Finite Group Databasebeta

← C2×C31×#1511 #21474311

Subgroups — C2×C31×#1511 #21474311

Order 7936.

1446 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C211yesmaximal, characteristic
2C2×C31×#407C211yesmaximal, characteristic
3C2×C31×#381C211yesmaximal, characteristic
4C31×#1511C218yesmaximal, direct product factor
5C2×C31×#328C214yesmaximal, complemented
6C23×Q8×C31C411yescharacteristic, complemented
7C24×C4×C31C2211yescharacteristic
8C22×M16×C31—21no—
9C2×C31×C22⋊Q8—21no—
10C2×C31×C22⋊Q8—22no—
11C2×C31×C22⋊Q8—21no—
12C2×C31×C22⋊Q8—22no—
13C2×C31×C22⋊Q8C412yescomplemented
14C2×C31×C22⋊Q8C414yescomplemented
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×C22⋊Q8C2214yes—
17C22×C4⋊C4×C31—21no—
18C22×C4⋊C4×C31C411yescharacteristic, complemented
19C22×C22⋊C4×C31—21no—
20C22×C22⋊C4×C31—21no—
21C31×#407—22no—
22C31×#407C2214yes—
23C2×C31×C22⋊C8—21no—
24C2×C31×C22⋊C8C2212yes—
25C31×#381—22no—
26C31×#381C2214yes—
27C2×C31×#95—21no—
28C2×C31×#95C2212yes—
29C31×#328C22116yescomplemented
30C25×C31D411yescharacteristic
31C22×Q8×C31—22no—
32C22×Q8×C31—21no—
33C22×Q8×C31—44no—
34C22×Q8×C31C2×C412yes—
35C22×Q8×C31C2×C414yescomplemented
36C23×C4×C31—21no—
37C23×C4×C31—21no—
38C23×C4×C31—22no—
39C23×C4×C31—44no—
40C23×C4×C31C2×C411yescharacteristic
41C23×C4×C31C2×C412yes—
42C23×C4×C31C2×C411yescharacteristic
43C23×C4×C31D411yescharacteristic
44C23×C4×C31D411yescharacteristic
45C23×C4×C31D412yes—
46C23×C4×C31D411yescharacteristic
47C23×C4×C31D412yes—
48C23×C4×C31C2311yescharacteristic
49C23×C4×C31C2312yes—
50C2×M16×C31—24no—
51C2×M16×C31—44no—
52C22×C8×C31—22no—
53C31×C22⋊Q8—24no—
54C31×C22⋊Q8—24no—
55C31×C22⋊Q8—24no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—24no—
60C31×C22⋊Q8C2×C414yes—
61C31×C22⋊Q8C2×C418yescomplemented
62C31×C22⋊Q8C2314yes—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—24no—
69C2×C4⋊C4×C31C2×C412yes—
70C2×C4⋊C4×C31C2×C414yescomplemented
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—22no—
73C2×C22⋊C4×C31—22no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—24no—
76C2×C22⋊C4×C31—44no—
77C31×C22⋊C8—24no—
78C31×C22⋊C8—22no—
79C31×C22⋊C8C2314yes—
80C31×#95—24no—
81C31×#95—22no—
82C31×#95C2314yes—
83C24×C31—24no—
84C24×C31—24no—
85C24×C31—42no—
86C24×C31C22⋊C411yescharacteristic
87C24×C31C22⋊C412yes—
88C24×C31C22⋊C411yescharacteristic
89C24×C31C2×D412yes—
90C24×C31C2×D411yescharacteristic
91C2×Q8×C31—22no—
92C2×Q8×C31—24no—
93C2×Q8×C31—48no—
94C2×Q8×C31—44no—
95C2×Q8×C31—44no—
96C2×Q8×C31—48no—
97C2×Q8×C31C22×C414yes—
98C22×C4×C31—24no—
99C22×C4×C31—24no—
100C22×C4×C31—24no—
101C22×C4×C31—24no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—24no—
105C22×C4×C31—44no—
106C22×C4×C31—42no—
107C22×C4×C31—42no—
108C22×C4×C31—44no—
