Finite Group Databasebeta

← C31×#9997 #21466756

Subgroups — C31×#9997 #21466756

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3100C211yesmaximal, characteristic
2C31×#3037C211yesmaximal, characteristic, complemented
3C31×#2536C211yesmaximal, characteristic, complemented
4C31×#1517C211yesmaximal, characteristic, complemented
5C31×#1486C211yesmaximal, characteristic, complemented
6C31×#1479C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#532C2211yescharacteristic
10C2×C31×C22⋊Q8—21no—
11C2×C4×Q8×C31C2211yescharacteristic
12C2×C4×D4×C31C2211yescharacteristic
13C31×#480—21no—
14C31×#474—21no—
15C31×#464—22no—
16C31×#462—21no—
17C31×#460—21no—
18C31×#459—21no—
19C31×#447—21no—
20C4×D8×C31—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C31×#417—21no—
25C31×#398—21no—
26C31×#394—21no—
27C31×#388—22no—
28C31×#382C2211yescharacteristic
29C31×#380—21no—
30C31×#327—22no—
31C31×Q8⋊C8—22no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C22×D4×C31D411yescharacteristic
36C23×C4×C31—21no—
37C31×#137—42no—
38C31×#137—42no—
39C2×Q16×C31—22no—
40C2×SD16×C31—22no—
41C2×D8×C31—22no—
42C2×M16×C31—21no—
43C31×C4⋊Q8D411yescharacteristic
44C31×#127D411yescharacteristic
45C31×#124—21no—
46C31×C22⋊Q8—42no—
47C31×C22⋊Q8—42no—
48C31×#121—21no—
49C31×#121—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C4×D4×C31—21no—
59C4×D4×C31—21no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—21no—
63C2×C4⋊C4×C31D411yescharacteristic
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—21no—
68C2×C42×C31—21no—
69C2×C42×C31C2311yescharacteristic
70C31×#107—22no—
71C31×C4⋊C8—21no—
72C31×C4⋊C8—21no—
73C31×C4⋊C8—21no—
74C31×C4⋊C8D412yes—
75C31×#103—22no—
76C31×#102—22no—
77C31×#102—22no—
78C31×#102—22no—
79C31×C22⋊C8—21no—
80C31×C22⋊C8—21no—
81C4×C8×C31—21no—
82C4×C8×C31D412yes—
83C31×#95—22no—
84C31×#95—21no—
85C31×#95D412yes—
86C24×C31—21no—
87#41×C31—41no—
88#41×C31—41no—
89#41×C31—41no—
90#41×C31—41no—
91C2×Q8×C31—21no—
92C2×Q8×C31—21no—
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—21no—
96C2×Q8×C31—21no—
97C2×Q8×C31D812yescomplemented
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31—21no—
102C2×D4×C31—21no—
103C2×D4×C31—42no—
104C2×D4×C31—41no—
105C22×C4×C31—21no—
106C22×C4×C31—22no—
107C22×C4×C31—21no—
108C22×C4×C31—21no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31—41no—
112C22×C4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31—41no—
115C22×C4×C31—41no—
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31C2×D411yescharacteristic
118C22×C4×C31C2×D411yescharacteristic
119Q16×C31—44no—
120SD16×C31—44no—
121D8×C31—44no—
122M16×C31—41no—
123M16×C31—41no—
124C2×C8×C31—22no—
125C2×C8×C31—22no—
126C2×C8×C31—41no—
127C4⋊C4×C31—21no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—44no—
134C4⋊C4×C31—41no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31—41no—
138C22⋊C4×C31—44no—
139C22⋊C4×C31—44no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C42×C31—41no—
145C42×C31—41no—
146C42×C31C2×D411yescharacteristic
147C42×C31C2×D411yescharacteristic
148C42×C31C2×D411yescharacteristic
149C42×C31#4111yescharacteristic
150#9997C3111yesmaximal, characteristic, direct product factor, Sylow
151C23×C31—22no—
152C23×C31—41no—
153C23×C31—41no—
154C23×C31—41no—
155C23×C31—41no—
156C23×C31—41no—
157C23×C31—41no—
158C23×C31—41no—
159C23×C31#12011yescharacteristic
160Q8×C31—22no—
161Q8×C31—22no—
162Q8×C31—41no—
163Q8×C31—41no—
164Q8×C31—41no—
165Q8×C31—41no—
166D4×C31—22no—
167D4×C31—22no—
168D4×C31—41no—
169D4×C31—41no—
170D4×C31—42no—
171D4×C31—41no—
172D4×C31—41no—
173D4×C31—81no—
174D4×C31—81no—
175D4×C31—81no—
176C2×C4×C31—22no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—42no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—41no—
185C2×C4×C31—41no—
186C2×C4×C31—41no—
187C2×C4×C31—42no—
188C2×C4×C31—41no—
189C2×C4×C31—41no—
190C2×C4×C31—81no—
191C2×C4×C31—81no—
192C2×C4×C31—81no—
193C2×C4×C31—81no—
194C2×C4×C31—81no—
195C2×C4×C31—81no—
196C2×C4×C31#12011yescharacteristic
197C2×C4×C31#12011yescharacteristic
198C2×C4×C31#12111yescharacteristic
199C2×C4×C31#12111yescharacteristic
200C2×C4×C31#12111yescharacteristic
201C2×C4×C31#12711yescharacteristic
202C2×C4×C31C2×D811yescharacteristic
203C2×C4×C31#13511yescharacteristic
204C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
206C8×C31—42no—
207C8×C31—42no—
208C8×C31—81no—
209#3100C2×C3111yescharacteristic
210#3037C2×C3111yescharacteristic, complemented
211#2536C2×C3111yescharacteristic, complemented
212#1517C2×C3111yescharacteristic, complemented
213#1486C2×C3111yescharacteristic, complemented
214#1479C2×C3111yescharacteristic, complemented
215#1286C2×C3111yescharacteristic, complemented
