Finite Group Databasebeta

← C2×C31×#1635 #21475246

Subgroups — C2×C31×#1635 #21475246

Order 7936.

1206 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#547C211yesmaximal, characteristic
2C2×C31×#487C211yesmaximal, characteristic, complemented
3C2×C31×#476C211yesmaximal, characteristic, complemented
4C2×C31×#476C211yesmaximal, characteristic, complemented
5C31×#1635C218yesmaximal, direct product factor
6C2×C31×#332C211yesmaximal, characteristic, complemented
7C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×C31×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#41×C31—21no—
10C31×#547—22no—
11C31×#547—22no—
12C31×#547—22no—
13C31×#547C2214yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#123—22no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×#121—21no—
19C2×C4×D4×C31C2211yescharacteristic
20C2×C4×D4×C31C2211yescharacteristic
21C22×C4⋊C4×C31—21no—
22C31×#487—22no—
23C31×#487C2214yescomplemented
24C31×#476—22no—
25C31×#476—22no—
26C31×#476C2214yescomplemented
27C31×#476C2214yescomplemented
28C2×C31×#106—21no—
29C2×C31×#106—21no—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#103—22no—
33C2×C31×#102—21no—
34C2×C31×#102—21no—
35C2×C4×C8×C31C2211yescharacteristic
36C31×#332C2214yescomplemented
37C31×D4⋊C8C2214yescomplemented
38C31×D4⋊C8C2214yescomplemented
39C2×#41×C31—24no—
40C2×#41×C31—24no—
41C2×#41×C31—44no—
42C2×#41×C31—44no—
43C22×Q8×C31—21no—
44C22×Q8×C31—21no—
45C22×D4×C31—22no—
46C22×D4×C31D411yescharacteristic
47C22×D4×C31D411yescharacteristic
48C23×C4×C31—22no—
49C23×C4×C31—21no—
50C23×C4×C31—21no—
51C22×C8×C31—21no—
52C22×C8×C31—21no—
53C22×C8×C31—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—21no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8D412yes—
60C31×C4⋊Q8C2312yes—
61C31×C4⋊Q8C2312yes—
62C31×#123—28no—
63C31×#123—28no—
64C31×C22⋊Q8—24no—
65C31×C22⋊Q8—24no—
66C31×#121—24no—
67C31×#121—24no—
68C4×D4×C31—21no—
69C4×D4×C31—21no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31D412yes—
73C4×D4×C31D412yes—
74C4×D4×C31C2312yes—
75C4×D4×C31C2312yes—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—24no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—24no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31D412yes—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C22⋊C4×C31—21no—
88C2×C42×C31C2311yescharacteristic
89C31×#106—24no—
90C31×#106—24no—
91C31×C4⋊C8—21no—
92C31×C4⋊C8—21no—
93C31×C4⋊C8C2312yes—
94C31×C4⋊C8C2312yes—
95C31×#103—28no—
96C31×#102—24no—
97C31×#102—24no—
98C4×C8×C31—21no—
99C4×C8×C31C2312yes—
100C24×C31—21no—
101C24×C31—21no—
102#41×C31—48no—
103#41×C31—416no—
104#41×C31—48no—
105C2×Q8×C31—22no—
106C2×Q8×C31—22no—
107C2×Q8×C31—22no—
108C2×Q8×C31—44no—
109C2×Q8×C31—41no—
110C2×Q8×C31—42no—
111C2×Q8×C31—42no—
112C2×D4×C31—28no—
113C2×D4×C31—21no—
114C2×D4×C31—21no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—48no—
119C2×D4×C31SD1614yes—
120C2×D4×C31C2×D412yes—
121C2×D4×C31C2×D412yes—
122C22×C4×C31—28no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—22no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—48no—
131C22×C4×C31—42no—
132C22×C4×C31—42no—
133C22×C4×C31—42no—
134C22×C4×C31—42no—
135C22×C4×C31—41no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C2×C8×C31—22no—
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—42no—
143C2×C8×C31—42no—
144C2×C8×C31—42no—
145C4⋊C4×C31—28no—
146C4⋊C4×C31—21no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—21no—
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×D414yes—
156C4⋊C4×C31C2×D412yes—
157C4⋊C4×C31C2×D412yes—
158C22⋊C4×C31—28no—
159C22⋊C4×C31—22no—
160C22⋊C4×C31—22no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—22no—