109C22×C4×C31—42no—
110C22×C4×C31—44no—
111C22×C4×C31—42no—
112C22×C4×C31—44no—
113C22×C4×C31—44no—
114C22×C4×C31—44no—
115C22×C4×C31—44no—
116C22×C4×C31—44no—
117C22×C4×C31—44no—
118C22×C4×C31C22⋊C412yes—
119C22×C4×C31C22⋊C412yes—
120C22×C4×C31C22⋊C411yescharacteristic
121C22×C4×C31C22⋊C411yescharacteristic
122C22×C4×C31C22⋊C412yes—
123C22×C4×C31C22⋊C412yes—
124C22×C4×C31C22⋊C412yes—
125C22×C4×C31C22×C412yes—
126C22×C4×C31C22×C411yescharacteristic
127C22×C4×C31C22×C411yescharacteristic
128C22×C4×C31C22×C412yes—
129C22×C4×C31C2×D412yes—
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D412yes—
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D412yes—
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2411yescharacteristic
136M16×C31—48no—
137C2×C8×C31—24no—
138C2×C8×C31—44no—
139C4⋊C4×C31—24no—
140C4⋊C4×C31—28no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31C22×C414yes—
145C22⋊C4×C31—24no—
146C22⋊C4×C31—28no—
147C22⋊C4×C31—24no—
148C22⋊C4×C31—48no—
149C2×#1511C3111yesmaximal, characteristic, direct product factor, Sylow
150C23×C31—22no—
151C23×C31—22no—
152C23×C31—22no—
153C23×C31—22no—
154C23×C31—24no—
155C23×C31—24no—
156C23×C31—24no—
157C23×C31—22no—
158C23×C31—24no—
159C23×C31—22no—
160C23×C31—22no—
161C23×C31—42no—
162C23×C31—42no—
163C23×C31—416no—
164C23×C31#9914yes—
165C23×C31C2×C22⋊C412yes—
166C23×C31C2×C22⋊C412yes—
167C23×C31C2×C22⋊C411yescharacteristic
168C23×C31C2×C22⋊C411yescharacteristic
169C23×C31#13714yes—
170C23×C31C22×D411yescharacteristic
171Q8×C31—44no—
172Q8×C31—44no—
173Q8×C31—48no—
174C2×C4×C31—22no—
175C2×C4×C31—24no—
176C2×C4×C31—24no—
177C2×C4×C31—22no—
178C2×C4×C31—24no—
179C2×C4×C31—22no—
180C2×C4×C31—24no—
181C2×C4×C31—22no—
182C2×C4×C31—24no—
183C2×C4×C31—24no—
184C2×C4×C31—42no—
185C2×C4×C31—42no—
186C2×C4×C31—42no—
187C2×C4×C31—41no—
188C2×C4×C31—42no—
189C2×C4×C31—41no—
190C2×C4×C31—42no—
191C2×C4×C31—41no—
192C2×C4×C31—42no—
193C2×C4×C31—41no—
194C2×C4×C31—48no—
195C2×C4×C31—48no—
196C2×C4×C31—44no—
197C2×C4×C31—44no—
198C2×C4×C31—44no—
199C2×C4×C31—44no—
200C2×C4×C31—44no—
201C2×C4×C31—88no—
202C2×C4×C31C2×C22⋊C411yescharacteristic
203C2×C4×C31C2×C22⋊C412yes—
204C2×C4×C31C2×C22⋊C411yescharacteristic
205C2×C4×C31C2×C22⋊C412yes—
206C2×C4×C31C23×C411yescharacteristic
207C2×C4×C31C22×D411yescharacteristic
208C8×C31—44no—
209C22×C22⋊Q8C2×C3111yescharacteristic
210C2×#407C2×C3111yescharacteristic
211C2×#381C2×C3111yescharacteristic
212#1511C2×C3118yesdirect product factor
213C2×#328C2×C3114yescomplemented
214C22×C31—21no—
215C22×C31—21no—
216C22×C31—22no—
217C22×C31—22no—
218C22×C31—21no—
219C22×C31—21no—
220C22×C31—28no—
221C22×C31—28no—
222C22×C31—22no—
223C22×C31—22no—
224C22×C31—22no—
225C22×C31—48no—
