Finite Group Databasebeta

← C31×#18535 #21475294

Subgroups — C31×#18535 #21475294

Order 7936.

908 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#557C211yesmaximal, characteristic, complemented
2C31×#3141C211yesmaximal, characteristic
3C31×#3112C211yesmaximal, characteristic
4C31×#3112C211yesmaximal, characteristic
5C31×#1636C214yesmaximal, complemented
6C31×#1626C214yesmaximal, complemented
7C31×#1556C211yesmaximal, characteristic, complemented
8C31×#1491C211yesmaximal, characteristic, complemented
9C31×#1491C211yesmaximal, characteristic, complemented
10C31×#557—22no—
11C31×#557—22no—
12C31×#557—22no—
13C31×#557C2214yes—
14C31×#532C2211yescharacteristic
15C2×C31×C4⋊Q8—21no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#125—22no—
18C2×C31×#125—21no—
19C2×C4×Q8×C31—21no—
20C2×C4×Q8×C31—21no—
21C2×C4×Q8×C31C2211yescharacteristic
22C2×C4×Q8×C31C2211yescharacteristic
23C31×#489—22no—
24C31×#487C2212yes—
25C31×#486C2212yes—
26C31×#480—21no—
27C31×#480—21no—
28C31×#477C2212yes—
29C31×#477C2212yes—
30C31×#460C2212yes—
31C31×#460C2212yes—
32C31×#448—21no—
33C31×#448—21no—
34C31×#440—22no—
35C31×#440—22no—
36C31×#432C2211yescharacteristic
37C31×#432C2211yescharacteristic
38C31×#417—22no—
39C4×M16×C31C2211yescharacteristic
40C31×#332C2214yes—
41C31×Q8⋊C8C2214yes—
42C31×Q8⋊C8C2214yes—
43C2×#41×C31—21no—
44C22×Q8×C31—21no—
45C22×Q8×C31D411yescharacteristic
46C22×Q8×C31D411yescharacteristic
47C2×SD16×C31—22no—
48C2×SD16×C31—22no—
49C2×M16×C31—21no—
50C31×C4⋊Q8—24no—
51C31×C4⋊Q8—24no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8D412yes—
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2312yes—
57C31×#127C2311yescharacteristic
58C31×#125—28no—
59C31×#125—24no—
60C31×#125—28no—
61C31×#125—24no—
62C31×#124—21no—
63C31×#121—21no—
64C31×#121—21no—
65C4×Q8×C31—24no—
66C4×Q8×C31—24no—
67C4×Q8×C31—24no—
68C4×Q8×C31—24no—
69C4×Q8×C31—21no—
70C4×Q8×C31—21no—
71C4×Q8×C31—21no—
72C4×Q8×C31—21no—
73C4×Q8×C31D412yes—
74C4×Q8×C31D412yes—
75C4×Q8×C31C2312yes—
76C4×Q8×C31C2312yes—
77C4×D4×C31—21no—
78C4×D4×C31—21no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31D412yes—
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C42×C31—22no—
89C2×C42×C31—21no—
90C2×C42×C31—21no—
91C2×C42×C31C2311yescharacteristic
92C31×#107—22no—
93C31×#107—22no—
94C31×C4⋊C8C2312yes—
95C31×C4⋊C8C2312yes—
96C31×#103—22no—
97C31×#103—22no—
98C31×#103—24no—
99C31×#102—24no—
100C31×#102—22no—
101C31×#102—22no—
102C31×C22⋊C8—21no—
103C31×C22⋊C8—21no—
104C31×#97—21no—
105C31×#97—21no—
106C31×#97—21no—
107C4×C8×C31C2312yes—
108#41×C31—44no—
109C2×Q8×C31—24no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×Q8×C31—21no—
115C2×Q8×C31—42no—
116C2×Q8×C31—44no—
117C2×Q8×C31C2×D412yes—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—21no—
120C2×D4×C31—21no—
121C2×D4×C31—21no—
122C2×D4×C31—41no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—41no—
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132SD16×C31—44no—
133SD16×C31—44no—
134M16×C31—42no—