163C42×C31C2×D411yescharacteristic
164C42×C31C2×D411yescharacteristic
165C42×C31C2×D411yescharacteristic
166C42×C31C2411yescharacteristic
167C2×#1635C3111yesmaximal, characteristic, direct product factor, Sylow
168C23×C31—22no—
169C23×C31—24no—
170C23×C31—22no—
171C23×C31—42no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—42no—
176C23×C31#12011yescharacteristic
177Q8×C31—44no—
178Q8×C31—42no—
179Q8×C31—42no—
180Q8×C31—44no—
181Q8×C31—82no—
182D4×C31—22no—
183D4×C31—22no—
184D4×C31—22no—
185D4×C31—24no—
186D4×C31—416no—
187D4×C31—42no—
188D4×C31C2×SD1614yes—
189C2×C4×C31—22no—
190C2×C4×C31—22no—
191C2×C4×C31—22no—
192C2×C4×C31—24no—
193C2×C4×C31—416no—
194C2×C4×C31—42no—
195C2×C4×C31—44no—
196C2×C4×C31—42no—
197C2×C4×C31—42no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—44no—
202C2×C4×C31—44no—
203C2×C4×C31—42no—
204C2×C4×C31—82no—
205C2×C4×C31#12011yescharacteristic
206C2×C4×C31#12011yescharacteristic
207C2×C4×C31#12011yescharacteristic
208C2×C4×C31C2×SD1612yes—
209C2×C4×C31#13512yes—
210C2×C4×C31AGL(1,ℤ/8)12yes—
211C2×C4×C31#13712yes—
212C2×C4×C31C22×D411yescharacteristic
213C2×C4×C31C22×D411yescharacteristic
214C2×C4×C31C22×D411yescharacteristic
215C8×C31—42no—
216C8×C31—42no—
217C8×C31—42no—
218C2×#547C2×C3111yescharacteristic
219C2×#487C2×C3111yescharacteristic, complemented
220C2×#476C2×C3111yescharacteristic, complemented
221C2×#476C2×C3111yescharacteristic, complemented
222#1635C2×C3118yesdirect product factor
223C2×#332C2×C3111yescharacteristic, complemented
224C2×D4⋊C8C2×C3111yescharacteristic, complemented
225C2×D4⋊C8C2×C3111yescharacteristic, complemented
226C22×C31—24no—
227C22×C31—48no—
228C22×C31—42no—
229C22×C31—42no—
230C22×C31—42no—
231C22×C31—42no—
232C22×C31—42no—
233C22×C31—84no—
234C22×C31#45012yes—
235C22×C31#45212yes—
236C22×C31#45412yes—
237C22×C31C2×#12011yescharacteristic
238C4×C31—21no—
239C4×C31—21no—
240C4×C31—42no—
241C4×C31—44no—
242C4×C31—42no—
243C4×C31—82no—
244C4×C31#45012yes—
245C4×C31#45212yes—
246C4×C31C22×SD1611yescharacteristic
247C4×C31C2×#13511yescharacteristic
248C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
249C4×C31C2×#13711yescharacteristic
250C22×#41—21no—
251#547—22no—
252#547—22no—
253#547—22no—
254#547C22×C3114yes—
255C2×C4⋊Q8C22×C3111yescharacteristic
256C2×C4⋊Q8C22×C3111yescharacteristic
257C2×#123—22no—
258C2×C22⋊Q8—21no—
259C2×#121—21no—
260C2×C4×D4C22×C3111yescharacteristic
261C2×C4×D4C22×C3111yescharacteristic
262C22×C4⋊C4—21no—
263#487—22no—
264#487C22×C3114yescomplemented
265#476—22no—
266#476—22no—
267#476C22×C3114yescomplemented
268#476C22×C3114yescomplemented
269C2×#106—21no—
270C2×#106—21no—
271C2×C4⋊C8C22×C3111yescharacteristic
272C2×C4⋊C8C22×C3111yescharacteristic
273C2×#103—22no—
274C2×#102—21no—
275C2×#102—21no—
276C2×C4×C8C22×C3111yescharacteristic
277#332C22×C3114yescomplemented
278D4⋊C8C22×C3114yescomplemented
279D4⋊C8C22×C3114yescomplemented
280C2×C31—44no—
281C2×C31—82no—
282C2×C31#163514yesdirect product factor
283C2×C31C2×#45011yescharacteristic
284C2×C31C2×#45211yescharacteristic
285C2×C31C2×#45411yescharacteristic
286C2×#41—24no—
287C2×#41—24no—
288C2×#41—44no—
289C2×#41—44no—
290C22×Q8—21no—
291C22×Q8—21no—
292C22×D4—22no—
293C22×D4D4×C3111yescharacteristic
294C22×D4D4×C3111yescharacteristic
295C23×C4—22no—
296C23×C4—21no—
297C23×C4—21no—
298C22×C8—21no—
299C22×C8—21no—
300C22×C8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8D4×C3112yes—
307C4⋊Q8C23×C3112yes—
308C4⋊Q8C23×C3112yes—
309#123—28no—
310#123—28no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313#121—24no—
314#121—24no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4D4×C3112yes—
