Finite Group Databasebeta

← C31×#8537 #21465296

Subgroups — C31×#8537 #21465296

Order 7936.

794 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic
2C31×#1901C212yesmaximal, complemented
3C31×#1870C211yesmaximal, characteristic
4C31×#1472C212yesmaximal, complemented
5C31×#1465C211yesmaximal, characteristic
6C22×M16×C31C2211yescharacteristic
7C31×#554—24no—
8C31×#554—22no—
9D4×Q8×C31—22no—
10C31×#544—21no—
11C31×#543—21no—
12C31×#543C412yes—
13C31×#533—21no—
14C2×C31×C22⋊Q8—21no—
15C2×C31×C22⋊Q8C2212yes—
16C31×#518C412yes—
17C2×C4×D4×C31C2211yescharacteristic
18C2×C31×#101—22no—
19C31×#407C2212yes—
20C2×C31×C22⋊C8—21no—
21C2×C31×C22⋊C8C2211yescharacteristic
22C31×#343—22no—
23C31×#336—21no—
24C31×#324—21no—
25C31×#324—24no—
26C31×#323—21no—
27C22×Q8×C31—21no—
28C22×Q8×C31D412yes—
29C22×D4×C31C2×C411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C23×C4×C31C2311yescharacteristic
32C2×M16×C31—22no—
33C2×M16×C31—22no—
34C2×M16×C31—42no—
35C22×C8×C31—21no—
36C22×C8×C31—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—24no—
40C31×#125—22no—
41C31×#125—24no—
42C31×C22⋊Q8—22no—
43C31×C22⋊Q8—22no—
44C31×C22⋊Q8—24no—
45C31×C22⋊Q8—24no—
46C31×C22⋊Q8—22no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—24no—
49C31×C22⋊Q8—24no—
50C31×C22⋊Q8—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—22no—
53C31×C22⋊Q8—22no—
54C4×Q8×C31—22no—
55C4×Q8×C31—22no—
56C4×D4×C31—24no—
57C4×D4×C31—24no—
58C4×D4×C31—22no—
59C4×D4×C31—22no—
60C31×#117—22no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31D411yescharacteristic
64C2×C4⋊C4×C31D414yes—
65C2×C4⋊C4×C31D412yes—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31C2×C411yescharacteristic
68C2×C22⋊C4×C31C2×C414yes—
69C2×C22⋊C4×C31D411yescharacteristic
70C2×C42×C31D411yescharacteristic
71C31×#101—44no—
72C31×C22⋊C8—22no—
73C31×C22⋊C8—22no—
74C31×C22⋊C8—22no—
75C31×C22⋊C8—42no—
76C24×C31C22⋊C411yescharacteristic
77C24×C31C22×C411yescharacteristic
78C2×Q8×C31—22no—
79C2×Q8×C31—22no—
80C2×Q8×C31—22no—
81C2×Q8×C31—22no—
82C2×Q8×C31—42no—
83C2×Q8×C31—42no—
84C2×Q8×C31—81no—
85C2×Q8×C31—81no—
86C2×D4×C31—22no—
87C2×D4×C31—22no—
88C2×D4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—41no—
94C22×C4×C31—41no—
95C22×C4×C31—41no—
96C22×C4×C31—41no—
97C22×C4×C31C22⋊C411yescharacteristic
98C22×C4×C31C22⋊C411yescharacteristic
99C22×C4×C31C22⋊C414yes—
100C22×C4×C31C22⋊C414yes—
101C22×C4×C31C22⋊C411yescharacteristic
102C22×C4×C31C2×D411yescharacteristic
103C22×C4×C31C2×D412yes—
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D412yes—
106M16×C31—24no—
107M16×C31—42no—
108M16×C31—42no—
109M16×C31—42no—
110C2×C8×C31—22no—
111C2×C8×C31—41no—
112C2×C8×C31—41no—
113C2×C8×C31—41no—
114C2×C8×C31—41no—
115C2×C8×C31—41no—
116C2×C8×C31—42no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—24no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—24no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—24no—
128C22⋊C4×C31—24no—
129C22⋊C4×C31—24no—
130C22⋊C4×C31—22no—
131C22⋊C4×C31—24no—
132C22⋊C4×C31—22no—
133C42×C31—24no—
134C42×C31—21no—
135C42×C31—21no—
136#8537C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—21no—
138C23×C31—21no—
139C23×C31—21no—
140C23×C31—21no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31#10112yescomplemented
146C23×C31C2×C22⋊C412yes—
147C23×C31C2×C22⋊C411yescharacteristic
148Q8×C31—42no—
149Q8×C31—42no—
150Q8×C31—82no—
151Q8×C31—82no—
152Q8×C31—82no—
153D4×C31—28no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—22no—
158C2×C4×C31—21no—
159C2×C4×C31—22no—
160C2×C4×C31—21no—
161C2×C4×C31—24no—
162C2×C4×C31—24no—
163C2×C4×C31—21no—
164C2×C4×C31—42no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—44no—
170C2×C4×C31—41no—
171C2×C4×C31—44no—
172C2×C4×C31—81no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31#10112yescomplemented
176C2×C4×C31#12011yescharacteristic
177C2×C4×C31#12012yes—
178C2×C4×C31#12011yescharacteristic
179C8×C31—42no—
180C8×C31—81no—
181C8×C31—82no—
