Finite Group Databasebeta

← C31×#18202 #21474961

Subgroups — C31×#18202 #21474961

Order 7936.

1282 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic
2C2×C31×#480C212yesmaximal, complemented
3C31×#2486C211yesmaximal, characteristic, complemented
4C31×#1610C214yesmaximal, complemented
5C31×#1607C214yesmaximal, complemented
6C31×#1519C212yesmaximal, complemented
7C31×#1459C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)—21no—
9C31×#547—22no—
10C31×#546—24no—
11D42×C31—22no—
12C31×#537C2212yes—
13C31×#534—21no—
14C31×#534C2212yes—
15C31×#532—21no—
16C2×C31×#123—21no—
17C2×C31×#121—21no—
18C2×C31×#121C2212yes—
19C2×C4×D4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic, complemented
21C31×#480—24no—
22C31×#480C2218yescomplemented
23C2×C31×#107—22no—
24C2×C31×#102—22no—
25C2×C31×C22⋊C8C2212yescomplemented
26C31×#398C2212yescomplemented
27C31×#397C2212yescomplemented
28C31×#389—21no—
29C31×#381—21no—
30C31×#380—21no—
31C31×#380C2211yescharacteristic, complemented
32C31×#327C2218yescomplemented
33C31×#324C2212yescomplemented
34C31×#323C2212yescomplemented
35C31×H(5,𝔽2)—44no—
36C31×H(5,𝔽2)—44no—
37C2×#41×C31—24no—
38C2×#41×C31—22no—
39C22×D4×C31—24no—
40C22×D4×C31—22no—
41C22×D4×C31—22no—
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C22×D4×C31D412yes—
45C23×C4×C31—21no—
46C23×C4×C31D411yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C22×C8×C31—22no—
49C31×C4⋊Q8—21no—
50C31×#127—21no—
51C31×#124—22no—
52C31×#123—24no—
53C31×#123—24no—
54C31×#123—44no—
55C31×C22⋊Q8—24no—
56C31×#121—22no—
57C31×#121—22no—
58C31×#121—24no—
59C31×#121—22no—
60C31×#121—22no—
61C31×#121—22no—
62C31×#121—22no—
63C31×#121D414yes—
64C31×#121C2314yes—
65C31×#120—24no—
66C31×#120—24no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31D412yes—
78C2×C4⋊C4×C31C2312yes—
79C2×C4⋊C4×C31C2311yescharacteristic
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31D411yescharacteristic
85C2×C22⋊C4×C31C2311yescharacteristic
86C2×C42×C31D411yescharacteristic
87C31×#107—28no—
88C31×#102—28no—
89C31×C22⋊C8—22no—
90C31×C22⋊C8C2314yes—
91C31×#95—22no—
92C31×#95—22no—
93C31×#95C2312yes—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31C2×D411yescharacteristic
97C24×C31C2×D411yescharacteristic
98#41×C31—44no—
99#41×C31—416no—
100#41×C31—44no—
101C2×Q8×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—22no—
104C2×D4×C31—24no—
105C2×D4×C31—22no—
106C2×D4×C31—24no—
107C2×D4×C31—24no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—22no—
111C2×D4×C31—44no—
112C2×D4×C31—416no—
113C2×D4×C31—44no—
114C2×D4×C31—44no—
115C2×D4×C31—44no—
116C2×D4×C31—44no—
117C2×D4×C31—82no—
118C2×D4×C31D814yes—
119C2×D4×C31C2×D414yes—
120C22×C4×C31—22no—
121C22×C4×C31—24no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—42no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D412yes—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2411yescharacteristic
139C2×C8×C31—24no—
140C2×C8×C31—44no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—42no—
147C4⋊C4×C31—41no—
148C4⋊C4×C31—41no—
149C4⋊C4×C31—44no—
150C4⋊C4×C31C2×D414yes—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—24no—
153C22⋊C4×C31—22no—
154C22⋊C4×C31—24no—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—22no—
157C22⋊C4×C31—42no—
158C22⋊C4×C31—44no—
159C42×C31—21no—
160C42×C31—21no—
161#18202C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—21no—
