Finite Group Databasebeta

← C31×#18286 #21475045

Subgroups — C31×#18286 #21475045

Order 7936.

1142 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#543C211yesmaximal, characteristic, complemented
2C31×#3187C212yesmaximal, complemented
3C31×#2617C211yesmaximal, characteristic
4C31×#1608C214yesmaximal, complemented
5C31×#1596C214yesmaximal, complemented
6C31×#1520C212yesmaximal, complemented
7C2×C31×#323C211yesmaximal, characteristic, complemented
8C31×#543—24no—
9C31×#543—22no—
10C31×#543C2214yes—
11C2×C31×C22⋊Q8—22no—
12C2×C31×C22⋊Q8—21no—
13C2×C31×C22⋊Q8—22no—
14C2×C31×C22⋊Q8C2212yes—
15C2×C31×#121C2211yescharacteristic
16C2×C31×#120—21no—
17C22×C22⋊C4×C31—21no—
18C22×C22⋊C4×C31C2211yescharacteristic
19C31×#483C2214yescomplemented
20C31×#481—22no—
21C31×#465—24no—
22C31×#450C2214yescomplemented
23C31×#448—22no—
24C2×C31×C22⋊C8C2212yes—
25C31×#397C2212yes—
26C31×#394—22no—
27C31×#392C2212yes—
28C31×#386—21no—
29C31×#380C2211yescharacteristic
30C31×#328C2218yescomplemented
31C31×#323C2214yes—
32C25×C31D411yescharacteristic
33C22×Q8×C31—21no—
34C22×Q8×C31D412yescomplemented
35C22×D4×C31—21no—
36C22×D4×C31—22no—
37C22×D4×C31C2311yescharacteristic
38C23×C4×C31—21no—
39C23×C4×C31—21no—
40C23×C4×C31D411yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C2×SD16×C31—24no—
43C22×C8×C31—22no—
44C31×C22⋊Q8—28no—
45C31×C22⋊Q8—28no—
46C31×C22⋊Q8—24no—
47C31×C22⋊Q8—28no—
48C31×C22⋊Q8—24no—
49C31×C22⋊Q8—22no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—28no—
52C31×C22⋊Q8D414yescomplemented
53C31×C22⋊Q8C2314yes—
54C31×#121—22no—
55C31×#121—22no—
56C31×#120—44no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D412yescomplemented
62C2×C4⋊C4×C31C2311yescharacteristic
63C2×C22⋊C4×C31—22no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31D412yes—
74C2×C22⋊C4×C31C2312yes—
75C31×#106—24no—
76C31×#103—24no—
77C31×#102—24no—
78C31×C22⋊C8—22no—
79C31×C22⋊C8C2314yes—
80C31×#95—22no—
81C31×#95C2312yes—
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—22no—
85C24×C31—21no—
86C24×C31—81no—
87C24×C31—81no—
88C24×C31SD1614yescomplemented
89C24×C31C2×D412yes—
90C24×C31C2×D411yescharacteristic
91C2×Q8×C31—24no—
92C2×Q8×C31—22no—
93C2×Q8×C31—82no—
94C2×Q8×C31—82no—
95C2×Q8×C31C2×D414yes—
96C2×D4×C31—22no—
97C2×D4×C31—41no—
98C2×D4×C31—41no—
99C2×D4×C31—44no—
100C2×D4×C31—42no—
101C2×D4×C31—42no—
102C2×D4×C31—81no—
103C2×D4×C31—82no—
104C2×D4×C31—82no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—22no—
108C22×C4×C31—24no—
109C22×C4×C31—24no—
110C22×C4×C31—21no—
111C22×C4×C31—21no—
112C22×C4×C31—22no—
113C22×C4×C31—81no—
114C22×C4×C31—81no—
115C22×C4×C31—82no—
116C22×C4×C31—82no—
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2×D411yescharacteristic
119C22×C4×C31C2×D412yes—
120C22×C4×C31C2×D412yes—
121C22×C4×C31C2411yescharacteristic
122SD16×C31—84no—
123C2×C8×C31—24no—
124C2×C8×C31—82no—
125C4⋊C4×C31—24no—
126C4⋊C4×C31—28no—
127C4⋊C4×C31—24no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—81no—
132C4⋊C4×C31C2×D414yes—
133C22⋊C4×C31—28no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—24no—
