Finite Group Databasebeta

← C31×#18285 #21475044

Subgroups — C31×#18285 #21475044

Order 7936.

1408 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#534C211yesmaximal, characteristic
2C31×#3187C212yesmaximal, complemented
3C31×#2626C211yesmaximal, characteristic, complemented
4C31×#1613C214yesmaximal, complemented
5C31×#1601C214yesmaximal, complemented
6C31×#1519C212yesmaximal, complemented
7C2×C31×#323C211yesmaximal, characteristic, complemented
8C23×D4×C31—21no—
9C31×#534—24no—
10C31×#534—22no—
11C31×#534C2214yes—
12C2×C31×#123—22no—
13C2×C31×C22⋊Q8C2211yescharacteristic, complemented
14C2×C31×#121—21no—
15C2×C31×#121C2212yes—
16C2×C31×#120—22no—
17C22×C22⋊C4×C31C2211yescharacteristic
18C31×#483—22no—
19C31×#481C2214yescomplemented
20C31×#465—24no—
21C31×#450—22no—
22C31×#448C2214yescomplemented
23C2×C31×C22⋊C8C2212yescomplemented
24C31×#399C2212yescomplemented
25C31×#396—22no—
26C31×#393C2212yescomplemented
27C31×#385—21no—
28C31×#380C2211yescharacteristic, complemented
29C31×#379—21no—
30C31×#327C2218yescomplemented
31C31×#323C2214yescomplemented
32C25×C31—21no—
33C25×C31D411yescharacteristic
34C22×Q8×C31C2311yescharacteristic, complemented
35C22×D4×C31—22no—
36C22×D4×C31—24no—
37C22×D4×C31—22no—
38C22×D4×C31—24no—
39C22×D4×C31—21no—
40C22×D4×C31—44no—
41C22×D4×C31—44no—
42C22×D4×C31D412yes—
43C23×C4×C31—21no—
44C23×C4×C31D411yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C2×SD16×C31—24no—
47C22×C8×C31—22no—
48C31×#123—28no—
49C31×#123—28no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—22no—
52C31×#121—24no—
53C31×#121—24no—
54C31×#121—22no—
55C31×#121—22no—
56C31×#121D414yes—
57C31×#121C2314yes—
58C31×#120—28no—
59C31×#120—28no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31D412yes—
65C2×C4⋊C4×C31C2311yescharacteristic, complemented
66C2×C22⋊C4×C31—22no—
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×C31—22no—
74C2×C22⋊C4×C31D412yes—
75C2×C22⋊C4×C31C2312yes—
76C31×#106—24no—
77C31×#103—24no—
78C31×#102—24no—
79C31×C22⋊C8—22no—
80C31×C22⋊C8C2314yescomplemented
81C31×#95—22no—
82C31×#95C2312yescomplemented
83C24×C31—21no—
84C24×C31—21no—
85C24×C31—22no—
86C24×C31—24no—
87C24×C31—22no—
88C24×C31—44no—
89C24×C31—81no—
90C24×C31—81no—
91C24×C31—82no—
92C24×C31—82no—
93C24×C31SD1614yescomplemented
94C24×C31C2×D412yes—
95C24×C31C2×D411yescharacteristic
96C2×Q8×C31—22no—
97C2×Q8×C31—81no—
98C2×D4×C31—28no—
99C2×D4×C31—24no—
100C2×D4×C31—24no—
101C2×D4×C31—24no—
102C2×D4×C31—22no—
103C2×D4×C31—48no—
104C2×D4×C31—416no—
105C2×D4×C31—48no—
106C2×D4×C31—48no—
107C2×D4×C31—44no—
108C2×D4×C31—44no—
109C2×D4×C31—82no—
110C2×D4×C31—82no—
111C2×D4×C31C2×D414yes—
112C22×C4×C31—24no—
113C22×C4×C31—21no—
114C22×C4×C31—22no—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—22no—
118C22×C4×C31—21no—
119C22×C4×C31—44no—
120C22×C4×C31—81no—
121C22×C4×C31—81no—
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2×D411yescharacteristic
124C22×C4×C31C2×D412yes—
125C22×C4×C31C2×D412yes—
126C22×C4×C31C2411yescharacteristic
127SD16×C31—84no—
128C2×C8×C31—24no—
129C2×C8×C31—82no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—42no—
134C4⋊C4×C31—81no—
135C4⋊C4×C31—82no—
136C4⋊C4×C31C2×D414yes—
137C22⋊C4×C31—28no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—24no—
140C22⋊C4×C31—24no—
141C22⋊C4×C31—24no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C22⋊C4×C31—42no—
145C22⋊C4×C31—82no—
146C22⋊C4×C31—82no—