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C2×C8×C31—22no—
138C4⋊C4×C31—28no—
139C4⋊C4×C31—28no—
140C4⋊C4×C31—28no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31C2×D414yes—
153C4⋊C4×C31C2×D412yes—
154C4⋊C4×C31C2×D412yes—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—41no—
157C42×C31—28no—
158C42×C31—22no—
159C42×C31—22no—
160C42×C31—22no—
161C42×C31—22no—
162C42×C31C2×D411yescharacteristic
163C42×C31C2×D411yescharacteristic
164C42×C31C2×D411yescharacteristic
165C42×C31C2411yescharacteristic
166#18535C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—41no—
168C23×C31#12011yescharacteristic
169Q8×C31—24no—
170Q8×C31—22no—
171Q8×C31—22no—
172Q8×C31—22no—
173Q8×C31—42no—
174Q8×C31—44no—
175Q8×C31—42no—
176Q8×C31—44no—
177Q8×C31#13714yes—
178D4×C31—42no—
179D4×C31—42no—
180D4×C31—42no—
181D4×C31—82no—
182C2×C4×C31—24no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—24no—
191C2×C4×C31—42no—
192C2×C4×C31—44no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—41no—
196C2×C4×C31—82no—
197C2×C4×C31#12011yescharacteristic
198C2×C4×C31#12011yescharacteristic
199C2×C4×C31#12011yescharacteristic
200C2×C4×C31C22×D411yescharacteristic
201C2×C4×C31C22×D411yescharacteristic
202C2×C4×C31C22×D411yescharacteristic
203C8×C31—42no—
204C8×C31—42no—
205C8×C31—42no—
206C2×#557C2×C3111yescharacteristic, complemented
207#3141C2×C3111yescharacteristic
208#3112C2×C3111yescharacteristic
209#3112C2×C3111yescharacteristic
210#1636C2×C3114yescomplemented
211#1626C2×C3114yescomplemented
212#1556C2×C3111yescharacteristic, complemented
213#1491C2×C3111yescharacteristic, complemented
214#1491C2×C3111yescharacteristic, complemented
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—81no—
218C22×C31—81no—
219C22×C31—81no—
220C22×C31#45412yes—
221C22×C31C2×#12011yescharacteristic
222C4×C31—24no—
223C4×C31—21no—
224C4×C31—21no—
225C4×C31—21no—
226C4×C31—21no—
227C4×C31—42no—
228C4×C31—44no—
229C4×C31—42no—
230C4×C31—42no—
231C4×C31—81no—
232C4×C31C2×#13712yes—
233C4×C31#57512yes—
234#557—22no—
235#557—22no—
236#557—22no—
237#557C22×C3114yes—
238#532C22×C3111yescharacteristic
239C2×C4⋊Q8—21no—
240C2×C4⋊Q8C22×C3111yescharacteristic
241C2×#125—22no—
242C2×#125—21no—
243C2×C4×Q8—21no—
244C2×C4×Q8—21no—
245C2×C4×Q8C22×C3111yescharacteristic
246C2×C4×Q8C22×C3111yescharacteristic
247#489—22no—
248#487C22×C3112yes—
249#486C22×C3112yes—
250#480—21no—
251#480—21no—
252#477C22×C3112yes—
253#477C22×C3112yes—
254#460C22×C3112yes—
255#460C22×C3112yes—
256#448—21no—
257#448—21no—
258#440—22no—
259#440—22no—
260#432C22×C3111yescharacteristic
261#432C22×C3111yescharacteristic
262#417—22no—
263C4×M16C22×C3111yescharacteristic
264#332C22×C3114yes—
265Q8⋊C8C22×C3114yes—
266Q8⋊C8C22×C3114yes—
267C2×C31—22no—
268C2×C31—161no—
269C2×C31#300411yescharacteristic
270C2×C31#300511yescharacteristic
271C2×C31C2×#45411yescharacteristic
272C2×#41—21no—
273C22×Q8—21no—
274C22×Q8D4×C3111yescharacteristic
275C22×Q8D4×C3111yescharacteristic
276C2×SD16—22no—
277C2×SD16—22no—
278C2×M16—21no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8D4×C3112yes—
