Finite Group Databasebeta

← C31×#24076 #21480835

Subgroups — C31×#24076 #21480835

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic
2C31×#3092C212yesmaximal, complemented
3C2×C31×#384C211yesmaximal, characteristic, complemented
4C31×#2060C212yesmaximal, complemented
5C31×#2039C212yesmaximal, complemented
6C31×#1896C212yesmaximal, complemented
7C31×#1859C214yesmaximal, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C31×#554—22no—
10C31×#553—22no—
11C31×#552—22no—
12C31×#547—22no—
13C31×#544C2211yescharacteristic
14C31×#539—22no—
15C31×#537C2211yescharacteristic
16C31×#525—22no—
17C31×#525C2212yes—
18C31×#520C2212yes—
19C22×C4⋊C4×C31—22no—
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C31×#484—22no—
24C31×#483—22no—
25C31×#481—22no—
26C31×#480—22no—
27C31×#428C2214yes—
28C2×C31×#104C2214yes—
29C31×#417C2214yes—
30C31×#407C2212yes—
31C31×#384—22no—
32C31×#384C2212yes—
33C31×#384C2212yes—
34C31×#343C2212yes—
35C31×#343C2212yes—
36C31×#339C2214yes—
37C2×#41×C31—22no—
38C2×#41×C31C2312yes—
39C23×C4×C31—21no—
40C23×C4×C31—21no—
41C23×C4×C31D412yes—
42C23×C4×C31C2311yescharacteristic
43C2×M16×C31C2314yes—
44C31×C4⋊Q8—22no—
45C31×#126—24no—
46C31×#125—24no—
47C31×#125—22no—
48C31×#123—24no—
49C31×#123—24no—
50C31×#123—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—24no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8D412yes—
56C31×C22⋊Q8D412yes—
57C31×#121—22no—
58C31×#121—22no—
59C31×#121—22no—
60C31×#121D412yes—
61C31×#121D412yes—
62C31×#120—22no—
63C4×Q8×C31—22no—
64C4×D4×C31—24no—
65C4×D4×C31—22no—
66C4×D4×C31—24no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117C2312yes—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—48no—
81C2×C4⋊C4×C31D412yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2311yescharacteristic
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C42×C31—21no—
86C2×C42×C31—24no—
87C2×C42×C31C2312yes—
88C31×#107—24no—
89C31×#106—24no—
90C31×#104—28no—
91C31×#103—24no—
92C31×#102—24no—
93C31×C22⋊C8—22no—
94C31×C22⋊C8—22no—
95C24×C31C2×D411yescharacteristic
96#41×C31—24no—
97#41×C31—24no—
98#41×C31—84no—
99C2×Q8×C31—22no—
100C2×Q8×C31C2×D412yes—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31C2×D412yes—
106C22×C4×C31—24no—
107C22×C4×C31—24no—
108C22×C4×C31—24no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—24no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—44no—
118C22×C4×C31—44no—
119C22×C4×C31—41no—
120C22×C4×C31—41no—
121C22×C4×C31C2×D411yescharacteristic
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2×D412yes—
124C22×C4×C31C2×D412yes—
125C22×C4×C31C2×D412yes—
126C22×C4×C31C2411yescharacteristic
127M16×C31—24no—
128M16×C31—24no—
129C2×C8×C31—24no—
130C4⋊C4×C31—24no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—24no—
134C4⋊C4×C31—24no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—22no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—48no—
141C4⋊C4×C31—48no—
142C4⋊C4×C31—42no—
143C4⋊C4×C31C2×D412yes—
144C4⋊C4×C31C2×D412yes—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—24no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—22no—
150C22⋊C4×C31—22no—
151C22⋊C4×C31—22no—
152C42×C31—24no—
153C42×C31—24no—
154C42×C31—22no—
155C42×C31—22no—
156C42×C31#4114yescomplemented
157#24076C3111yesmaximal, characteristic, direct product factor, Sylow
158C23×C31—21no—
159C23×C31—21no—
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—22no—
163C23×C31#12011yescharacteristic
164C23×C31#12011yescharacteristic
165C23×C31C22×D411yescharacteristic
166Q8×C31—24no—
167Q8×C31—82no—
168D4×C31—24no—
169D4×C31—42no—
170D4×C31—42no—
171D4×C31—44no—
172D4×C31—44no—
173D4×C31—82no—
174C2×C4×C31—24no—
175C2×C4×C31—216no—
176C2×C4×C31—24no—
177C2×C4×C31—21no—
178C2×C4×C31—24no—
179C2×C4×C31—21no—
180C2×C4×C31—24no—
181C2×C4×C31—24no—
182C2×C4×C31—22no—
183C2×C4×C31—22no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—44no—
188C2×C4×C31—44no—
189C2×C4×C31—42no—
190C2×C4×C31—42no—
191C2×C4×C31—44no—
192C2×C4×C31—42no—
193C2×C4×C31—82no—
194C2×C4×C31—84no—
195C2×C4×C31—82no—
196C2×C4×C31—82no—
197C2×C4×C31#12012yes—
198C2×C4×C31C22×D412yes—
199C2×C4×C31C2×#4112yes—
200C2×C4×C312−1+412yes—
201C8×C31—44no—
202#3492C2×C3111yescharacteristic
203#3092C2×C3112yescomplemented
204C2×#384C2×C3111yescharacteristic, complemented
205#2060C2×C3112yescomplemented
206#2039C2×C3112yescomplemented
207#1896C2×C3112yescomplemented
208#1859C2×C3114yescomplemented
209#1821C2×C3111yescharacteristic, complemented