163C23×C31—21no—
164C23×C31—24no—
165C23×C31—41no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—41no—
169C23×C31—44no—
170C23×C31—44no—
171C23×C31—82no—
172C23×C31—82no—
173C23×C31—82no—
174C23×C31—82no—
175C23×C31#12012yes—
176C23×C31C2×D812yes—
177C23×C31C2×D812yes—
178C23×C31AGL(1,ℤ/8)12yes—
179C23×C31C22×D411yescharacteristic
180Q8×C31—44no—
181D4×C31—28no—
182D4×C31—44no—
183D4×C31—44no—
184D4×C31—48no—
185D4×C31—44no—
186D4×C31—44no—
187D4×C31—48no—
188D4×C31—44no—
189D4×C31—82no—
190D4×C31—82no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—22no—
194C2×C4×C31—24no—
195C2×C4×C31—21no—
196C2×C4×C31—21no—
197C2×C4×C31—24no—
198C2×C4×C31—42no—
199C2×C4×C31—41no—
200C2×C4×C31—48no—
201C2×C4×C31—44no—
202C2×C4×C31—44no—
203C2×C4×C31—42no—
204C2×C4×C31—41no—
205C2×C4×C31—41no—
206C2×C4×C31—42no—
207C2×C4×C31—42no—
208C2×C4×C31—42no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—81no—
213C2×C4×C31—84no—
214C2×C4×C31#12011yescharacteristic
215C2×C4×C31#12011yescharacteristic
216C2×C4×C31C2×D812yes—
217C2×C4×C31C22×D412yes—
218C8×C31—44no—
219#3496C2×C3111yescharacteristic
220C2×#480C2×C3112yescomplemented
221#2486C2×C3111yescharacteristic, complemented
222#1610C2×C3114yescomplemented
223#1607C2×C3114yescomplemented
224#1519C2×C3112yescomplemented
225#1459C2×C3111yescharacteristic, complemented
226C22×C31—21no—
227C22×C31—22no—
228C22×C31—21no—
229C22×C31—22no—
230C22×C31—22no—
231C22×C31—42no—
232C22×C31—42no—
233C22×C31—42no—
234C22×C31—41no—
235C22×C31—41no—
236C22×C31—44no—
237C22×C31—44no—
238C22×C31—44no—
239C22×C31—81no—
240C22×C31—164no—
241C22×C31—164no—
242C22×C31#44714yes—
243C22×C31C2×#12011yescharacteristic
244C22×C31C22×D811yescharacteristic
245C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C31—22no—
247C4×C31—22no—
248C4×C31—21no—
249C4×C31—44no—
250C4×C31—41no—
251C4×C31—41no—
252C4×C31—82no—
253C4×C31—82no—
254C4×C31#45812yes—
255C2×H(5,𝔽2)—21no—
256#547—22no—
257#546—24no—
258D42—22no—
259#537C22×C3112yes—
260#534—21no—
261#534C22×C3112yes—
262#532—21no—
263C2×#123—21no—
264C2×#121—21no—
265C2×#121C22×C3112yes—
266C2×C4×D4C22×C3111yescharacteristic
267C22×C4⋊C4C22×C3111yescharacteristic, complemented
268#480—24no—
269#480C22×C3118yescomplemented
270C2×#107—22no—
271C2×#102—22no—
272C2×C22⋊C8C22×C3112yescomplemented
273#398C22×C3112yescomplemented
274#397C22×C3112yescomplemented
275#389—21no—
276#381—21no—
277#380—21no—
278#380C22×C3111yescharacteristic, complemented
279#327C22×C3118yescomplemented
280#324C22×C3112yescomplemented
281#323C22×C3112yescomplemented
282C2×C31—21no—
283C2×C31—21no—
284C2×C31—24no—
285C2×C31—42no—
286C2×C31—84no—
287C2×C31C2×#44711yescharacteristic
288C2×C31#302211yescharacteristic
289C2×C31C2×#45811yescharacteristic
290H(5,𝔽2)—44no—
291H(5,𝔽2)—44no—
292C2×#41—24no—
293C2×#41—22no—
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—21no—
298C22×D4D4×C3111yescharacteristic
299C22×D4D4×C3112yes—
300C23×C4—21no—
301C23×C4D4×C3111yescharacteristic
302C23×C4C23×C3111yescharacteristic
303C22×C8—22no—
304C4⋊Q8—21no—
305#127—21no—
306#124—22no—
307#123—24no—
308#123—24no—
309#123—44no—
310C22⋊Q8—24no—
311#121—22no—
312#121—22no—
313#121—24no—
314#121—22no—
315#121—22no—
316#121—22no—
317#121—22no—
318#121D4×C3114yes—
319#121C23×C3114yes—
320#120—24no—
