Finite Group Databasebeta

← C31×#10246 #21467005

Subgroups — C31×#10246 #21467005

Order 7936.

740 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3208C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C31×#2541C211yesmaximal, characteristic
4C31×#1545C211yesmaximal, characteristic, complemented
5C31×#1503C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic, complemented
7C31×#1303C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#127—21no—
10C2×C31×#127C2211yescharacteristic
11C2×C31×#121—22no—
12C2×C4×D4×C31C2211yescharacteristic
13C31×#492—21no—
14C31×#486—21no—
15C31×#478—22no—
16C31×#476—21no—
17C31×#474—21no—
18C31×#460—22no—
19C31×#450—22no—
20C31×#431C2211yescharacteristic
21C31×#431C2211yescharacteristic
22C31×#428—21no—
23C31×#423C2211yescharacteristic
24C31×#418—21no—
25C31×#392—22no—
26C31×#390—21no—
27C4×C4⋊C4×C31C2211yescharacteristic
28C31×#331—22no—
29C31×#328—22no—
30C31×D4⋊C8—22no—
31C22×D4×C31—24no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31D411yescharacteristic
35C22×D4×C31D412yes—
36C23×C4×C31—21no—
37C2×SD16×C31—24no—
38C2×M16×C31—21no—
39C31×C4⋊Q8D411yescharacteristic, complemented
40C31×C4⋊Q8D411yescharacteristic, complemented
41C31×#127—21no—
42C31×#127—21no—
43C31×#127—21no—
44C31×#127—21no—
45C31×#127—42no—
46C31×#127—42no—
47C31×#125—21no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C31×#121—48no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C31×#117—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—21no—
62C2×C42×C31—21no—
63C2×C42×C31C2311yescharacteristic
64C31×#107—22no—
65C31×#106—22no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8D412yescomplemented
69C31×C4⋊C8D412yescomplemented
70C31×#102—22no—
71C31×#102—22no—
72C31×#102—24no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C4×C8×C31—21no—
76C4×C8×C31—21no—
77C31×#95D412yes—
78C24×C31—21no—
79C24×C31—22no—
80C2×Q8×C31—21no—
81C2×Q8×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—22no—
84C2×D4×C31—21no—
85C2×D4×C31—22no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—44no—
90C2×D4×C31—48no—
91C2×D4×C31—48no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C2×D4×C31—41no—
97C2×D4×C31—41no—
98C2×D4×C31—41no—
99C2×D4×C31—41no—
100C2×D4×C31—82no—
101C2×D4×C31—84no—
102C2×D4×C31SD1612yescomplemented
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—44no—
106C22×C4×C31—41no—
107C22×C4×C31—41no—
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111SD16×C31—84no—
112M16×C31—41no—
113M16×C31—41no—
114C2×C8×C31—22no—
115C2×C8×C31—22no—
116C2×C8×C31—41no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C22⋊C4×C31—48no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C42×C31—41no—
131C42×C31—41no—
132C42×C31—41no—
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31C2×D411yescharacteristic
136C42×C31#4111yescharacteristic
137#10246C3111yesmaximal, characteristic, direct product factor, Sylow
138C23×C31—22no—
139C23×C31—42no—
140C23×C31—42no—
141C23×C31—44no—
142C23×C31—44no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—42no—
146C23×C31—42no—
147C23×C31—42no—
148C23×C31#12011yescharacteristic
149Q8×C31—42no—
150Q8×C31—81no—
151D4×C31—22no—
152D4×C31—22no—
153D4×C31—41no—
154D4×C31—44no—
155D4×C31—41no—
156D4×C31—82no—
157D4×C31—82no—
158D4×C31—82no—
159D4×C31—816no—
160D4×C31—81no—
161D4×C31—81no—
162D4×C31—81no—
163D4×C31—81no—
164C2×C4×C31—22no—
165C2×C4×C31—24no—
166C2×C4×C31—41no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12011yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C2×C4×C31#12111yescharacteristic
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12711yescharacteristic
181C2×C4×C31C2×SD1611yescharacteristic
182C2×C4×C31#13511yescharacteristic
183C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C31#13711yescharacteristic
185C8×C31—42no—
186C8×C31—42no—
187C8×C31—81no—
188#3208C2×C3111yescharacteristic, complemented
189#3077C2×C3111yescharacteristic, complemented
190#2541C2×C3111yescharacteristic
191#1545C2×C3111yescharacteristic, complemented
192#1503C2×C3111yescharacteristic, complemented
193#1478C2×C3111yescharacteristic, complemented
194#1303C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—44no—
198C22×C31—42no—
199C22×C31—82no—
200C22×C31—84no—
201C22×C31—84no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—82no—
205C22×C31—82no—
206C22×C31—82no—
207C22×C31#39211yescharacteristic
208C22×C31#45011yescharacteristic
209C22×C31#45211yescharacteristic
210C4×C31—21no—
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—44no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31—81no—