216C22×C31—21no—
217C22×C31—21no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31—81no—
222C22×C31—81no—
223C22×C31—81no—
224C22×C31—81no—
225C22×C31—81no—
226C22×C31—81no—
227C22×C31—81no—
228C22×C31#39211yescharacteristic
229C22×C31#44711yescharacteristic
230C22×C31#44911yescharacteristic
231C4×C31—21no—
232C4×C31—21no—
233C4×C31—21no—
234C4×C31—21no—
235C4×C31—42no—
236C4×C31—82no—
237C4×C31—81no—
238C4×C31—81no—
239C4×C31—81no—
240C4×C31#44711yescharacteristic
241C4×C31#44911yescharacteristic
242C4×C31#46611yescharacteristic
243C4×C31#46911yescharacteristic
244C2×#137—21no—
245#532C22×C3111yescharacteristic
246C2×C22⋊Q8—21no—
247C2×C4×Q8C22×C3111yescharacteristic
248C2×C4×D4C22×C3111yescharacteristic
249#480—21no—
250#474—21no—
251#464—22no—
252#462—21no—
253#460—21no—
254#459—21no—
255#447—21no—
256C4×D8—22no—
257#431C22×C3111yescharacteristic
258#431C22×C3111yescharacteristic
259#423C22×C3111yescharacteristic
260#417—21no—
261#398—21no—
262#394—21no—
263#388—22no—
264#382C22×C3111yescharacteristic
265#380—21no—
266#327—22no—
267Q8⋊C8—22no—
268D4⋊C8—22no—
269C2×C31—21no—
270C2×C31—21no—
271C2×C31—82no—
272C2×C31—161no—
273C2×C31#199911yescharacteristic
274C2×C31#200911yescharacteristic
275C2×C31#201011yescharacteristic
276C2×#41—21no—
277C22×Q8D4×C3111yescharacteristic
278C22×D4D4×C3111yescharacteristic
279C23×C4—21no—
280#137—42no—
281#137—42no—
282C2×Q16—22no—
283C2×SD16—22no—
284C2×D8—22no—
285C2×M16—21no—
286C4⋊Q8D4×C3111yescharacteristic
287#127D4×C3111yescharacteristic
288#124—21no—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291#121—21no—
292#121—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4D4×C3111yescharacteristic
307C2×C4⋊C4D4×C3111yescharacteristic
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C42—21no—
312C2×C42C23×C3111yescharacteristic
313#107—22no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8D4×C3112yes—
318#103—22no—
319#102—22no—
320#102—22no—
321#102—22no—
322C22⋊C8—21no—
323C22⋊C8—21no—
324C4×C8—21no—
325C4×C8D4×C3112yes—
326#95—22no—
327#95—21no—
328#95D4×C3112yes—
329C31#999711yescharacteristic, direct product factor, Sylow
330C24—21no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8D8×C3112yescomplemented
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—42no—
348C2×D4—41no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C2×D4×C3111yescharacteristic
361C22×C4C2×D4×C3111yescharacteristic
362C22×C4C2×D4×C3111yescharacteristic
363Q16—44no—
364SD16—44no—
365D8—44no—
366M16—41no—
367M16—41no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—41no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C42—41no—
389C42—41no—
390C42C2×D4×C3111yescharacteristic
391C42C2×D4×C3111yescharacteristic
392C42C2×D4×C3111yescharacteristic
393C42#41×C3111yescharacteristic
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C31×#12011yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—41no—
409D4—22no—
410D4—22no—
411D4—41no—
412D4—41no—
413D4—42no—
414D4—41no—
415D4—41no—
416D4—81no—
417D4—81no—
418D4—81no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4C31×#12011yescharacteristic
440C2×C4C31×#12011yescharacteristic
441C2×C4C31×#12111yescharacteristic
442C2×C4C31×#12111yescharacteristic
443C2×C4C31×#12111yescharacteristic
444C2×C4C31×#12711yescharacteristic
445C2×C4C2×D8×C3111yescharacteristic
446C2×C4C31×#13511yescharacteristic
447C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
449C8—42no—
450C8—42no—
451C8—81no—
452C22—21no—
453C22—21no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C31×#39211yescharacteristic
465C22C31×#44711yescharacteristic
466C22C31×#44911yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—82no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4C31×#44711yescharacteristic
477C4C31×#44911yescharacteristic
478C4C31×#46611yescharacteristic
479C4C31×#46911yescharacteristic
480C2—21no—
481C2—21no—
482C2—82no—
483C2—161no—
484C2C31×#199911yescharacteristic
485C2C31×#200911yescharacteristic
486C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3100C2
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
2. C31×#3037C2semidirect
3. C31×#2536C2semidirect
4. C31×#1517C2semidirect
5. C31×#1486C2semidirect
6. C31×#1479C2semidirect
7. C31×#1286C2semidirect
97. C2×Q8×C31D8semidirect
150. #9997C31direct (N × H)
210. #3037C2×C31semidirect
211. #2536C2×C31semidirect
212. #1517C2×C31semidirect
213. #1486C2×C31semidirect
214. #1479C2×C31semidirect
215. #1286C2×C31semidirect
329. C31#9997direct (N × H)
341. C2×Q8D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31002
C31×#30372
C31×#25362
C31×#15172
C31×#14862
C31×#14792
C31×#12862
#999731