Finite Group Databasebeta

← C31×#18325 #21475084

Subgroups — C31×#18325 #21475084

Order 7936.

1230 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C31×#3193C212yesmaximal, complemented
3C31×#2583C211yesmaximal, characteristic, complemented
4C31×#1610C214yesmaximal, complemented
5C31×#1604C214yesmaximal, complemented
6C31×#1507C212yesmaximal, complemented
7C31×#1463C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#537—24no—
10C31×#537—22no—
11C31×#537C2214yes—
12C2×C31×C22⋊Q8—22no—
13C2×C31×C22⋊Q8C2211yescharacteristic
14C2×C31×#121—22no—
15C2×C31×#121—21no—
16C2×C31×#121C2212yes—
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#484—22no—
19C31×#480C2214yescomplemented
20C31×#468—24no—
21C31×#450—22no—
22C31×#448C2214yescomplemented
23C31×#407C2212yes—
24C31×#398C2212yescomplemented
25C31×#395C2212yescomplemented
26C31×#389—22no—
27C31×#385—21no—
28C31×#385—21no—
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#327C2218yescomplemented
31C31×#324C2214yescomplemented
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—48no—
35C22×Q8×C31—21no—
36C22×Q8×C31C2311yescharacteristic
37C22×D4×C31—21no—
38C22×D4×C31—22no—
39C22×D4×C31—21no—
40C22×D4×C31D412yes—
41C23×C4×C31—22no—
42C23×C4×C31D411yescharacteristic
43C23×C4×C31D411yescharacteristic
44C23×C4×C31C2311yescharacteristic
45C2×SD16×C31—24no—
46C2×M16×C31—22no—
47C31×C22⋊Q8—28no—
48C31×C22⋊Q8—28no—
49C31×C22⋊Q8—22no—
50C31×C22⋊Q8—22no—
51C31×#121—24no—
52C31×#121—28no—
53C31×#121—28no—
54C31×#121—24no—
55C31×#121—22no—
56C31×#121—22no—
57C31×#121D414yes—
58C31×#121C2314yes—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31D412yes—
66C2×C4⋊C4×C31D412yes—
67C2×C4⋊C4×C31C2311yescharacteristic
68C2×C4⋊C4×C31C2312yes—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—22no—
72C2×C22⋊C4×C31—21no—
73C2×C42×C31—21no—
74C31×#107—24no—
75C31×#103—24no—
76C31×#102—24no—
77C31×C22⋊C8—22no—
78C31×C22⋊C8C2314yes—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95C2312yes—
82C24×C31—22no—
83C24×C31C2×D411yescharacteristic
84#41×C31—416no—
85#41×C31—48no—
86#41×C31—48no—
87C2×Q8×C31—24no—
88C2×Q8×C31—22no—
89C2×Q8×C31—44no—
90C2×Q8×C31—81no—
91C2×D4×C31—24no—
92C2×D4×C31—24no—
93C2×D4×C31—24no—
94C2×D4×C31—24no—
95C2×D4×C31—22no—
96C2×D4×C31—44no—
97C2×D4×C31—48no—
98C2×D4×C31—82no—
99C2×D4×C31—82no—
100C2×D4×C31C2×D414yes—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—24no—
104C22×C4×C31—28no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—21no—
109C22×C4×C31—21no—
110C22×C4×C31—21no—
111C22×C4×C31—48no—
112C22×C4×C31—81no—
113C22×C4×C31—81no—
114C22×C4×C31C2×D412yes—
115C22×C4×C31C2×D412yes—
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D412yes—
118C22×C4×C31C2×D412yes—
119C22×C4×C31C2411yescharacteristic
120SD16×C31—84no—
121M16×C31—82no—
122C2×C8×C31—24no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—42no—
129C4⋊C4×C31—44no—
130C4⋊C4×C31—82no—
131C4⋊C4×C31C2×D414yes—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—28no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—24no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C42×C31—81no—
139#18325C3111yesmaximal, characteristic, direct product factor, Sylow
140C23×C31—22no—
141C23×C31—22no—
142C23×C31—24no—
143C23×C31—41no—
144C23×C31—44no—
145C23×C31—44no—
146C23×C31—44no—
147C23×C31#12011yescharacteristic
148C23×C31#12011yescharacteristic
149C23×C31C22×D411yescharacteristic
150Q8×C31—48no—
151Q8×C31—44no—
152D4×C31—44no—
153D4×C31—48no—
154D4×C31—48no—
155D4×C31—48no—
156D4×C31—82no—
157D4×C31—84no—
158C2×C4×C31—24no—
159C2×C4×C31—24no—
160C2×C4×C31—22no—
161C2×C4×C31—22no—
162C2×C4×C31—24no—
163C2×C4×C31—22no—
164C2×C4×C31—24no—
165C2×C4×C31—41no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—41no—
169C2×C4×C31—416no—
170C2×C4×C31—42no—
171C2×C4×C31—44no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—42no—
175C2×C4×C31—42no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—42no—
179C2×C4×C31—81no—
180C2×C4×C31#12012yes—
181C2×C4×C31AGL(1,ℤ/8)14yes—
182C2×C4×C31C22×D412yes—
183C8×C31—44no—
184C2×#537C2×C3111yescharacteristic
185#3193C2×C3112yescomplemented
186#2583C2×C3111yescharacteristic, complemented
187#1610C2×C3114yescomplemented
188#1604C2×C3114yescomplemented
189#1507C2×C3112yescomplemented
190#1463C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—44no—
199C22×C31—44no—
200C22×C31—44no—
