Finite Group Databasebeta

← C31×#10241 #21467000

Subgroups — C31×#10241 #21467000

Order 7936.

590 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
1C31×#3227C211yesmaximal, characteristic, complemented
2C31×#3036C211yesmaximal, characteristic
3C31×#2563C211yesmaximal, characteristic, complemented
4C31×#1549C211yesmaximal, characteristic, complemented
5C31×#1502C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic, complemented
7C31×#1303C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#498—21no—
13C31×#486—21no—
14C31×#481—22no—
15C31×#476—22no—
16C31×#463—22no—
17C31×#461—21no—
18C31×#459—21no—
19C31×#431C2211yescharacteristic
20C31×#431C2211yescharacteristic
21C31×#423C2211yescharacteristic
22C31×#417—21no—
23C31×#393—22no—
24C31×#390—21no—
25C31×#382—21no—
26C31×#380—22no—
27C4×C4⋊C4×C31C2211yescharacteristic
28C31×#332—22no—
29C31×#327—22no—
30C31×D4⋊C8—22no—
31C2×#41×C31—21no—
32C22×Q8×C31—21no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C31×AGL(1,ℤ/8)—42no—
36C31×AGL(1,ℤ/8)—42no—
37C2×SD16×C31—22no—
38C2×D8×C31—22no—
39C2×M16×C31—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8D411yescharacteristic
45C31×#127D411yescharacteristic
46C31×#124—21no—
47C31×#121—21no—
48C31×#121—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D412yes—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—24no—
61C2×C42×C31—21no—
62C2×C42×C31C2311yescharacteristic
63C31×#106—24no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8D412yes—
67C31×C4⋊C8D412yes—
68C31×#103—22no—
69C31×#102—22no—
70C31×#102—24no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C4×C8×C31—21no—
74C4×C8×C31—21no—
75C31×#95—24no—
76C31×#95D412yes—
77C24×C31—21no—
78#41×C31—41no—
79#41×C31—41no—
80#41×C31—41no—
81#41×C31—41no—
82C2×Q8×C31—21no—
83C2×Q8×C31—41no—
84C2×Q8×C31—41no—
85C2×Q8×C31—82no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—41no—
95C2×D4×C31SD1612yes—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—44no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106SD16×C31—44no—
107D8×C31—44no—
108M16×C31—41no—
109M16×C31—41no—
110C2×C8×C31—22no—
111C2×C8×C31—22no—
112C2×C8×C31—41no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—42no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—84no—
127C42×C31—41no—
128C42×C31—41no—
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132C42×C31#4111yescharacteristic
133#10241C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—22no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31#12011yescharacteristic
141Q8×C31—41no—
142Q8×C31—41no—
143Q8×C31—82no—
144Q8×C31—82no—
145D4×C31—22no—
146D4×C31—22no—
147D4×C31—41no—
148D4×C31—41no—
149D4×C31—41no—
150D4×C31—42no—
151D4×C31—41no—
152D4×C31—81no—
153D4×C31—81no—
154D4×C31—81no—
155C2×C4×C31—22no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—44no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—82no—
166C2×C4×C31—82no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12711yescharacteristic
175C2×C4×C31C2×SD1611yescharacteristic
176C2×C4×C31#13511yescharacteristic
177C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C31#13711yescharacteristic
179C8×C31—42no—
180C8×C31—42no—
181C8×C31—81no—
182#3227C2×C3111yescharacteristic, complemented
183#3036C2×C3111yescharacteristic
184#2563C2×C3111yescharacteristic, complemented
185#1549C2×C3111yescharacteristic, complemented
186#1502C2×C3111yescharacteristic, complemented
187#1478C2×C3111yescharacteristic, complemented
188#1303C2×C3111yescharacteristic, complemented
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—44no—
192C22×C31—42no—
193C22×C31—82no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31#39211yescharacteristic
200C22×C31#44811yescharacteristic
201C22×C31#44911yescharacteristic
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—82no—
