Finite Group Databasebeta

← C31×#18398 #21475157

Subgroups — C31×#18398 #21475157

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3507C211yesmaximal, characteristic
2C31×#3229C211yesmaximal, characteristic, complemented
3C31×#3226C211yesmaximal, characteristic
4C31×#2546C211yesmaximal, characteristic, complemented
5C31×#1609C218yesmaximal, complemented
6C31×#1503C211yesmaximal, characteristic, complemented
7C31×#1502C211yesmaximal, characteristic
8C31×#1461C211yesmaximal, characteristic, complemented
9C31×#559—21no—
10C31×#557—21no—
11C31×#556—21no—
12C31×#556—22no—
13C31×#554—21no—
14C31×#552—22no—
15C31×#539C2214yes—
16C31×#536—21no—
17C31×#533C2211yescharacteristic
18C31×#532C2211yescharacteristic
19C2×C31×C4⋊Q8—21no—
20C2×C31×C4⋊Q8C2211yescharacteristic
21C31×#529—22no—
22C31×#520—21no—
23C2×C4×Q8×C31C2211yescharacteristic
24C31×#500—21no—
25C31×#498—21no—
26C31×#487—21no—
27C31×#486—21no—
28C31×#483C2214yes—
29C31×#480C2214yes—
30C31×#475—22no—
31C31×#474—22no—
32C31×#431C2211yescharacteristic
33C31×#431C2211yescharacteristic
34C31×#398C2214yes—
35C31×#391—21no—
36C31×#389—22no—
37C4×C4⋊C4×C31C2211yescharacteristic
38C31×#328C2214yes—
39C31×#327C2214yes—
40C31×#324C2214yes—
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8—21no—
47C31×C4⋊Q8—21no—
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—44no—
50C31×C4⋊Q8D411yescharacteristic
51C31×C4⋊Q8D411yescharacteristic
52C31×C4⋊Q8D411yescharacteristic
53C31×#127D411yescharacteristic
54C31×#126—24no—
55C31×#126—24no—
56C31×#125—22no—
57C31×#125—24no—
58C31×#125—21no—
59C31×#124—22no—
60C31×#124—22no—
61C31×#124—21no—
62C31×#123—24no—
63C31×#123—24no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—22no—
66C31×C22⋊Q8—24no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8C2312yes—
69C31×#121—21no—
70C31×#121C2312yes—
71C4×Q8×C31—22no—
72C4×Q8×C31—22no—
73C4×Q8×C31—21no—
74C4×Q8×C31—21no—
75C4×Q8×C31—21no—
76C4×Q8×C31—21no—
77C4×Q8×C31—21no—
78C4×Q8×C31D412yes—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—24no—
82C4×D4×C31—21no—
83C4×D4×C31D412yes—
84C31×#117—22no—
85C31×#117—24no—
86C31×#117—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—24no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31D411yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2312yes—
94C2×C42×C31—21no—
95C2×C42×C31D412yes—
96C2×C42×C31C2311yescharacteristic
97C31×#107—24no—
98C31×#106—24no—
99C31×C4⋊C8—21no—
100C31×C4⋊C8—21no—
101C31×#103—24no—
102C31×#102—24no—
103C31×C22⋊C8C2312yes—
104C31×C22⋊C8C2312yes—
105C4×C8×C31—21no—
106C4×C8×C31—21no—
107C31×#95—24no—
108C31×#95C2312yes—
109#41×C31—42no—
110#41×C31—42no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31C2×D412yes—
122C2×D4×C31—22no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31C2×D412yes—
126C22×C4×C31—22no—
127C22×C4×C31—24no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2411yescharacteristic
135C2×C8×C31—22no—
136C2×C8×C31—22no—
137C4⋊C4×C31—24no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—21no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—21no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—48no—
154C4⋊C4×C31—41no—
155C4⋊C4×C31—41no—
156C4⋊C4×C31C2×D412yes—
157C4⋊C4×C31C2×D412yes—
158C22⋊C4×C31—24no—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C42×C31—24no—
165C42×C31—22no—
166C42×C31—22no—
167C42×C31—22no—
168C42×C31—42no—
169C42×C31C2×D411yescharacteristic
170C42×C31C2×D411yescharacteristic
171C42×C31C2×D411yescharacteristic
172C42×C31C2×D411yescharacteristic
173#18398C3111yesmaximal, characteristic, direct product factor, Sylow
174C23×C31—22no—
175C23×C31C22×D411yescharacteristic
176Q8×C31—42no—
177Q8×C31—42no—
178Q8×C31—42no—
179Q8×C31—42no—
180Q8×C31—42no—
181Q8×C31—82no—
182Q8×C31—82no—
183D4×C31—42no—
184D4×C31—42no—
185D4×C31—42no—
186D4×C31—42no—
187D4×C31—42no—
188C2×C4×C31—22no—
189C2×C4×C31—24no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—22no—
193C2×C4×C31—22no—
194C2×C4×C31—22no—
195C2×C4×C31—22no—
196C2×C4×C31—22no—
197C2×C4×C31—41no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—44no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—42no—
205C2×C4×C31—84no—
206C2×C4×C31#12011yescharacteristic
207C2×C4×C31#12011yescharacteristic
208C2×C4×C31#12011yescharacteristic
209C2×C4×C31#12011yescharacteristic
210C2×C4×C31#13512yes—
211C2×C4×C31AGL(1,ℤ/8)12yes—
212C2×C4×C31C22×D411yescharacteristic
213C2×C4×C31C22×D411yescharacteristic
214C8×C31—42no—
215C8×C31—42no—
216#3507C2×C3111yescharacteristic
217#3229C2×C3111yescharacteristic, complemented
218#3226C2×C3111yescharacteristic
219#2546C2×C3111yescharacteristic, complemented
220#1609C2×C3118yescomplemented
