Finite Group Databasebeta

← C31×#10000 #21466759

Subgroups — C31×#10000 #21466759

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3101C211yesmaximal, characteristic, complemented
2C31×#3077C211yesmaximal, characteristic, complemented
3C31×#2535C211yesmaximal, characteristic
4C31×#1518C211yesmaximal, characteristic, complemented
5C31×#1486C211yesmaximal, characteristic, complemented
6C31×#1478C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#121—21no—
10C2×C4×D4×C31C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#483—21no—
13C31×#478—22no—
14C31×#476—21no—
15C31×#476—21no—
16C31×#474—21no—
17C31×#461—21no—
18C31×#450—21no—
19C4×SD16×C31—22no—
20C31×#431C2211yescharacteristic
21C31×#431C2211yescharacteristic
22C31×#428—21no—
23C31×#423C2211yescharacteristic
24C31×#418—21no—
25C31×#397—21no—
26C31×#392—21no—
27C31×#386—22no—
28C31×#382C2211yescharacteristic
29C31×#380—21no—
30C31×#328—22no—
31C31×D4⋊C8—22no—
32C31×D4⋊C8—22no—
33C22×D4×C31—22no—
34C22×D4×C31D411yescharacteristic
35C22×D4×C31D411yescharacteristic
36C23×C4×C31—21no—
37C23×C4×C31—21no—
38C2×SD16×C31—22no—
39C2×M16×C31—21no—
40C31×C4⋊Q8D411yescharacteristic, complemented
41C31×C4⋊Q8D411yescharacteristic, complemented
42C31×#125—21no—
43C31×C22⋊Q8—21no—
44C31×C22⋊Q8—21no—
45C31×#121—42no—
46C31×#121—42no—
47C4×Q8×C31—21no—
48C4×Q8×C31—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—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C31×#117—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C4⋊C4×C31D411yescharacteristic
61C2×C22⋊C4×C31—22no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—22no—
65C2×C42×C31C2311yescharacteristic
66C31×#107—22no—
67C31×#106—22no—
68C31×#106—22no—
69C31×C4⋊C8—21no—
70C31×C4⋊C8—21no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8D412yescomplemented
73C31×#103—22no—
74C31×#102—22no—
75C31×#102—22no—
76C31×#102—22no—
77C31×C22⋊C8—21no—
78C31×C22⋊C8—21no—
79C4×C8×C31—21no—
80C4×C8×C31D412yescomplemented
81C31×#95—22no—
82C31×#95—21no—
83C31×#95D412yes—
84C24×C31—21no—
85C24×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×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—21no—
95C2×D4×C31—42no—
96C2×D4×C31—44no—
97C2×D4×C31—44no—
98C2×D4×C31—82no—
99C2×D4×C31SD1612yescomplemented
100C22×C4×C31—22no—
101C22×C4×C31—21no—
102C22×C4×C31—21no—
103C22×C4×C31—41no—
104C22×C4×C31—41no—
105C22×C4×C31—41no—
106C22×C4×C31—41no—
107C22×C4×C31—41no—
108C22×C4×C31—41no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31—82no—
112C22×C4×C31C2×D411yescharacteristic
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115SD16×C31—44no—
116M16×C31—41no—
117M16×C31—41no—
118C2×C8×C31—22no—
119C2×C8×C31—22no—
120C2×C8×C31—41no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31—41no—
131C4⋊C4×C31—41no—
132C4⋊C4×C31—41no—
133C22⋊C4×C31—44no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138C22⋊C4×C31—41no—
139C22⋊C4×C31—82no—
140C42×C31—41no—
141C42×C31C2×D411yescharacteristic
142C42×C31C2×D411yescharacteristic
143C42×C31C2×D411yescharacteristic
144C42×C31#4111yescharacteristic
145#10000C3111yesmaximal, characteristic, direct product factor, Sylow
146C23×C31—22no—
147C23×C31—22no—
148C23×C31—41no—
149C23×C31—42no—
150C23×C31—42no—
151C23×C31—41no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31#12011yescharacteristic
159Q8×C31—42no—
160Q8×C31—81no—
161D4×C31—22no—
162D4×C31—22no—
163D4×C31—22no—
164D4×C31—22no—
165D4×C31—41no—
166D4×C31—41no—
167D4×C31—41no—
168D4×C31—41no—
169D4×C31—88no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—42no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—42no—
181C2×C4×C31—41no—
182C2×C4×C31—82no—
183C2×C4×C31—82no—
184C2×C4×C31—81no—
185C2×C4×C31—81no—
186C2×C4×C31—81no—
187C2×C4×C31—81no—
188C2×C4×C31#12011yescharacteristic
189C2×C4×C31#12011yescharacteristic
190C2×C4×C31#12111yescharacteristic
191C2×C4×C31#12111yescharacteristic
192C2×C4×C31#12111yescharacteristic
193C2×C4×C31#12711yescharacteristic
194C2×C4×C31C2×SD1611yescharacteristic
195C2×C4×C31#13511yescharacteristic
196C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C31#13711yescharacteristic
198C8×C31—42no—
199C8×C31—42no—
200C8×C31—81no—
201#3101C2×C3111yescharacteristic, complemented
202#3077C2×C3111yescharacteristic, complemented
203#2535C2×C3111yescharacteristic
204#1518C2×C3111yescharacteristic, complemented
205#1486C2×C3111yescharacteristic, complemented
206#1478C2×C3111yescharacteristic, complemented
207#1286C2×C3111yescharacteristic, complemented
208C22×C31—21no—
209C22×C31—21no—
210C22×C31—42no—
211C22×C31—44no—
212C22×C31—42no—
213C22×C31—42no—
214C22×C31—42no—
215C22×C31—81no—
216C22×C31—82no—
217C22×C31—81no—
218C22×C31—81no—
219C22×C31—81no—