137C22⋊C4×C31—24no—
138C22⋊C4×C31—44no—
139C22⋊C4×C31—42no—
140C22⋊C4×C31—42no—
141C22⋊C4×C31—82no—
142#18286C3111yesmaximal, characteristic, direct product factor, Sylow
143C23×C31—22no—
144C23×C31—24no—
145C23×C31—22no—
146C23×C31—24no—
147C23×C31—41no—
148C23×C31—41no—
149C23×C31—42no—
150C23×C31—42no—
151C23×C31—42no—
152C23×C31—42no—
153C23×C31—42no—
154C23×C31—42no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31—81no—
159C23×C31—81no—
160C23×C31—84no—
161C23×C31—84no—
162C23×C31#12011yescharacteristic
163C23×C31#12011yescharacteristic
164C23×C31C2×SD1614yes—
165C23×C31C2×SD1612yes—
166C23×C31#13712yes—
167C23×C31C22×D411yescharacteristic
168Q8×C31—44no—
169Q8×C31—82no—
170Q8×C31—84no—
171D4×C31—44no—
172D4×C31—88no—
173D4×C31—82no—
174D4×C31—82no—
175C2×C4×C31—24no—
176C2×C4×C31—22no—
177C2×C4×C31—24no—
178C2×C4×C31—41no—
179C2×C4×C31—42no—
180C2×C4×C31—42no—
181C2×C4×C31—41no—
182C2×C4×C31—42no—
183C2×C4×C31—44no—
184C2×C4×C31—44no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—41no—
189C2×C4×C31—42no—
190C2×C4×C31—41no—
191C2×C4×C31—41no—
192C2×C4×C31—81no—
193C2×C4×C31—84no—
194C2×C4×C31—84no—
195C2×C4×C31#12012yes—
196C2×C4×C31C22×D412yes—
197C8×C31—44no—
198C2×#543C2×C3111yescharacteristic, complemented
199#3187C2×C3112yescomplemented
200#2617C2×C3111yescharacteristic
201#1608C2×C3114yescomplemented
202#1596C2×C3114yescomplemented
203#1520C2×C3112yescomplemented
204C2×#323C2×C3111yescharacteristic, complemented
205C22×C31—21no—
206C22×C31—21no—
207C22×C31—21no—
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—21no—
211C22×C31—24no—
212C22×C31—42no—
213C22×C31—42no—
214C22×C31—44no—
215C22×C31—48no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31—82no—
219C22×C31—84no—
220C22×C31—82no—
221C22×C31—82no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31#44812yes—
226C22×C31#44812yes—
227C22×C31C2×#12011yescharacteristic
228C22×C31C22×SD1611yescharacteristic
229C22×C31C2×#13711yescharacteristic
230C4×C31—22no—
231C4×C31—22no—
232C4×C31—42no—
233C4×C31—44no—
234C4×C31—44no—
235C4×C31—82no—
236C4×C31—81no—
237#543—24no—
238#543—22no—
239#543C22×C3114yes—
240C2×C22⋊Q8—22no—
241C2×C22⋊Q8—21no—
242C2×C22⋊Q8—22no—
243C2×C22⋊Q8C22×C3112yes—
244C2×#121C22×C3111yescharacteristic
245C2×#120—21no—
246C22×C22⋊C4—21no—
247C22×C22⋊C4C22×C3111yescharacteristic
248#483C22×C3114yescomplemented
249#481—22no—
250#465—24no—
251#450C22×C3114yescomplemented
252#448—22no—
253C2×C22⋊C8C22×C3112yes—
254#397C22×C3112yes—
255#394—22no—
256#392C22×C3112yes—
257#386—21no—
258#380C22×C3111yescharacteristic
259#328C22×C3118yescomplemented
260#323C22×C3114yes—
261C2×C31—21no—
262C2×C31—22no—
263C2×C31—21no—
264C2×C31—22no—
265C2×C31—44no—
266C2×C31—161no—
267C2×C31C2×#44811yescharacteristic
268C2×C31#301911yescharacteristic
269C2×C31#302611yescharacteristic
270C25D4×C3111yescharacteristic
271C22×Q8—21no—
272C22×Q8D4×C3112yescomplemented
273C22×D4—21no—
274C22×D4—22no—
275C22×D4C23×C3111yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4D4×C3111yescharacteristic
279C23×C4C23×C3111yescharacteristic
280C2×SD16—24no—