226C22×C31—48no—
227C22×C31—44no—
228C22×C31C2×#9914yes—
229C22×C31C2×#9912yes—
230C22×C31#41912yes—
231C22×C31#41912yes—
232C22×C31#42112yes—
233C22×C31C22×C22⋊C411yescharacteristic
234C22×C31C2×#13712yes—
235C4×C31—21no—
236C4×C31—21no—
237C4×C31—22no—
238C4×C31—21no—
239C4×C31—22no—
240C4×C31—21no—
241C4×C31—44no—
242C4×C31—44no—
243C4×C31—84no—
244C23×Q8C4×C3111yescharacteristic, complemented
245C24×C4C22×C3111yescharacteristic
246C22×M16—21no—
247C2×C22⋊Q8—21no—
248C2×C22⋊Q8—22no—
249C2×C22⋊Q8—21no—
250C2×C22⋊Q8—22no—
251C2×C22⋊Q8C4×C3112yescomplemented
252C2×C22⋊Q8C4×C3114yescomplemented
253C2×C22⋊Q8C22×C3112yes—
254C2×C22⋊Q8C22×C3114yes—
255C22×C4⋊C4—21no—
256C22×C4⋊C4C4×C3111yescharacteristic, complemented
257C22×C22⋊C4—21no—
258C22×C22⋊C4—21no—
259#407—22no—
260#407C22×C3114yes—
261C2×C22⋊C8—21no—
262C2×C22⋊C8C22×C3112yes—
263#381—22no—
264#381C22×C3114yes—
265C2×#95—21no—
266C2×#95C22×C3112yes—
267#328C22×C31116yescomplemented
268C2×C31—22no—
269C2×C31—24no—
270C2×C31—22no—
271C2×C31—42no—
272C2×C31#151114yesdirect product factor
273C2×C31C22×#9911yescharacteristic
274C2×C31C2×#41911yescharacteristic
275C2×C31C2×#42111yescharacteristic
276C25D4×C3111yescharacteristic
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—44no—
280C22×Q8C2×C4×C3112yes—
281C22×Q8C2×C4×C3114yescomplemented
282C23×C4—21no—
283C23×C4—21no—
284C23×C4—22no—
285C23×C4—44no—
286C23×C4C2×C4×C3111yescharacteristic
287C23×C4C2×C4×C3111yescharacteristic
288C23×C4C2×C4×C3112yes—
289C23×C4D4×C3111yescharacteristic
290C23×C4D4×C3111yescharacteristic
291C23×C4D4×C3112yes—
292C23×C4D4×C3111yescharacteristic
293C23×C4D4×C3112yes—
294C23×C4C23×C3111yescharacteristic
295C23×C4C23×C3112yes—
296C2×M16—24no—
297C2×M16—44no—
298C22×C8—22no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—22no—
303C22⋊Q8—24no—
304C22⋊Q8—22no—
305C22⋊Q8—24no—
306C22⋊Q8C2×C4×C3114yes—
307C22⋊Q8C2×C4×C3118yescomplemented
308C22⋊Q8C23×C3114yes—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4C2×C4×C3112yes—
316C2×C4⋊C4C2×C4×C3114yescomplemented
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C22⋊C8—24no—
324C22⋊C8—22no—
325C22⋊C8C23×C3114yes—
326#95—24no—
327#95—22no—
328#95C23×C3114yes—
329C31C2×#151111yescharacteristic, direct product factor, Sylow
330C24—24no—
331C24—24no—
332C24—42no—
333C24C22⋊C4×C3111yescharacteristic
334C24C22⋊C4×C3112yes—
335C24C22⋊C4×C3111yescharacteristic
336C24C2×D4×C3112yes—
337C24C2×D4×C3111yescharacteristic
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—48no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8C22×C4×C3114yes—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4C22⋊C4×C3112yes—
366C22×C4C22⋊C4×C3111yescharacteristic
367C22×C4C22⋊C4×C3112yes—
