Finite Group Databasebeta

← C31×#18147 #21474906

Subgroups — C31×#18147 #21474906

Order 7936.

1210 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#537C211yesmaximal, characteristic
2C31×#3094C212yesmaximal, complemented
3C31×#2484C211yesmaximal, characteristic, complemented
4C31×#1616C214yesmaximal, complemented
5C31×#1610C214yesmaximal, complemented
6C31×#1510C212yesmaximal, complemented
7C31×#1463C211yesmaximal, characteristic, complemented
8C22×#41×C31—21no—
9C31×#537—24no—
10C31×#537—22no—
11C31×#537C2214yes—
12C2×C31×C22⋊Q8—22no—
13C2×C31×#121—22no—
14C2×C31×#121—21no—
15C2×C31×#121C2212yes—
16C2×C31×#117C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#482—24no—
19C31×#481C2214yescomplemented
20C31×#480C2214yescomplemented
21C31×#428—22no—
22C31×#418—22no—
23C31×#407C2212yes—
24C31×#400C2212yescomplemented
25C31×#398C2212yescomplemented
26C31×#387—22no—
27C31×#385—21no—
28C31×#381—21no—
29C31×#381C2211yescharacteristic
30C31×#327C2218yescomplemented
31C31×#324C2214yescomplemented
32C2×#41×C31—24no—
33C2×#41×C31—24no—
34C2×#41×C31—48no—
35C22×Q8×C31—21no—
36C22×D4×C31—21no—
37C22×D4×C31—22no—
38C22×D4×C31—21no—
39C22×D4×C31D412yes—
40C23×C4×C31—22no—
41C23×C4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C2×M16×C31—22no—
45C31×C22⋊Q8—28no—
46C31×C22⋊Q8—28no—
47C31×#121—24no—
48C31×#121—28no—
49C31×#121—28no—
50C31×#121—24no—
51C31×#121—22no—
52C31×#121—22no—
53C31×#121D414yes—
54C31×#121C2314yes—
55C31×#117—22no—
56C31×#117—22no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31D412yes—
62C2×C4⋊C4×C31D412yes—
63C2×C4⋊C4×C31C2311yescharacteristic
64C2×C4⋊C4×C31C2311yescharacteristic
65C2×C4⋊C4×C31C2312yes—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C42×C31—21no—
71C31×#107—24no—
72C31×#106—24no—
73C31×#102—24no—
74C31×#102—24no—
75C31×C22⋊C8—22no—
76C31×C22⋊C8C2314yes—
77C31×#95—22no—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95—21no—
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—44no—
89C2×D4×C31—24no—
90C2×D4×C31—24no—
91C2×D4×C31—24no—
92C2×D4×C31—24no—
93C2×D4×C31—22no—
94C2×D4×C31—44no—
95C2×D4×C31—44no—
96C2×D4×C31—48no—
97C2×D4×C31—82no—
98C2×D4×C31C2×D414yes—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—24no—
102C22×C4×C31—28no—
103C22×C4×C31—21no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—42no—
110C22×C4×C31—48no—
111C22×C4×C31—42no—
112C22×C4×C31C2×D412yes—
113C22×C4×C31C2×D412yes—
114C22×C4×C31C2×D412yes—
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D412yes—
117C22×C4×C31C2411yescharacteristic
118M16×C31—44no—
119C2×C8×C31—24no—
120C4⋊C4×C31—24no—
121C4⋊C4×C31—24no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—44no—
127C4⋊C4×C31C2×D414yes—
128C22⋊C4×C31—28no—
129C22⋊C4×C31—24no—
130C22⋊C4×C31—24no—
131C22⋊C4×C31—42no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C42×C31—42no—
135#18147C3111yesmaximal, characteristic, direct product factor, Sylow
136C23×C31—22no—
137C23×C31—22no—
138C23×C31—24no—
139C23×C31—41no—
140C23×C31—44no—
141C23×C31—44no—
142C23×C31—44no—
143C23×C31#12011yescharacteristic
144C23×C31#12011yescharacteristic
145C23×C31C22×D411yescharacteristic
146Q8×C31—48no—
147D4×C31—44no—
148D4×C31—48no—
149D4×C31—48no—
150D4×C31—48no—
151D4×C31—82no—
152D4×C31—84no—
153C2×C4×C31—24no—
154C2×C4×C31—24no—
155C2×C4×C31—22no—
156C2×C4×C31—22no—
157C2×C4×C31—24no—
158C2×C4×C31—22no—
159C2×C4×C31—24no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—416no—
164C2×C4×C31—44no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—81no—
175C2×C4×C31#12012yes—
176C2×C4×C31AGL(1,ℤ/8)14yes—
177C2×C4×C31C22×D412yes—
178C8×C31—44no—
179C2×#537C2×C3111yescharacteristic
180#3094C2×C3112yescomplemented
181#2484C2×C3111yescharacteristic, complemented
182#1616C2×C3114yescomplemented
183#1610C2×C3114yescomplemented
184#1510C2×C3112yescomplemented
185#1463C2×C3111yescharacteristic, complemented
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—44no—
194C22×C31—44no—
195C22×C31—44no—
196C22×C31—44no—
197C22×C31—88no—
198C22×C31#45812yes—
199C22×C31C2×#12011yescharacteristic
200C22×C31C2×AGL(1,ℤ/8)12yes—
201C4×C31—24no—
