Finite Group Databasebeta

← C2×C31×#1664 #21475745

Subgroups — C2×C31×#1664 #21475745

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#563C211yesmaximal, characteristic
2C2×C31×#501C211yesmaximal, characteristic, complemented
3C2×C31×#491C211yesmaximal, characteristic, complemented
4C2×C31×#489C211yesmaximal, characteristic
5C31×#1664C218yesmaximal, direct product factor
6C2×C31×#330C211yesmaximal, characteristic, complemented
7C2×C31×#330C211yesmaximal, characteristic, complemented
8C2×C31×#329C211yesmaximal, characteristic
9C31×#563—22no—
10C31×#563—22no—
11C31×#563—22no—
12C31×#563C2214yes—
13C2×C31×C4⋊Q8—21no—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#126—21no—
16C2×C31×#126—21no—
17C2×C31×#125C2211yescharacteristic
18C2×C31×#125C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8—21no—
23C31×#501—22no—
24C31×#501C2214yescomplemented
25C31×#491—22no—
26C31×#491C2214yescomplemented
27C31×#489—22no—
28C31×#489C2214yes—
29C2×C31×#107—21no—
30C2×C31×#106—21no—
31C2×C31×#103—21no—
32C2×C31×#103—21no—
33C2×C31×#103—21no—
34C2×C31×#102—21no—
35C2×C31×#97C2211yescharacteristic
36C2×C31×#97C2211yescharacteristic
37C2×C31×#97C2211yescharacteristic
38C31×#330C2214yescomplemented
39C31×#330C2214yescomplemented
40C31×#329C2214yes—
41C22×Q8×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C22×D4×C31D411yescharacteristic
45C23×C4×C31—21no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C31×C4⋊Q8—24no—
50C31×C4⋊Q8—24no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8C2312yes—
55C31×#126—24no—
56C31×#126—24no—
57C31×#126—24no—
58C31×#126—24no—
59C31×#125—21no—
60C31×#125—21no—
61C31×#125—21no—
62C31×#125—21no—
63C31×#125D412yes—
64C31×#125D412yes—
65C31×#125C2312yes—
66C31×#125C2312yes—
67C31×#124—21no—
68C31×#124—21no—
69C31×#124D412yes—
70C31×#124C2312yes—
71C31×#123—24no—
72C31×#123—24no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—24no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—24no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31D411yescharacteristic
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C22⋊C4×C31—21no—
90C2×C22⋊C4×C31—21no—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C42×C31—21no—
95C2×C42×C31C2311yescharacteristic
96C31×#107—24no—
97C31×#106—24no—
98C31×#103—24no—
99C31×#103—24no—
100C31×#103—24no—
101C31×#102—24no—
102C31×#97—21no—
103C31×#97—21no—
104C31×#97—21no—
105C31×#97C2312yes—
106C31×#97C2312yes—
107C31×#97C2312yes—
108C24×C31—21no—
109C2×Q8×C31—24no—
110C2×Q8×C31—21no—
111C2×Q8×C31—22no—
112C2×Q8×C31—42no—
113C2×Q8×C31—44no—
114C2×Q8×C31—41no—
115C2×Q8×C31—82no—
116C2×Q8×C31C2×D412yes—
117C2×D4×C31—21no—
118C2×D4×C31—42no—
119C2×D4×C31C2×D412yes—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—24no—
126C22×C4×C31—41no—
127C22×C4×C31—82no—
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D411yescharacteristic
130C22×C4×C31C2×D411yescharacteristic
131C2×C8×C31—24no—
132C2×C8×C31—22no—
133C2×C8×C31—22no—
134C2×C8×C31—22no—
135C2×C8×C31—42no—
136C2×C8×C31—42no—
137C4⋊C4×C31—24no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—21no—