368C22×C4C22⋊C4×C3111yescharacteristic
369C22×C4C22⋊C4×C3112yes—
370C22×C4C22⋊C4×C3112yes—
371C22×C4C22⋊C4×C3112yes—
372C22×C4C22×C4×C3111yescharacteristic
373C22×C4C22×C4×C3112yes—
374C22×C4C22×C4×C3111yescharacteristic
375C22×C4C22×C4×C3112yes—
376C22×C4C2×D4×C3111yescharacteristic
377C22×C4C2×D4×C3112yes—
378C22×C4C2×D4×C3112yes—
379C22×C4C2×D4×C3111yescharacteristic
380C22×C4C2×D4×C3112yes—
381C22×C4C2×D4×C3111yescharacteristic
382C22×C4C24×C3111yescharacteristic
383M16—48no—
384C2×C8—24no—
385C2×C8—44no—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4C22×C4×C3114yes—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—24no—
395C22⋊C4—48no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—416no—
410C23C31×#9914yes—
411C23C2×C22⋊C4×C3112yes—
412C23C2×C22⋊C4×C3112yes—
413C23C2×C22⋊C4×C3111yescharacteristic
414C23C2×C22⋊C4×C3111yescharacteristic
415C23C31×#13714yes—
416C23C22×D4×C3111yescharacteristic
417Q8—44no—
418Q8—44no—
419Q8—48no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—88no—
448C2×C4C2×C22⋊C4×C3111yescharacteristic
449C2×C4C2×C22⋊C4×C3112yes—
450C2×C4C2×C22⋊C4×C3111yescharacteristic
451C2×C4C2×C22⋊C4×C3112yes—
452C2×C4C23×C4×C3111yescharacteristic
453C2×C4C22×D4×C3111yescharacteristic
454C8—44no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22—28no—
462C22—28no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—48no—
467C22—48no—
468C22—44no—
469C22C2×C31×#9914yes—
470C22C2×C31×#9912yes—
471C22C31×#41912yes—
472C22C31×#41912yes—
473C22C31×#42112yes—
474C22C22×C22⋊C4×C3111yescharacteristic
475C22C2×C31×#13712yes—
476C4—21no—
477C4—21no—
478C4—22no—
479C4—21no—
480C4—22no—
481C4—21no—
482C4—44no—
483C4—44no—
484C4—84no—
485C2—22no—
486C2—24no—
487C2—22no—
488C2—42no—
489C2C31×#151114yesdirect product factor
490C2C22×C31×#9911yescharacteristic
491C2C2×C31×#41911yescharacteristic
492C2C2×C31×#42111yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1511C2direct (N × H)
5. C2×C31×#328C2semidirect
6. C23×Q8×C31C4semidirect
13. C2×C31×C22⋊Q8C4semidirect
14. C2×C31×C22⋊Q8C4semidirect
18. C22×C4⋊C4×C31C4semidirect
29. C31×#328C22semidirect
35. C22×Q8×C31C2×C4semidirect
61. C31×C22⋊Q8C2×C4semidirect
70. C2×C4⋊C4×C31C2×C4semidirect
149. C2×#1511C31direct (N × H)
212. #1511C2×C31direct (N × H)
213. C2×#328C2×C31semidirect
244. C23×Q8C4×C31semidirect
251. C2×C22⋊Q8C4×C31semidirect
252. C2×C22⋊Q8C4×C31semidirect
256. C22×C4⋊C4C4×C31semidirect
267. #328C22×C31semidirect
272. C2×C31#1511direct (N × H)
281. C22×Q8C2×C4×C31semidirect
307. C22⋊Q8C2×C4×C31semidirect
316. C2×C4⋊C4C2×C4×C31semidirect
329. C31C2×#1511direct (N × H)
489. C2C31×#1511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C2×C31×#4072
C2×C31×#3812
C31×#15112
C2×C31×#3282
C2×#151131