201C22×C31—44no—
202C22×C31—88no—
203C22×C31#45612yes—
204C22×C31C2×#12011yescharacteristic
205C22×C31C2×AGL(1,ℤ/8)12yes—
206C4×C31—24no—
207C4×C31—22no—
208C4×C31—22no—
209C4×C31—44no—
210C4×C31—44no—
211C4×C31—42no—
212C4×C31—81no—
213C4×C31—81no—
214C22×#41—21no—
215#537—24no—
216#537—22no—
217#537C22×C3114yes—
218C2×C22⋊Q8—22no—
219C2×C22⋊Q8C22×C3111yescharacteristic
220C2×#121—22no—
221C2×#121—21no—
222C2×#121C22×C3112yes—
223C22×C4⋊C4C22×C3111yescharacteristic
224#484—22no—
225#480C22×C3114yescomplemented
226#468—24no—
227#450—22no—
228#448C22×C3114yescomplemented
229#407C22×C3112yes—
230#398C22×C3112yescomplemented
231#395C22×C3112yescomplemented
232#389—22no—
233#385—21no—
234#385—21no—
235C4×C22⋊C4C22×C3111yescharacteristic
236#327C22×C3118yescomplemented
237#324C22×C3114yescomplemented
238C2×C31—22no—
239C2×C31—22no—
240C2×C31—41no—
241C2×C31—84no—
242C2×C31#300311yescharacteristic
243C2×C31C2×#45611yescharacteristic
244C2×C31#302611yescharacteristic
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—48no—
248C22×Q8—21no—
249C22×Q8C23×C3111yescharacteristic
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4D4×C3112yes—
254C23×C4—22no—
255C23×C4D4×C3111yescharacteristic
256C23×C4D4×C3111yescharacteristic
257C23×C4C23×C3111yescharacteristic
258C2×SD16—24no—
259C2×M16—22no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264#121—24no—
265#121—28no—
266#121—28no—
267#121—24no—
268#121—22no—
269#121—22no—
270#121D4×C3114yes—
271#121C23×C3114yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4D4×C3112yes—
279C2×C4⋊C4D4×C3112yes—
280C2×C4⋊C4C23×C3111yescharacteristic
281C2×C4⋊C4C23×C3112yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C42—21no—
287#107—24no—
288#103—24no—
289#102—24no—
290C22⋊C8—22no—
291C22⋊C8C23×C3114yes—
292#95—22no—
293#95—22no—
294#95C23×C3112yes—
295C31#1832511yescharacteristic, direct product factor, Sylow
296C24—22no—
297C24C2×D4×C3111yescharacteristic
298#41—416no—
299#41—48no—
300#41—48no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×Q8—81no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×D4×C3114yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—48no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4C2×D4×C3112yes—
329C22×C4C2×D4×C3112yes—
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3112yes—
332C22×C4C2×D4×C3112yes—
333C22×C4C24×C3111yescharacteristic
334SD16—84no—
335M16—82no—
336C2×C8—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—82no—
345C4⋊C4C2×D4×C3114yes—
346C22⋊C4—24no—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—81no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—41no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23C31×#12011yescharacteristic
361C23C31×#12011yescharacteristic
362C23C22×D4×C3111yescharacteristic
363Q8—48no—
364Q8—44no—
365D4—44no—
366D4—48no—
367D4—48no—
368D4—48no—
369D4—82no—
370D4—84no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—416no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—81no—
393C2×C4C31×#12012yes—
394C2×C4C31×AGL(1,ℤ/8)14yes—
395C2×C4C22×D4×C3112yes—
396C8—44no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—88no—
409C22C31×#45612yes—
410C22C2×C31×#12011yescharacteristic
411C22C2×C31×AGL(1,ℤ/8)12yes—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—44no—
417C4—42no—
418C4—81no—
419C4—81no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—84no—
424C2C31×#300311yescharacteristic
425C2C2×C31×#45611yescharacteristic
426C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#537C2
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×#3193C2semidirect
3. C31×#2583C2semidirect
4. C31×#1610C2semidirect
5. C31×#1604C2semidirect
6. C31×#1507C2semidirect
7. C31×#1463C2semidirect
19. C31×#480C22semidirect
22. C31×#448C22semidirect
24. C31×#398C22semidirect
25. C31×#395C22semidirect
30. C31×#327C22semidirect
31. C31×#324C22semidirect
139. #18325C31direct (N × H)
185. #3193C2×C31semidirect
186. #2583C2×C31semidirect
187. #1610C2×C31semidirect
188. #1604C2×C31semidirect
189. #1507C2×C31semidirect
190. #1463C2×C31semidirect
225. #480C22×C31semidirect
228. #448C22×C31semidirect
230. #398C22×C31semidirect
231. #395C22×C31semidirect
236. #327C22×C31semidirect
237. #324C22×C31semidirect
295. C31#18325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5372
C31×#31932
C31×#25832
C31×#16102
C31×#16042
C31×#15072
C31×#14632
#1832531