144C4⋊C4×C31—21no—
145C4⋊C4×C31—21no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—41no—
154C4⋊C4×C31—41no—
155C4⋊C4×C31C2×D412yes—
156C4⋊C4×C31C2×D412yes—
157C4⋊C4×C31C2×D412yes—
158C4⋊C4×C31C2×D412yes—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—24no—
162C22⋊C4×C31—22no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—22no—
166C42×C31—24no—
167C42×C31C2×D411yescharacteristic
168C42×C31C2×D411yescharacteristic
169C42×C31C2×D411yescharacteristic
170C42×C31C2411yescharacteristic
171C2×#1664C3111yesmaximal, characteristic, direct product factor, Sylow
172C23×C31—22no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31#12011yescharacteristic
177Q8×C31—42no—
178Q8×C31—44no—
179Q8×C31—81no—
180Q8×C31—82no—
181Q8×C31—84no—
182D4×C31—42no—
183D4×C31—81no—
184C2×C4×C31—22no—
185C2×C4×C31—22no—
186C2×C4×C31—22no—
187C2×C4×C31—22no—
188C2×C4×C31—22no—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—84no—
199C2×C4×C31—82no—
200C2×C4×C31#12011yescharacteristic
201C2×C4×C31#12011yescharacteristic
202C2×C4×C31#12011yescharacteristic
203C2×C4×C31C22×D411yescharacteristic
204C2×C4×C31C22×D411yescharacteristic
205C2×C4×C31C22×D411yescharacteristic
206C8×C31—24no—
207C8×C31—42no—
208C8×C31—42no—
209C2×#563C2×C3111yescharacteristic
210C2×#501C2×C3111yescharacteristic, complemented
211C2×#491C2×C3111yescharacteristic, complemented
212C2×#489C2×C3111yescharacteristic
213#1664C2×C3118yesdirect product factor
214C2×#330C2×C3111yescharacteristic, complemented
215C2×#330C2×C3111yescharacteristic, complemented
216C2×#329C2×C3111yescharacteristic
217C22×C31—42no—
218C22×C31—42no—
219C22×C31—42no—
220C22×C31—84no—
221C22×C31#45412yes—
222C22×C31#45512yes—
223C22×C31#45612yes—
224C22×C31C2×#12011yescharacteristic
225C4×C31—21no—
226C4×C31—21no—
227C4×C31—21no—
228C4×C31—21no—
229C4×C31—21no—
230C4×C31—21no—
231C4×C31—42no—
232C4×C31—42no—
233C4×C31—42no—
234C4×C31—42no—
235C4×C31—42no—
236C4×C31—82no—
237#563—22no—
238#563—22no—
239#563—22no—
240#563C22×C3114yes—
241C2×C4⋊Q8—21no—
242C2×C4⋊Q8C22×C3111yescharacteristic
243C2×#126—21no—
244C2×#126—21no—
245C2×#125C22×C3111yescharacteristic
246C2×#125C22×C3111yescharacteristic
247C2×#124C22×C3111yescharacteristic
248C2×#123—21no—
249C2×C22⋊Q8—21no—
250C2×C22⋊Q8—21no—
251#501—22no—
252#501C22×C3114yescomplemented
253#491—22no—
254#491C22×C3114yescomplemented
255#489—22no—
256#489C22×C3114yes—
257C2×#107—21no—
258C2×#106—21no—
259C2×#103—21no—
260C2×#103—21no—
261C2×#103—21no—
262C2×#102—21no—
263C2×#97C22×C3111yescharacteristic
264C2×#97C22×C3111yescharacteristic
265C2×#97C22×C3111yescharacteristic
266#330C22×C3114yescomplemented
267#330C22×C3114yescomplemented
268#329C22×C3114yes—
269C2×C31—82no—
270C2×C31#166414yesdirect product factor
271C2×C31C2×#45511yescharacteristic
272C2×C31C2×#45411yescharacteristic
273C2×C31C2×#45611yescharacteristic
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8D4×C3111yescharacteristic
277C22×D4D4×C3111yescharacteristic
278C23×C4—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8C23×C3112yes—
288#126—24no—