182#3501C2×C3111yescharacteristic
183#1901C2×C3112yescomplemented
184#1870C2×C3111yescharacteristic
185#1472C2×C3112yescomplemented
186#1465C2×C3111yescharacteristic
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—22no—
190C22×C31—22no—
191C22×C31—22no—
192C22×C31—22no—
193C22×C31—42no—
194C22×C31—41no—
195C22×C31—41no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31#37911yescharacteristic
200C22×C31C2×#10111yescharacteristic
201C22×C31C2×#10111yescharacteristic
202C4×C31—22no—
203C4×C31—21no—
204C4×C31—22no—
205C4×C31—41no—
206C4×C31—44no—
207C4×C31—44no—
208C4×C31—41no—
209C4×C31B(4,𝔽2)11yescharacteristic
210C4×C31#45811yescharacteristic
211C22×M16C22×C3111yescharacteristic
212#554—24no—
213#554—22no—
214D4×Q8—22no—
215#544—21no—
216#543—21no—
217#543C4×C3112yes—
218#533—21no—
219C2×C22⋊Q8—21no—
220C2×C22⋊Q8C22×C3112yes—
221#518C4×C3112yes—
222C2×C4×D4C22×C3111yescharacteristic
223C2×#101—22no—
224#407C22×C3112yes—
225C2×C22⋊C8—21no—
226C2×C22⋊C8C22×C3111yescharacteristic
227#343—22no—
228#336—21no—
229#324—21no—
230#324—24no—
231#323—21no—
232C2×C31—21no—
233C2×C31—21no—
234C2×C31—42no—
235C2×C31—42no—
236C2×C31#178411yescharacteristic
237C2×C31#179011yescharacteristic
238C2×C31#179111yescharacteristic
239C22×Q8—21no—
240C22×Q8D4×C3112yes—
241C22×D4C2×C4×C3111yescharacteristic
242C23×C4D4×C3111yescharacteristic
243C23×C4C23×C3111yescharacteristic
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—42no—
247C22×C8—21no—
248C22×C8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—24no—
252#125—22no—
253#125—24no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—22no—
271C4×D4—22no—
272#117—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4D4×C3111yescharacteristic
276C2×C4⋊C4D4×C3114yes—
277C2×C4⋊C4D4×C3112yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C2×C4×C3111yescharacteristic
280C2×C22⋊C4C2×C4×C3114yes—
281C2×C22⋊C4D4×C3111yescharacteristic
282C2×C42D4×C3111yescharacteristic
283#101—44no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—42no—
288C31#853711yescharacteristic, direct product factor, Sylow
289C24C22⋊C4×C3111yescharacteristic
290C24C22×C4×C3111yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C22⋊C4×C3111yescharacteristic
311C22×C4C22⋊C4×C3111yescharacteristic
312C22×C4C22⋊C4×C3114yes—
313C22×C4C22⋊C4×C3114yes—
314C22×C4C22⋊C4×C3111yescharacteristic
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3112yes—
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3112yes—
319M16—24no—
320M16—42no—
321M16—42no—
322M16—42no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—24no—
345C22⋊C4—22no—
346C42—24no—
347C42—21no—
348C42—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C31×#10112yescomplemented
358C23C2×C22⋊C4×C3112yes—
359C23C2×C22⋊C4×C3111yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365D4—28no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4C31×#10112yescomplemented
388C2×C4C31×#12011yescharacteristic
389C2×C4C31×#12012yes—
390C2×C4C31×#12011yescharacteristic
391C8—42no—
392C8—81no—
393C8—82no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C31×#37911yescharacteristic
407C22C2×C31×#10111yescharacteristic
408C22C2×C31×#10111yescharacteristic
409C4—22no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—44no—
414C4—44no—
415C4—41no—
416C4C31×B(4,𝔽2)11yescharacteristic
417C4C31×#45811yescharacteristic
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2C31×#178411yescharacteristic
423C2C31×#179011yescharacteristic
424C2C31×#179111yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C24 ▷ 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×#1901C2semidirect
4. C31×#1472C2semidirect
136. #8537C31direct (N × H)
145. C23×C31#101semidirect
175. C2×C4×C31#101semidirect
183. #1901C2×C31semidirect
185. #1472C2×C31semidirect
288. C31#8537direct (N × H)
357. C23C31×#101semidirect
387. C2×C4C31×#101semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#19012
C31×#18702
C31×#14722
C31×#14652
#853731