147#18285C3111yesmaximal, characteristic, direct product factor, Sylow
148C23×C31—22no—
149C23×C31—24no—
150C23×C31—22no—
151C23×C31—24no—
152C23×C31—24no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—42no—
156C23×C31—42no—
157C23×C31—42no—
158C23×C31—42no—
159C23×C31—48no—
160C23×C31—44no—
161C23×C31—44no—
162C23×C31—44no—
163C23×C31—81no—
164C23×C31—81no—
165C23×C31—84no—
166C23×C31—84no—
167C23×C31—84no—
168C23×C31—84no—
169C23×C31—816no—
170C23×C31#12011yescharacteristic
171C23×C31#12011yescharacteristic
172C23×C31C2×SD1614yes—
173C23×C31C2×SD1612yes—
174C23×C31AGL(1,ℤ/8)12yes—
175C23×C31C22×D411yescharacteristic
176Q8×C31—44no—
177D4×C31—44no—
178D4×C31—416no—
179D4×C31—44no—
180D4×C31—44no—
181D4×C31—48no—
182D4×C31—82no—
183D4×C31—84no—
184C2×C4×C31—22no—
185C2×C4×C31—24no—
186C2×C4×C31—41no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—41no—
190C2×C4×C31—48no—
191C2×C4×C31—42no—
192C2×C4×C31—44no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—41no—
199C2×C4×C31—41no—
200C2×C4×C31—42no—
201C2×C4×C31—81no—
202C2×C4×C31#12012yes—
203C2×C4×C31C22×D412yes—
204C8×C31—44no—
205C2×#534C2×C3111yescharacteristic
206#3187C2×C3112yescomplemented
207#2626C2×C3111yescharacteristic, complemented
208#1613C2×C3114yescomplemented
209#1601C2×C3114yescomplemented
210#1519C2×C3112yescomplemented
211C2×#323C2×C3111yescharacteristic, complemented
212C22×C31—21no—
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—21no—
218C22×C31—24no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—44no—
222C22×C31—48no—
223C22×C31—42no—
224C22×C31—42no—
225C22×C31—44no—
226C22×C31—44no—
227C22×C31—44no—
228C22×C31—82no—
229C22×C31—84no—
230C22×C31—88no—
231C22×C31—816no—
232C22×C31#45012yes—
233C22×C31#45012yes—
234C22×C31C2×#12011yescharacteristic
235C22×C31C22×SD1611yescharacteristic
236C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C31—22no—
238C4×C31—22no—
239C4×C31—44no—
240C4×C31—42no—
241C4×C31—82no—
242C4×C31—81no—
243C4×C31—81no—
244C23×D4—21no—
245#534—24no—
246#534—22no—
247#534C22×C3114yes—
248C2×#123—22no—
249C2×C22⋊Q8C22×C3111yescharacteristic, complemented
250C2×#121—21no—
251C2×#121C22×C3112yes—
252C2×#120—22no—
253C22×C22⋊C4C22×C3111yescharacteristic
254#483—22no—
255#481C22×C3114yescomplemented
256#465—24no—
257#450—22no—
258#448C22×C3114yescomplemented
259C2×C22⋊C8C22×C3112yescomplemented
260#399C22×C3112yescomplemented
261#396—22no—
262#393C22×C3112yescomplemented
263#385—21no—
264#380C22×C3111yescharacteristic, complemented
265#379—21no—
266#327C22×C3118yescomplemented
267#323C22×C3114yescomplemented
268C2×C31—21no—
269C2×C31—22no—
270C2×C31—21no—
271C2×C31—22no—
272C2×C31—44no—
273C2×C31—84no—
274C2×C31C2×#45011yescharacteristic
275C2×C31#301911yescharacteristic
276C2×C31#302611yescharacteristic
277C25—21no—
278C25D4×C3111yescharacteristic
279C22×Q8C23×C3111yescharacteristic, complemented
280C22×D4—22no—
281C22×D4—24no—
282C22×D4—22no—
283C22×D4—24no—
284C22×D4—21no—
285C22×D4—44no—
286C22×D4—44no—
287C22×D4D4×C3112yes—
288C23×C4—21no—
289C23×C4D4×C3111yescharacteristic
290C23×C4C23×C3111yescharacteristic
291C2×SD16—24no—
292C22×C8—22no—
293#123—28no—
294#123—28no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—24no—
298#121—24no—
299#121—22no—
300#121—22no—
301#121D4×C3114yes—
302#121C23×C3114yes—
303#120—28no—
304#120—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4D4×C3112yes—
310C2×C4⋊C4C23×C3111yescharacteristic, complemented
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4D4×C3112yes—