221#1503C2×C3111yescharacteristic, complemented
222#1502C2×C3111yescharacteristic
223#1461C2×C3111yescharacteristic, complemented
224C22×C31—21no—
225C22×C31—21no—
226C22×C31—42no—
227C22×C31—42no—
228C22×C31—42no—
229C22×C31C2×#12011yescharacteristic
230C22×C31C2×#13511yescharacteristic
231C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C31—21no—
233C4×C31—21no—
234C4×C31—21no—
235C4×C31—44no—
236C4×C31—42no—
237C4×C31—42no—
238C4×C31—42no—
239C4×C31—42no—
240C4×C31—42no—
241C4×C31—82no—
242C4×C31#44912yes—
243C4×C31#44912yes—
244C4×C31#45612yes—
245#559—21no—
246#557—21no—
247#556—21no—
248#556—22no—
249#554—21no—
250#552—22no—
251#539C22×C3114yes—
252#536—21no—
253#533C22×C3111yescharacteristic
254#532C22×C3111yescharacteristic
255C2×C4⋊Q8—21no—
256C2×C4⋊Q8C22×C3111yescharacteristic
257#529—22no—
258#520—21no—
259C2×C4×Q8C22×C3111yescharacteristic
260#500—21no—
261#498—21no—
262#487—21no—
263#486—21no—
264#483C22×C3114yes—
265#480C22×C3114yes—
266#475—22no—
267#474—22no—
268#431C22×C3111yescharacteristic
269#431C22×C3111yescharacteristic
270#398C22×C3114yes—
271#391—21no—
272#389—22no—
273C4×C4⋊C4C22×C3111yescharacteristic
274#328C22×C3114yes—
275#327C22×C3114yes—
276#324C22×C3114yes—
277C2×C31—21no—
278C2×C31—21no—
279C2×C31—82no—
280C2×C31C2×#44911yescharacteristic
281C2×C31C2×#45611yescharacteristic
282C2×C31#302811yescharacteristic
283C2×#41—21no—
284C22×Q8—21no—
285C22×Q8D4×C3111yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—44no—
292C4⋊Q8D4×C3111yescharacteristic
293C4⋊Q8D4×C3111yescharacteristic
294C4⋊Q8D4×C3111yescharacteristic
295#127D4×C3111yescharacteristic
296#126—24no—
297#126—24no—
298#125—22no—
299#125—24no—
300#125—21no—
301#124—22no—
302#124—22no—
303#124—21no—
304#123—24no—
305#123—24no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—24no—
309C22⋊Q8—21no—
310C22⋊Q8C23×C3112yes—
311#121—21no—
312#121C23×C3112yes—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8D4×C3112yes—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—24no—
324C4×D4—21no—
325C4×D4D4×C3112yes—
326#117—22no—
327#117—24no—
328#117—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4D4×C3111yescharacteristic
334C2×C4⋊C4C23×C3112yes—
335C2×C4⋊C4C23×C3112yes—
336C2×C42—21no—
337C2×C42D4×C3112yes—
338C2×C42C23×C3111yescharacteristic
339#107—24no—
340#106—24no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343#103—24no—
344#102—24no—
345C22⋊C8C23×C3112yes—
346C22⋊C8C23×C3112yes—
347C4×C8—21no—
348C4×C8—21no—
349#95—24no—
350#95C23×C3112yes—
351C31#1839811yescharacteristic, direct product factor, Sylow
352#41—42no—
353#41—42no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C2×D4×C3112yes—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C2×D4×C3112yes—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4C2×D4×C3112yes—
374C22×C4C2×D4×C3111yescharacteristic
375C22×C4C2×D4×C3112yes—
376C22×C4C2×D4×C3111yescharacteristic
377C22×C4C24×C3111yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4C2×D4×C3112yes—
400C4⋊C4C2×D4×C3112yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C42—24no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—42no—
412C42C2×D4×C3111yescharacteristic
413C42C2×D4×C3111yescharacteristic
414C42C2×D4×C3111yescharacteristic
415C42C2×D4×C3111yescharacteristic
416C23—22no—
417C23C22×D4×C3111yescharacteristic
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—82no—
424Q8—82no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—84no—
448C2×C4C31×#12011yescharacteristic
449C2×C4C31×#12011yescharacteristic
450C2×C4C31×#12011yescharacteristic
451C2×C4C31×#12011yescharacteristic
452C2×C4C31×#13512yes—
453C2×C4C31×AGL(1,ℤ/8)12yes—
454C2×C4C22×D4×C3111yescharacteristic
455C2×C4C22×D4×C3111yescharacteristic
456C8—42no—
457C8—42no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22C2×C31×#12011yescharacteristic
464C22C2×C31×#13511yescharacteristic
465C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4C31×#44912yes—
477C4C31×#44912yes—
478C4C31×#45612yes—
479C2—21no—
480C2—21no—
481C2—82no—
482C2C2×C31×#44911yescharacteristic
483C2C2×C31×#45611yescharacteristic
484C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#532C22
LCSG ▷ C22×C4 ▷ 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×#3229C2semidirect
4. C31×#2546C2semidirect
5. C31×#1609C2semidirect
6. C31×#1503C2semidirect
8. C31×#1461C2semidirect
173. #18398C31direct (N × H)
217. #3229C2×C31semidirect
219. #2546C2×C31semidirect
220. #1609C2×C31semidirect
221. #1503C2×C31semidirect
223. #1461C2×C31semidirect
351. C31#18398direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35072
C31×#32292
C31×#32262
C31×#25462
C31×#16092
C31×#15032
C31×#15022
C31×#14612
#1839831