284C4⋊Q8C23×C3111yescharacteristic
285C4⋊Q8C23×C3112yes—
286#127C23×C3111yescharacteristic
287#125—28no—
288#125—24no—
289#125—28no—
290#125—24no—
291#124—21no—
292#121—21no—
293#121—21no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8D4×C3112yes—
303C4×Q8D4×C3112yes—
304C4×Q8C23×C3112yes—
305C4×Q8C23×C3112yes—
306C4×D4—21no—
307C4×D4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4D4×C3111yescharacteristic
315C2×C4⋊C4D4×C3111yescharacteristic
316C2×C4⋊C4D4×C3112yes—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42C23×C3111yescharacteristic
321#107—22no—
322#107—22no—
323C4⋊C8C23×C3112yes—
324C4⋊C8C23×C3112yes—
325#103—22no—
326#103—22no—
327#103—24no—
328#102—24no—
329#102—22no—
330#102—22no—
331C22⋊C8—21no—
332C22⋊C8—21no—
333#97—21no—
334#97—21no—
335#97—21no—
336C4×C8C23×C3112yes—
337C31#1853511yescharacteristic, direct product factor, Sylow
338#41—44no—
339C2×Q8—24no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8C2×D4×C3112yes—
348C2×Q8C2×D4×C3112yes—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4C2×D4×C3111yescharacteristic
360C22×C4C2×D4×C3111yescharacteristic
361C22×C4C2×D4×C3111yescharacteristic
362SD16—44no—
363SD16—44no—
364M16—42no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4C2×D4×C3114yes—
383C4⋊C4C2×D4×C3112yes—
384C4⋊C4C2×D4×C3112yes—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C42—28no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42C2×D4×C3111yescharacteristic
393C42C2×D4×C3111yescharacteristic
394C42C2×D4×C3111yescharacteristic
395C42C24×C3111yescharacteristic
396C23—41no—
397C23C31×#12011yescharacteristic
398Q8—24no—
399Q8—22no—
400Q8—22no—
401Q8—22no—
402Q8—42no—
403Q8—44no—
404Q8—42no—
405Q8—44no—
406Q8C31×#13714yes—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—82no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—82no—
426C2×C4C31×#12011yescharacteristic
427C2×C4C31×#12011yescharacteristic
428C2×C4C31×#12011yescharacteristic
429C2×C4C22×D4×C3111yescharacteristic
430C2×C4C22×D4×C3111yescharacteristic
431C2×C4C22×D4×C3111yescharacteristic
432C8—42no—
433C8—42no—
434C8—42no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22C31×#45412yes—
441C22C2×C31×#12011yescharacteristic
442C4—24no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—44no—
449C4—42no—
450C4—42no—
451C4—81no—
452C4C2×C31×#13712yes—
453C4C31×#57512yes—
454C2—22no—
455C2—161no—
456C2C31×#300411yescharacteristic
457C2C31×#300511yescharacteristic
458C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#557C2semidirect
5. C31×#1636C2semidirect
6. C31×#1626C2semidirect
7. C31×#1556C2semidirect
8. C31×#1491C2semidirect
9. C31×#1491C2semidirect
166. #18535C31direct (N × H)
206. C2×#557C2×C31semidirect
210. #1636C2×C31semidirect
211. #1626C2×C31semidirect
212. #1556C2×C31semidirect
213. #1491C2×C31semidirect
214. #1491C2×C31semidirect
337. C31#18535direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5572
C31×#31412
C31×#31122
C31×#16362
C31×#16262
C31×#15562
C31×#14912
#1853531