210C22×C31—24no—
211C22×C31—24no—
212C22×C31—21no—
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—21no—
217C22×C31—21no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—42no—
221C22×C31#45312yes—
222C22×C31#45612yes—
223C22×C31C2×#12011yescharacteristic
224C22×C31#52511yescharacteristic
225C22×C31#53611yescharacteristic
226C4×C31—28no—
227C4×C31—22no—
228C4×C31—42no—
229C4×C31—42no—
230C4×C31—42no—
231C4×C31—42no—
232C4×C31—42no—
233C4×C31—82no—
234C4×C31#54714yes—
235#554—22no—
236#553—22no—
237#552—22no—
238#547—22no—
239#544C22×C3111yescharacteristic
240#539—22no—
241#537C22×C3111yescharacteristic
242#525—22no—
243#525C22×C3112yes—
244#520C22×C3112yes—
245C22×C4⋊C4—22no—
246C22×C4⋊C4C22×C3111yescharacteristic
247C22×C4⋊C4C22×C3111yescharacteristic
248C22×C42C22×C3111yescharacteristic
249#484—22no—
250#483—22no—
251#481—22no—
252#480—22no—
253#428C22×C3114yes—
254C2×#104C22×C3114yes—
255#417C22×C3114yes—
256#407C22×C3112yes—
257#384—22no—
258#384C22×C3112yes—
259#384C22×C3112yes—
260#343C22×C3112yes—
261#343C22×C3112yes—
262#339C22×C3114yes—
263C2×C31—21no—
264C2×C31—21no—
265C2×C31—22no—
266C2×C31—22no—
267C2×C31—82no—
268C2×C31#240611yescharacteristic
269C2×C31C2×#45311yescharacteristic
270C2×C31C2×#45611yescharacteristic
271C2×#41—22no—
272C2×#41C23×C3112yes—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4D4×C3112yes—
276C23×C4C23×C3111yescharacteristic
277C2×M16C23×C3114yes—
278C4⋊Q8—22no—
279#126—24no—
280#125—24no—
281#125—22no—
282#123—24no—
283#123—24no—
284#123—22no—
285C22⋊Q8—22no—
286C22⋊Q8—24no—
287C22⋊Q8—22no—
288C22⋊Q8—22no—
289C22⋊Q8D4×C3112yes—
290C22⋊Q8D4×C3112yes—
291#121—22no—
292#121—22no—
293#121—22no—
294#121D4×C3112yes—
295#121D4×C3112yes—
296#120—22no—
297C4×Q8—22no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—22no—
303#117—22no—
304#117—22no—
305#117C23×C3112yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4D4×C3112yes—
316C2×C4⋊C4C23×C3112yes—
317C2×C4⋊C4C23×C3111yescharacteristic
318C2×C4⋊C4C23×C3111yescharacteristic
319C2×C42—21no—
320C2×C42—24no—
321C2×C42C23×C3112yes—
322#107—24no—
323#106—24no—
324#104—28no—
325#103—24no—
326#102—24no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C31#2407611yescharacteristic, direct product factor, Sylow
330C24C2×D4×C3111yescharacteristic
331#41—24no—
332#41—24no—
333#41—84no—
334C2×Q8—22no—
335C2×Q8C2×D4×C3112yes—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4C2×D4×C3112yes—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4C2×D4×C3111yescharacteristic
357C22×C4C2×D4×C3112yes—
358C22×C4C2×D4×C3112yes—
359C22×C4C2×D4×C3112yes—
360C22×C4C2×D4×C3112yes—
361C22×C4C24×C3111yescharacteristic
362M16—24no—
363M16—24no—
364C2×C8—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4C2×D4×C3112yes—
379C4⋊C4C2×D4×C3112yes—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—24no—
388C42—24no—
389C42—22no—
390C42—22no—
391C42#41×C3114yescomplemented
392C23—22no—
393C23—22no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23C31×#12011yescharacteristic
398C23C31×#12011yescharacteristic
399C23C22×D4×C3111yescharacteristic
400Q8—24no—
401Q8—82no—
402D4—24no—
403D4—44no—
404D4—44no—
405D4—42no—
406D4—42no—
407D4—82no—
408C2×C4—24no—
409C2×C4—216no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4C31×#12012yes—
432C2×C4C22×D4×C3112yes—
433C2×C4C2×#41×C3112yes—
434C2×C4C31×2−1+412yes—
435C8—44no—
436C22—24no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22C31×#45312yes—
448C22C31×#45612yes—
449C22C2×C31×#12011yescharacteristic
450C22C31×#52511yescharacteristic
451C22C31×#53611yescharacteristic
452C4—28no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4C31×#54714yes—
461C2—21no—
462C2—22no—
463C2—22no—
464C2—21no—
465C2—82no—
466C2C31×#240611yescharacteristic
467C2C2×C31×#45311yescharacteristic
468C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3092C2semidirect
3. C2×C31×#384C2semidirect
4. C31×#2060C2semidirect
5. C31×#2039C2semidirect
6. C31×#1896C2semidirect
7. C31×#1859C2semidirect
8. C31×#1821C2semidirect
156. C42×C31#41semidirect
157. #24076C31direct (N × H)
203. #3092C2×C31semidirect
204. C2×#384C2×C31semidirect
205. #2060C2×C31semidirect
206. #2039C2×C31semidirect
207. #1896C2×C31semidirect
208. #1859C2×C31semidirect
209. #1821C2×C31semidirect
329. C31#24076direct (N × H)
391. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30922
C2×C31×#3842
C31×#20602
C31×#20392
C31×#18962
C31×#18592
C31×#18212
#2407631