C22×C31#37911yescharacteristic
203C22×C31C2×C22⋊C811yescharacteristic
204C22×C31#40711yescharacteristic
205C4×C31—21no—
206C4×C31—22no—
207C4×C31—24no—
208C4×C31—42no—
209C4×C31—42no—
210C4×C31—42no—
211C4×C31C2×C22⋊C812yes—
212C4×C31#40712yes—
213C4×C31#45311yescharacteristic
214C4×C31#45511yescharacteristic
215C2×H(5,𝔽2)C4×C3111yescharacteristic
216#520—22no—
217#520C4×C3112yes—
218#518—22no—
219#518—22no—
220#518C4×C3111yescharacteristic
221C4×#41—22no—
222C4×#41C22×C3112yes—
223C2×C4×D4—22no—
224C2×C4×D4—22no—
225C2×C4×D4C22×C3111yescharacteristic
226#431C22×C3112yescomplemented
227C2×C4⋊C8C22×C3111yescharacteristic, complemented
228C2×C22⋊C8—22no—
229C2×C4×C8C22×C3111yescharacteristic, complemented
230Q8⋊C8—22no—
231Q8⋊C8—22no—
232D4⋊C8—22no—
233D4⋊C8—22no—
234#323—22no—
235C2×C31—21no—
236C2×C31—21no—
237C2×C31—44no—
238C2×C31—44no—
239C2×C31#177911yescharacteristic
240C2×C31#179211yescharacteristic
241C2×C31#179411yescharacteristic
242H(5,𝔽2)—24no—
243H(5,𝔽2)—22no—
244H(5,𝔽2)C8×C3114yescomplemented
245C2×#41—22no—
246C2×#41C2×C4×C3112yes—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4C2×C4×C3111yescharacteristic
250C23×C4—22no—
251C23×C4D4×C3112yes—
252C22×C8—21no—
253C22×C8—21no—
254C4×Q8—22no—
255C4×Q8D4×C3112yes—
256C4×Q8D4×C3112yes—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4D4×C3112yes—
270C4×D4D4×C3112yes—
271#117—24no—
272#117—22no—
273#117—22no—
274#117C2×C4×C3112yes—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4C2×C4×C3111yescharacteristic
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4D4×C3112yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42C23×C3111yescharacteristic
285C4⋊C8—22no—
286C4⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—44no—
289C4×C8—22no—
290C4×C8—22no—
291C31#831811yescharacteristic, direct product factor, Sylow
292C24—22no—
293C24C22⋊C4×C3112yes—
294#41—28no—
295#41—28no—
296#41—22no—
297#41—22no—
298#41C2×C8×C3114yes—
299#41M16×C3114yes—
300C2×Q8C22⋊C4×C3112yes—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—22no—
306C2×D4—28no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4C22⋊C4×C3112yes—
310C2×D4C2×C8×C3112yes—
311C2×D4M16×C3112yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C22⋊C4×C3112yes—
326C22×C4C22×C4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3111yescharacteristic
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4C22⋊C4×C3112yes—
340C4⋊C4C22⋊C4×C3112yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—24no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42C2×D4×C3112yes—
352C42C2×D4×C3112yes—
353C23—28no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23C31×C22⋊C814yes—
362C23C2×C22⋊C4×C3111yescharacteristic
363Q8—22no—
364Q8C31×C22⋊C814yescomplemented
365D4—216no—
366D4—24no—
367D4—24no—
368D4—24no—
369D4—22no—
370D4—24no—
371D4C31×C22⋊C814yescomplemented
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4C2×C22⋊C4×C3112yes—
393C2×C4C31×#12012yes—
394C2×C4C31×#12011yescharacteristic
395C2×C4C31×#12011yescharacteristic
396C2×C4C22×C8×C3111yescharacteristic
397C2×C4C2×M16×C3112yes—
398C2×C4C2×M16×C3111yescharacteristic
399C8—42no—
400C8—82no—
401C8—81no—
402C22—21no—
403C22—21no—
404C22—24no—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—88no—
413C22C31×#37911yescharacteristic
414C22C2×C31×C22⋊C811yescharacteristic
415C22C31×#40711yescharacteristic
416C4—24no—
417C4—21no—
418C4—22no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4C2×C31×C22⋊C812yes—
423C4C31×#40712yes—
424C4C31×#45311yescharacteristic
425C4C31×#45511yescharacteristic
426C2—21no—
427C2—21no—
428C2—44no—
429C2—44no—
430C2C31×#177911yescharacteristic
431C2C31×#179211yescharacteristic
432C2C31×#179411yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#379
prime radicalC2×C31×H(5,𝔽2)C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1954C2semidirect
3. C31×#1477C2semidirect
4. C31×#1476C2semidirect
5. C31×#1460C2semidirect
17. C31×#431C22semidirect
18. C2×C31×C4⋊C8C22semidirect
20. C2×C4×C8×C31C22semidirect
28. C31×H(5,𝔽2)C8semidirect
136. #8318C31direct (N × H)
148. Q8×C31C22⋊C8semidirect
155. D4×C31C22⋊C8semidirect
187. #1954C2×C31semidirect
188. #1477C2×C31semidirect
189. #1476C2×C31semidirect
190. #1460C2×C31semidirect
226. #431C22×C31semidirect
227. C2×C4⋊C8C22×C31semidirect
229. C2×C4×C8C22×C31semidirect
244. H(5,𝔽2)C8×C31semidirect
291. C31#8318direct (N × H)
364. Q8C31×C22⋊C8semidirect
371. D4C31×C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×C31×H(5,𝔽2)2
C31×#19542
C31×#14772
C31×#14762
C31×#14602
#831831