320C2×C22⋊C4C23×C3112yes—
321#106—24no—
322#103—24no—
323#102—24no—
324C22⋊C8—22no—
325C22⋊C8C23×C3114yescomplemented
326#95—22no—
327#95C23×C3112yescomplemented
328C31#1828511yescharacteristic, direct product factor, Sylow
329C24—21no—
330C24—21no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—44no—
335C24—81no—
336C24—81no—
337C24—82no—
338C24—82no—
339C24SD16×C3114yescomplemented
340C24C2×D4×C3112yes—
341C24C2×D4×C3111yescharacteristic
342C2×Q8—22no—
343C2×Q8—81no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—416no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4C2×D4×C3114yes—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—44no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4C2×D4×C3112yes—
369C22×C4C2×D4×C3111yescharacteristic
370C22×C4C2×D4×C3112yes—
371C22×C4C2×D4×C3112yes—
372C22×C4C24×C3111yescharacteristic
373SD16—84no—
374C2×C8—24no—
375C2×C8—82no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4C2×D4×C3114yes—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—24no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—81no—
409C23—81no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23—816no—
415C23C31×#12011yescharacteristic
416C23C31×#12011yescharacteristic
417C23C2×SD16×C3114yes—
418C23C2×SD16×C3112yes—
419C23C31×AGL(1,ℤ/8)12yes—
420C23C22×D4×C3111yescharacteristic
421Q8—44no—
422D4—44no—
423D4—416no—
424D4—48no—
425D4—44no—
426D4—44no—
427D4—82no—
428D4—84no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4C31×#12012yes—
448C2×C4C22×D4×C3112yes—
449C8—44no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—82no—
467C22—84no—
468C22—88no—
469C22—816no—
470C22C31×#45012yes—
471C22C31×#45012yes—
472C22C2×C31×#12011yescharacteristic
473C22C22×SD16×C3111yescharacteristic
474C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
475C4—22no—
476C4—22no—
477C4—44no—
478C4—42no—
479C4—82no—
480C4—81no—
481C4—81no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—84no—
488C2C2×C31×#45011yescharacteristic
489C2C31×#301911yescharacteristic
490C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#534C2
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. C31×#3187C2semidirect
3. C31×#2626C2semidirect
4. C31×#1613C2semidirect
5. C31×#1601C2semidirect
6. C31×#1519C2semidirect
7. C2×C31×#323C2semidirect
13. C2×C31×C22⋊Q8C22semidirect
19. C31×#481C22semidirect
22. C31×#448C22semidirect
23. C2×C31×C22⋊C8C22semidirect
24. C31×#399C22semidirect
26. C31×#393C22semidirect
28. C31×#380C22semidirect
30. C31×#327C22semidirect
31. C31×#323C22semidirect
34. C22×Q8×C31C23semidirect
65. C2×C4⋊C4×C31C23semidirect
80. C31×C22⋊C8C23semidirect
82. C31×#95C23semidirect
93. C24×C31SD16semidirect
147. #18285C31direct (N × H)
206. #3187C2×C31semidirect
207. #2626C2×C31semidirect
208. #1613C2×C31semidirect
209. #1601C2×C31semidirect
210. #1519C2×C31semidirect
211. C2×#323C2×C31semidirect
249. C2×C22⋊Q8C22×C31semidirect
255. #481C22×C31semidirect
258. #448C22×C31semidirect
259. C2×C22⋊C8C22×C31semidirect
260. #399C22×C31semidirect
262. #393C22×C31semidirect
264. #380C22×C31semidirect
266. #327C22×C31semidirect
267. #323C22×C31semidirect
279. C22×Q8C23×C31semidirect
310. C2×C4⋊C4C23×C31semidirect
325. C22⋊C8C23×C31semidirect
327. #95C23×C31semidirect
328. C31#18285direct (N × H)
339. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5342
C31×#31872
C31×#26262
C31×#16132
C31×#16012
C31×#15192
C2×C31×#3232
#1828531