289#126—24no—
290#126—24no—
291#126—24no—
292#125—21no—
293#125—21no—
294#125—21no—
295#125—21no—
296#125D4×C3112yes—
297#125D4×C3112yes—
298#125C23×C3112yes—
299#125C23×C3112yes—
300#124—21no—
301#124—21no—
302#124D4×C3112yes—
303#124C23×C3112yes—
304#123—24no—
305#123—24no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4D4×C3111yescharacteristic
319C2×C4⋊C4D4×C3111yescharacteristic
320C2×C4⋊C4D4×C3111yescharacteristic
321C2×C4⋊C4D4×C3111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C42—21no—
328C2×C42C23×C3111yescharacteristic
329#107—24no—
330#106—24no—
331#103—24no—
332#103—24no—
333#103—24no—
334#102—24no—
335#97—21no—
336#97—21no—
337#97—21no—
338#97C23×C3112yes—
339#97C23×C3112yes—
340#97C23×C3112yes—
341C31C2×#166411yescharacteristic, direct product factor, Sylow
342C24—21no—
343C2×Q8—24no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—41no—
349C2×Q8—82no—
350C2×Q8C2×D4×C3112yes—
351C2×D4—21no—
352C2×D4—42no—
353C2×D4C2×D4×C3112yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—41no—
361C22×C4—82no—
362C22×C4C2×D4×C3111yescharacteristic
363C22×C4C2×D4×C3111yescharacteristic
364C22×C4C2×D4×C3111yescharacteristic
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4C2×D4×C3112yes—
390C4⋊C4C2×D4×C3112yes—
391C4⋊C4C2×D4×C3112yes—
392C4⋊C4C2×D4×C3112yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C42—24no—
401C42C2×D4×C3111yescharacteristic
402C42C2×D4×C3111yescharacteristic
403C42C2×D4×C3111yescharacteristic
404C42C24×C3111yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C31×#12011yescharacteristic
410Q8—42no—
411Q8—44no—
412Q8—81no—
413Q8—82no—
414Q8—84no—
415D4—42no—
416D4—81no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4C31×#12011yescharacteristic
434C2×C4C31×#12011yescharacteristic
435C2×C4C31×#12011yescharacteristic
436C2×C4C22×D4×C3111yescharacteristic
437C2×C4C22×D4×C3111yescharacteristic
438C2×C4C22×D4×C3111yescharacteristic
439C8—24no—
440C8—42no—
441C8—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—84no—
446C22C31×#45412yes—
447C22C31×#45512yes—
448C22C31×#45612yes—
449C22C2×C31×#12011yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—82no—
462C2—82no—
463C2C31×#166414yesdirect product factor
464C2C2×C31×#45511yescharacteristic
465C2C2×C31×#45411yescharacteristic
466C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#501C2semidirect
3. C2×C31×#491C2semidirect
5. C31×#1664C2direct (N × H)
6. C2×C31×#330C2semidirect
7. C2×C31×#330C2semidirect
24. C31×#501C22semidirect
26. C31×#491C22semidirect
38. C31×#330C22semidirect
39. C31×#330C22semidirect
171. C2×#1664C31direct (N × H)
210. C2×#501C2×C31semidirect
211. C2×#491C2×C31semidirect
213. #1664C2×C31direct (N × H)
214. C2×#330C2×C31semidirect
215. C2×#330C2×C31semidirect
252. #501C22×C31semidirect
254. #491C22×C31semidirect
266. #330C22×C31semidirect
267. #330C22×C31semidirect
270. C2×C31#1664direct (N × H)
341. C31C2×#1664direct (N × H)
463. C2C31×#1664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5632
C2×C31×#5012
C2×C31×#4912
C2×C31×#4892
C31×#16642
C2×C31×#3302
C2×C31×#3292
C2×#166431