321#120—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4D4×C3112yes—
333C2×C4⋊C4C23×C3112yes—
334C2×C4⋊C4C23×C3111yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4D4×C3111yescharacteristic
340C2×C22⋊C4C23×C3111yescharacteristic
341C2×C42D4×C3111yescharacteristic
342#107—28no—
343#102—28no—
344C22⋊C8—22no—
345C22⋊C8C23×C3114yes—
346#95—22no—
347#95—22no—
348#95C23×C3112yes—
349C31#1820211yescharacteristic, direct product factor, Sylow
350C24—22no—
351C24—22no—
352C24C2×D4×C3111yescharacteristic
353C24C2×D4×C3111yescharacteristic
354#41—44no—
355#41—416no—
356#41—44no—
357C2×Q8—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—44no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—82no—
374C2×D4D8×C3114yes—
375C2×D4C2×D4×C3114yes—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4C2×D4×C3111yescharacteristic
389C22×C4C2×D4×C3112yes—
390C22×C4C2×D4×C3111yescharacteristic
391C22×C4C2×D4×C3111yescharacteristic
392C22×C4C2×D4×C3111yescharacteristic
393C22×C4C2×D4×C3112yes—
394C22×C4C24×C3111yescharacteristic
395C2×C8—24no—
396C2×C8—44no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4C2×D4×C3114yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C42—21no—
416C42—21no—
417C23—21no—
418C23—21no—
419C23—24no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—44no—
425C23—44no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23C31×#12012yes—
431C23C2×D8×C3112yes—
432C23C2×D8×C3112yes—
433C23C31×AGL(1,ℤ/8)12yes—
434C23C22×D4×C3111yescharacteristic
435Q8—44no—
436D4—28no—
437D4—44no—
438D4—44no—
439D4—48no—
440D4—48no—
441D4—44no—
442D4—44no—
443D4—44no—
444D4—82no—
445D4—82no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—48no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—84no—
469C2×C4C31×#12011yescharacteristic
470C2×C4C31×#12011yescharacteristic
471C2×C4C2×D8×C3112yes—
472C2×C4C22×D4×C3112yes—
473C8—44no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—81no—
488C22—164no—
489C22—164no—
490C22C31×#44714yes—
491C22C2×C31×#12011yescharacteristic
492C22C22×D8×C3111yescharacteristic
493C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—44no—
498C4—41no—
499C4—41no—
500C4—82no—
501C4—82no—
502C4C31×#45812yes—
503C2—21no—
504C2—21no—
505C2—24no—
506C2—42no—
507C2—84no—
508C2C2×C31×#44711yescharacteristic
509C2C31×#302211yescharacteristic
510C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#480C2semidirect
3. C31×#2486C2semidirect
4. C31×#1610C2semidirect
5. C31×#1607C2semidirect
6. C31×#1519C2semidirect
7. C31×#1459C2semidirect
20. C22×C4⋊C4×C31C22semidirect
22. C31×#480C22semidirect
25. C2×C31×C22⋊C8C22semidirect
26. C31×#398C22semidirect
27. C31×#397C22semidirect
31. C31×#380C22semidirect
32. C31×#327C22semidirect
33. C31×#324C22semidirect
34. C31×#323C22semidirect
161. #18202C31direct (N × H)
220. C2×#480C2×C31semidirect
221. #2486C2×C31semidirect
222. #1610C2×C31semidirect
223. #1607C2×C31semidirect
224. #1519C2×C31semidirect
225. #1459C2×C31semidirect
267. C22×C4⋊C4C22×C31semidirect
269. #480C22×C31semidirect
272. C2×C22⋊C8C22×C31semidirect
273. #398C22×C31semidirect
274. #397C22×C31semidirect
278. #380C22×C31semidirect
279. #327C22×C31semidirect
280. #324C22×C31semidirect
281. #323C22×C31semidirect
349. C31#18202direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34962
C2×C31×#4802
C31×#24862
C31×#16102
C31×#16072
C31×#15192
C31×#14592
#1820231