206C4×C31—81no—
207C4×C31—81no—
208C4×C31—82no—
209C4×C31#45011yescharacteristic
210C4×C31#45211yescharacteristic
211C4×C31#45412yes—
212C4×C31#46511yescharacteristic
213C4×C31#46811yescharacteristic
214C2×AGL(1,ℤ/8)—21no—
215#532C22×C3111yescharacteristic
216C2×C4⋊Q8C22×C3111yescharacteristic
217C2×C4×D4C22×C3111yescharacteristic
218#498—21no—
219#486—21no—
220#481—22no—
221#476—22no—
222#463—22no—
223#461—21no—
224#459—21no—
225#431C22×C3111yescharacteristic
226#431C22×C3111yescharacteristic
227#423C22×C3111yescharacteristic
228#417—21no—
229#393—22no—
230#390—21no—
231#382—21no—
232#380—22no—
233C4×C4⋊C4C22×C3111yescharacteristic
234#332—22no—
235#327—22no—
236D4⋊C8—22no—
237C2×C31—21no—
238C2×C31—21no—
239C2×C31—82no—
240C2×C31—161no—
241C2×C31#200011yescharacteristic
242C2×C31#200311yescharacteristic
243C2×C31#200811yescharacteristic
244C2×#41—21no—
245C22×Q8—21no—
246C22×D4D4×C3111yescharacteristic
247C23×C4—21no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8D4×C3111yescharacteristic
258#127D4×C3111yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4D4×C3111yescharacteristic
271C2×C4⋊C4D4×C3112yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—24no—
274C2×C42—21no—
275C2×C42C23×C3111yescharacteristic
276#106—24no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8D4×C3112yes—
280C4⋊C8D4×C3112yes—
281#103—22no—
282#102—22no—
283#102—24no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288#95—24no—
289#95D4×C3112yes—
290C31#1024111yescharacteristic, direct product factor, Sylow
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—82no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4SD16×C3112yes—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320SD16—44no—
321D8—44no—
322M16—41no—
323M16—41no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—84no—
341C42—41no—
342C42—41no—
343C42C2×D4×C3111yescharacteristic
344C42C2×D4×C3111yescharacteristic
345C42C2×D4×C3111yescharacteristic
346C42#41×C3111yescharacteristic
347C23—22no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23C31×#12011yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—82no—
357Q8—82no—
358D4—22no—
359D4—22no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—41no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4C31×#12011yescharacteristic
383C2×C4C31×#12011yescharacteristic
384C2×C4C31×#12111yescharacteristic
385C2×C4C31×#12111yescharacteristic
386C2×C4C31×#12111yescharacteristic
387C2×C4C31×#12711yescharacteristic
388C2×C4C2×SD16×C3111yescharacteristic
389C2×C4C31×#13511yescharacteristic
390C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
391C2×C4C31×#13711yescharacteristic
392C8—42no—
393C8—42no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22—42no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C31×#39211yescharacteristic
406C22C31×#44811yescharacteristic
407C22C31×#44911yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—82no—
415C4C31×#45011yescharacteristic
416C4C31×#45211yescharacteristic
417C4C31×#45412yes—
418C4C31×#46511yescharacteristic
419C4C31×#46811yescharacteristic
420C2—21no—
421C2—21no—
422C2—82no—
423C2—161no—
424C2C31×#200011yescharacteristic
425C2C31×#200311yescharacteristic
426C2C31×#200811yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3036C2
LCSG ▷ C2×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. C31×#3227C2semidirect
3. C31×#2563C2semidirect
4. C31×#1549C2semidirect
5. C31×#1502C2semidirect
6. C31×#1478C2semidirect
7. C31×#1303C2semidirect
133. #10241C31direct (N × H)
182. #3227C2×C31semidirect
184. #2563C2×C31semidirect
185. #1549C2×C31semidirect
186. #1502C2×C31semidirect
187. #1478C2×C31semidirect
188. #1303C2×C31semidirect
290. C31#10241direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32272
C31×#30362
C31×#25632
C31×#15492
C31×#15022
C31×#14782
C31×#13032
#1024131