281C22×C8—22no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8—28no—
290C22⋊Q8D4×C3114yescomplemented
291C22⋊Q8C23×C3114yes—
292#121—22no—
293#121—22no—
294#120—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4D4×C3112yescomplemented
300C2×C4⋊C4C23×C3111yescharacteristic
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4D4×C3112yes—
312C2×C22⋊C4C23×C3112yes—
313#106—24no—
314#103—24no—
315#102—24no—
316C22⋊C8—22no—
317C22⋊C8C23×C3114yes—
318#95—22no—
319#95C23×C3112yes—
320C31#1828611yescharacteristic, direct product factor, Sylow
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—81no—
326C24—81no—
327C24SD16×C3114yescomplemented
328C24C2×D4×C3112yes—
329C24C2×D4×C3111yescharacteristic
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8C2×D4×C3114yes—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—82no—
343C2×D4—82no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4C2×D4×C3112yes—
357C22×C4C2×D4×C3111yescharacteristic
358C22×C4C2×D4×C3112yes—
359C22×C4C2×D4×C3112yes—
360C22×C4C24×C3111yescharacteristic
361SD16—84no—
362C2×C8—24no—
363C2×C8—82no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—81no—
371C4⋊C4C2×D4×C3114yes—
372C22⋊C4—28no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—82no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—81no—
397C23—81no—
398C23—84no—
399C23—84no—
400C23C31×#12011yescharacteristic
401C23C31×#12011yescharacteristic
402C23C2×SD16×C3114yes—
403C23C2×SD16×C3112yes—
404C23C31×#13712yes—
405C23C22×D4×C3111yescharacteristic
406Q8—44no—
407Q8—82no—
408Q8—84no—
409D4—44no—
410D4—88no—
411D4—82no—
412D4—82no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4C31×#12012yes—
434C2×C4C22×D4×C3112yes—
435C8—44no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C31×#44812yes—
457C22C31×#44812yes—
458C22C2×C31×#12011yescharacteristic
459C22C22×SD16×C3111yescharacteristic
460C22C2×C31×#13711yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—82no—
467C4—81no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—161no—
474C2C2×C31×#44811yescharacteristic
475C2C31×#301911yescharacteristic
476C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2617C2
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
1. C2×C31×#543C2semidirect
2. C31×#3187C2semidirect
4. C31×#1608C2semidirect
5. C31×#1596C2semidirect
6. C31×#1520C2semidirect
7. C2×C31×#323C2semidirect
19. C31×#483C22semidirect
22. C31×#450C22semidirect
30. C31×#328C22semidirect
34. C22×Q8×C31D4semidirect
52. C31×C22⋊Q8D4semidirect
61. C2×C4⋊C4×C31D4semidirect
88. C24×C31SD16semidirect
142. #18286C31direct (N × H)
198. C2×#543C2×C31semidirect
199. #3187C2×C31semidirect
201. #1608C2×C31semidirect
202. #1596C2×C31semidirect
203. #1520C2×C31semidirect
204. C2×#323C2×C31semidirect
248. #483C22×C31semidirect
251. #450C22×C31semidirect
259. #328C22×C31semidirect
272. C22×Q8D4×C31semidirect
290. C22⋊Q8D4×C31semidirect
299. C2×C4⋊C4D4×C31semidirect
320. C31#18286direct (N × H)
327. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5432
C31×#31872
C31×#26172
C31×#16082
C31×#15962
C31×#15202
C2×C31×#3232
#1828631