320C4×D4D4×C3112yes—
321C4×D4C23×C3112yes—
322C4×D4C23×C3112yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4D4×C3111yescharacteristic
330C2×C4⋊C4D4×C3111yescharacteristic
331C2×C4⋊C4D4×C3112yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C42C23×C3111yescharacteristic
336#106—24no—
337#106—24no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8C23×C3112yes—
341C4⋊C8C23×C3112yes—
342#103—28no—
343#102—24no—
344#102—24no—
345C4×C8—21no—
346C4×C8C23×C3112yes—
347C31C2×#163511yescharacteristic, direct product factor, Sylow
348C24—21no—
349C24—21no—
350#41—416no—
351#41—48no—
352#41—48no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×D4—28no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—48no—
367C2×D4SD16×C3114yes—
368C2×D4C2×D4×C3112yes—
369C2×D4C2×D4×C3112yes—
370C22×C4—28no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—28no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4C2×D4×C3114yes—
404C4⋊C4C2×D4×C3112yes—
405C4⋊C4C2×D4×C3112yes—
406C22⋊C4—28no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C42C2×D4×C3111yescharacteristic
412C42C2×D4×C3111yescharacteristic
413C42C2×D4×C3111yescharacteristic
414C42C24×C3111yescharacteristic
415C23—24no—
416C23—22no—
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23C31×#12011yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—42no—
427Q8—42no—
428Q8—82no—
429D4—24no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—416no—
434D4—42no—
435D4C2×SD16×C3114yes—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—416no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4C31×#12011yescharacteristic
453C2×C4C31×#12011yescharacteristic
454C2×C4C31×#12011yescharacteristic
455C2×C4C2×SD16×C3112yes—
456C2×C4C31×#13512yes—
457C2×C4C31×AGL(1,ℤ/8)12yes—
458C2×C4C31×#13712yes—
459C2×C4C22×D4×C3111yescharacteristic
460C2×C4C22×D4×C3111yescharacteristic
461C2×C4C22×D4×C3111yescharacteristic
462C8—42no—
463C8—42no—
464C8—42no—
465C22—24no—
466C22—48no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22C31×#45012yes—
474C22C31×#45212yes—
475C22C31×#45412yes—
476C22C2×C31×#12011yescharacteristic
477C4—21no—
478C4—21no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—82no—
483C4C31×#45012yes—
484C4C31×#45212yes—
485C4C22×SD16×C3111yescharacteristic
486C4C2×C31×#13511yescharacteristic
487C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
488C4C2×C31×#13711yescharacteristic
489C2—44no—
490C2—82no—
491C2C31×#163514yesdirect product factor
492C2C2×C31×#45011yescharacteristic
493C2C2×C31×#45211yescharacteristic
494C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#547C2
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×#487C2semidirect
3. C2×C31×#476C2semidirect
4. C2×C31×#476C2semidirect
5. C31×#1635C2direct (N × H)
6. C2×C31×#332C2semidirect
7. C2×C31×D4⋊C8C2semidirect
8. C2×C31×D4⋊C8C2semidirect
23. C31×#487C22semidirect
26. C31×#476C22semidirect
27. C31×#476C22semidirect
36. C31×#332C22semidirect
37. C31×D4⋊C8C22semidirect
38. C31×D4⋊C8C22semidirect
167. C2×#1635C31direct (N × H)
219. C2×#487C2×C31semidirect
220. C2×#476C2×C31semidirect
221. C2×#476C2×C31semidirect
222. #1635C2×C31direct (N × H)
223. C2×#332C2×C31semidirect
224. C2×D4⋊C8C2×C31semidirect
225. C2×D4⋊C8C2×C31semidirect
264. #487C22×C31semidirect
267. #476C22×C31semidirect
268. #476C22×C31semidirect
277. #332C22×C31semidirect
278. D4⋊C8C22×C31semidirect
279. D4⋊C8C22×C31semidirect
282. C2×C31#1635direct (N × H)
347. C31C2×#1635direct (N × H)
491. C2C31×#1635direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5472
C2×C31×#4872
C2×C31×#4762
C31×#16352
C2×C31×#3322
C2×C31×D4⋊C82
C2×#163531