202C4×C31—22no—
203C4×C31—22no—
204C4×C31—44no—
205C4×C31—42no—
206C4×C31—82no—
207C4×C31—81no—
208C4×C31—81no—
209C22×#41—21no—
210#537—24no—
211#537—22no—
212#537C22×C3114yes—
213C2×C22⋊Q8—22no—
214C2×#121—22no—
215C2×#121—21no—
216C2×#121C22×C3112yes—
217C2×#117C22×C3111yescharacteristic
218C22×C4⋊C4C22×C3111yescharacteristic
219#482—24no—
220#481C22×C3114yescomplemented
221#480C22×C3114yescomplemented
222#428—22no—
223#418—22no—
224#407C22×C3112yes—
225#400C22×C3112yescomplemented
226#398C22×C3112yescomplemented
227#387—22no—
228#385—21no—
229#381—21no—
230#381C22×C3111yescharacteristic
231#327C22×C3118yescomplemented
232#324C22×C3114yescomplemented
233C2×C31—22no—
234C2×C31—22no—
235C2×C31—41no—
236C2×C31—84no—
237C2×C31#300311yescharacteristic
238C2×C31#302011yescharacteristic
239C2×C31C2×#45811yescharacteristic
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—48no—
243C22×Q8—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4D4×C3112yes—
248C23×C4—22no—
249C23×C4D4×C3111yescharacteristic
250C23×C4D4×C3111yescharacteristic
251C23×C4C23×C3111yescharacteristic
252C2×M16—22no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255#121—24no—
256#121—28no—
257#121—28no—
258#121—24no—
259#121—22no—
260#121—22no—
261#121D4×C3114yes—
262#121C23×C3114yes—
263#117—22no—
264#117—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4D4×C3112yes—
270C2×C4⋊C4D4×C3112yes—
271C2×C4⋊C4C23×C3111yescharacteristic
272C2×C4⋊C4C23×C3111yescharacteristic
273C2×C4⋊C4C23×C3112yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279#107—24no—
280#106—24no—
281#102—24no—
282#102—24no—
283C22⋊C8—22no—
284C22⋊C8C23×C3114yes—
285#95—22no—
286#95—22no—
287#95—21no—
288#95—22no—
289#95C23×C3112yes—
290C31#1814711yescharacteristic, direct product factor, Sylow
291C24—22no—
292C24C2×D4×C3111yescharacteristic
293#41—416no—
294#41—48no—
295#41—48no—
296C2×Q8—24no—
297C2×Q8—44no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—82no—
307C2×D4C2×D4×C3114yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4C2×D4×C3112yes—
322C22×C4C2×D4×C3112yes—
323C22×C4C2×D4×C3112yes—
324C22×C4C2×D4×C3111yescharacteristic
325C22×C4C2×D4×C3112yes—
326C22×C4C24×C3111yescharacteristic
327M16—44no—
328C2×C8—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4C2×D4×C3114yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—42no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—41no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23C31×#12011yescharacteristic
352C23C31×#12011yescharacteristic
353C23C22×D4×C3111yescharacteristic
354Q8—48no—
355D4—44no—
356D4—48no—
357D4—48no—
358D4—48no—
359D4—82no—
360D4—84no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—416no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4C31×#12012yes—
384C2×C4C31×AGL(1,ℤ/8)14yes—
385C2×C4C22×D4×C3112yes—
386C8—44no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—88no—
399C22C31×#45812yes—
400C22C2×C31×#12011yescharacteristic
401C22C2×C31×AGL(1,ℤ/8)12yes—
402C4—24no—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—42no—
407C4—82no—
408C4—81no—
409C4—81no—
410C2—22no—
411C2—22no—
412C2—41no—
413C2—84no—
414C2C31×#300311yescharacteristic
415C2C31×#302011yescharacteristic
416C2C2×C31×#45811yescharacteristic
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×#3094C2semidirect
3. C31×#2484C2semidirect
4. C31×#1616C2semidirect
5. C31×#1610C2semidirect
6. C31×#1510C2semidirect
7. C31×#1463C2semidirect
19. C31×#481C22semidirect
20. C31×#480C22semidirect
24. C31×#400C22semidirect
25. C31×#398C22semidirect
30. C31×#327C22semidirect
31. C31×#324C22semidirect
135. #18147C31direct (N × H)
180. #3094C2×C31semidirect
181. #2484C2×C31semidirect
182. #1616C2×C31semidirect
183. #1610C2×C31semidirect
184. #1510C2×C31semidirect
185. #1463C2×C31semidirect
220. #481C22×C31semidirect
221. #480C22×C31semidirect
225. #400C22×C31semidirect
226. #398C22×C31semidirect
231. #327C22×C31semidirect
232. #324C22×C31semidirect
290. C31#18147direct (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×#30942
C31×#24842
C31×#16162
C31×#16102
C31×#15102
C31×#14632
#1814731