217C4×C31#44811yescharacteristic
218C4×C31#44911yescharacteristic
219C4×C31#45312yes—
220C4×C31#46511yescharacteristic
221C4×C31#46811yescharacteristic
222#533C22×C3111yescharacteristic
223C2×#127—21no—
224C2×#127C22×C3111yescharacteristic
225C2×#121—22no—
226C2×C4×D4C22×C3111yescharacteristic
227#492—21no—
228#486—21no—
229#478—22no—
230#476—21no—
231#474—21no—
232#460—22no—
233#450—22no—
234#431C22×C3111yescharacteristic
235#431C22×C3111yescharacteristic
236#428—21no—
237#423C22×C3111yescharacteristic
238#418—21no—
239#392—22no—
240#390—21no—
241C4×C4⋊C4C22×C3111yescharacteristic
242#331—22no—
243#328—22no—
244D4⋊C8—22no—
245C2×C31—21no—
246C2×C31—21no—
247C2×C31—82no—
248C2×C31—162no—
249C2×C31#200011yescharacteristic
250C2×C31#200211yescharacteristic
251C2×C31#200811yescharacteristic
252C22×D4—24no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4D4×C3111yescharacteristic
256C22×D4D4×C3112yes—
257C23×C4—21no—
258C2×SD16—24no—
259C2×M16—21no—
260C4⋊Q8D4×C3111yescharacteristic, complemented
261C4⋊Q8D4×C3111yescharacteristic, complemented
262#127—21no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127—42no—
267#127—42no—
268#125—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—48no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278#117—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4—24no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42C23×C3111yescharacteristic
285#107—22no—
286#106—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8D4×C3112yescomplemented
290C4⋊C8D4×C3112yescomplemented
291#102—22no—
292#102—22no—
293#102—24no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298#95D4×C3112yes—
299C31#1024611yescharacteristic, direct product factor, Sylow
300C24—21no—
301C24—22no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—44no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—82no—
323C2×D4—84no—
324C2×D4SD16×C3112yescomplemented
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—44no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333SD16—84no—
334M16—41no—
335M16—41no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—41no—
339C4⋊C4—24no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—48no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42C2×D4×C3111yescharacteristic
356C42C2×D4×C3111yescharacteristic
357C42C2×D4×C3111yescharacteristic
358C42#41×C3111yescharacteristic
359C23—22no—
360C23—42no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23C31×#12011yescharacteristic
370Q8—42no—
371Q8—81no—
372D4—22no—
373D4—22no—
374D4—41no—
375D4—44no—
376D4—41no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—816no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4C31×#12011yescharacteristic
397C2×C4C31×#12011yescharacteristic
398C2×C4C31×#12111yescharacteristic
399C2×C4C31×#12111yescharacteristic
400C2×C4C31×#12111yescharacteristic
401C2×C4C31×#12711yescharacteristic
402C2×C4C2×SD16×C3111yescharacteristic
403C2×C4C31×#13511yescharacteristic
404C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C31×#13711yescharacteristic
406C8—42no—
407C8—42no—
408C8—81no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—42no—
413C22—82no—
414C22—84no—
415C22—84no—
416C22—81no—
417C22—81no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22C31×#39211yescharacteristic
422C22C31×#45011yescharacteristic
423C22C31×#45211yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—44no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4C31×#44811yescharacteristic
432C4C31×#44911yescharacteristic
433C4C31×#45312yes—
434C4C31×#46511yescharacteristic
435C4C31×#46811yescharacteristic
436C2—21no—
437C2—21no—
438C2—82no—
439C2—162no—
440C2C31×#200011yescharacteristic
441C2C31×#200211yescharacteristic
442C2C31×#200811yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2541C2
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×#3208C2semidirect
2. C31×#3077C2semidirect
4. C31×#1545C2semidirect
5. C31×#1503C2semidirect
6. C31×#1478C2semidirect
7. C31×#1303C2semidirect
39. C31×C4⋊Q8D4semidirect
40. C31×C4⋊Q8D4semidirect
68. C31×C4⋊C8D4semidirect
69. C31×C4⋊C8D4semidirect
102. C2×D4×C31SD16semidirect
137. #10246C31direct (N × H)
188. #3208C2×C31semidirect
189. #3077C2×C31semidirect
191. #1545C2×C31semidirect
192. #1503C2×C31semidirect
193. #1478C2×C31semidirect
194. #1303C2×C31semidirect
260. C4⋊Q8D4×C31semidirect
261. C4⋊Q8D4×C31semidirect
289. C4⋊C8D4×C31semidirect
290. C4⋊C8D4×C31semidirect
299. C31#10246direct (N × H)
324. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32082
C31×#30772
C31×#25412
C31×#15452
C31×#15032
C31×#14782
C31×#13032
#1024631