220C22×C31—81no—
221C22×C31—81no—
222C22×C31#39211yescharacteristic
223C22×C31#45011yescharacteristic
224C22×C31#45211yescharacteristic
225C4×C31—21no—
226C4×C31—21no—
227C4×C31—21no—
228C4×C31—21no—
229C4×C31—82no—
230C4×C31—81no—
231C4×C31—81no—
232C4×C31—81no—
233C4×C31—81no—
234C4×C31#44811yescharacteristic
235C4×C31#44911yescharacteristic
236C4×C31#46511yescharacteristic
237C4×C31#46811yescharacteristic
238#533C22×C3111yescharacteristic
239C2×#121—21no—
240C2×C4×D4C22×C3111yescharacteristic
241C2×C4×D4C22×C3111yescharacteristic
242#483—21no—
243#478—22no—
244#476—21no—
245#476—21no—
246#474—21no—
247#461—21no—
248#450—21no—
249C4×SD16—22no—
250#431C22×C3111yescharacteristic
251#431C22×C3111yescharacteristic
252#428—21no—
253#423C22×C3111yescharacteristic
254#418—21no—
255#397—21no—
256#392—21no—
257#386—22no—
258#382C22×C3111yescharacteristic
259#380—21no—
260#328—22no—
261D4⋊C8—22no—
262D4⋊C8—22no—
263C2×C31—21no—
264C2×C31—21no—
265C2×C31—82no—
266C2×C31—82no—
267C2×C31#200011yescharacteristic
268C2×C31#200811yescharacteristic
269C2×C31#201011yescharacteristic
270C22×D4—22no—
271C22×D4D4×C3111yescharacteristic
272C22×D4D4×C3111yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C2×SD16—22no—
276C2×M16—21no—
277C4⋊Q8D4×C3111yescharacteristic, complemented
278C4⋊Q8D4×C3111yescharacteristic, complemented
279#125—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—42no—
283#121—42no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294#117—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4D4×C3111yescharacteristic
297C2×C4⋊C4D4×C3111yescharacteristic
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C42C23×C3111yescharacteristic
303#107—22no—
304#106—22no—
305#106—22no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8D4×C3112yescomplemented
310#103—22no—
311#102—22no—
312#102—22no—
313#102—22no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C4×C8—21no—
317C4×C8D4×C3112yescomplemented
318#95—22no—
319#95—21no—
320#95D4×C3112yes—
321C31#1000011yescharacteristic, direct product factor, Sylow
322C24—21no—
323C24—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—82no—
337C2×D4SD16×C3112yescomplemented
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—82no—
350C22×C4C2×D4×C3111yescharacteristic
351C22×C4C2×D4×C3111yescharacteristic
352C22×C4C2×D4×C3111yescharacteristic
353SD16—44no—
354M16—41no—
355M16—41no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C22⋊C4—44no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—82no—
378C42—41no—
379C42C2×D4×C3111yescharacteristic
380C42C2×D4×C3111yescharacteristic
381C42C2×D4×C3111yescharacteristic
382C42#41×C3111yescharacteristic
383C23—22no—
384C23—22no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C31×#12011yescharacteristic
396Q8—42no—
397Q8—81no—
398D4—22no—
399D4—22no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406D4—88no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4C31×#12011yescharacteristic
426C2×C4C31×#12011yescharacteristic
427C2×C4C31×#12111yescharacteristic
428C2×C4C31×#12111yescharacteristic
429C2×C4C31×#12111yescharacteristic
430C2×C4C31×#12711yescharacteristic
431C2×C4C2×SD16×C3111yescharacteristic
432C2×C4C31×#13511yescharacteristic
433C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C31×#13711yescharacteristic
435C8—42no—
436C8—42no—
437C8—81no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—82no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22C31×#39211yescharacteristic
453C22C31×#45011yescharacteristic
454C22C31×#45211yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4C31×#44811yescharacteristic
465C4C31×#44911yescharacteristic
466C4C31×#46511yescharacteristic
467C4C31×#46811yescharacteristic
468C2—21no—
469C2—21no—
470C2—82no—
471C2—82no—
472C2C31×#200011yescharacteristic
473C2C31×#200811yescharacteristic
474C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2535C2
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×#3101C2semidirect
2. C31×#3077C2semidirect
4. C31×#1518C2semidirect
5. C31×#1486C2semidirect
6. C31×#1478C2semidirect
7. C31×#1286C2semidirect
40. C31×C4⋊Q8D4semidirect
41. C31×C4⋊Q8D4semidirect
72. C31×C4⋊C8D4semidirect
80. C4×C8×C31D4semidirect
99. C2×D4×C31SD16semidirect
145. #10000C31direct (N × H)
201. #3101C2×C31semidirect
202. #3077C2×C31semidirect
204. #1518C2×C31semidirect
205. #1486C2×C31semidirect
206. #1478C2×C31semidirect
207. #1286C2×C31semidirect
277. C4⋊Q8D4×C31semidirect
278. C4⋊Q8D4×C31semidirect
309. C4⋊C8D4×C31semidirect
317. C4×C8D4×C31semidirect
321. C31#10000direct (N × H)
337. C2×D4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31012
C31×#30772
C31×#25352
C31×#15182
C31×#14862
C31×#14782
C31×#12862
#1000031