Finite Group Databasebeta

← C2×C31×#1644 #21475242

Subgroups — C2×C31×#1644 #21475242

Order 7936.

1022 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#554C211yesmaximal, characteristic
2C2×C31×#487C211yesmaximal, characteristic, complemented
3C2×C31×#474C211yesmaximal, characteristic
4C2×C31×#462C211yesmaximal, characteristic, complemented
5C31×#1644C218yesmaximal, direct product factor
6C2×C31×#333C211yesmaximal, characteristic, complemented
7C2×C31×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×C31×D4⋊C8C211yesmaximal, characteristic
9C22×Q16×C31—21no—
10C31×#554—22no—
11C31×#554—22no—
12C31×#554—22no—
13C31×#554C2214yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×#125—21no—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8—22no—
19C2×C4×Q8×C31C2211yescharacteristic
20C2×C4×D4×C31—21no—
21C2×C4×D4×C31C2211yescharacteristic
22C22×C4⋊C4×C31—21no—
23C31×#487—22no—
24C31×#487C2214yescomplemented
25C31×#474—22no—
26C31×#474C2214yes—
27C31×#462—22no—
28C31×#462C2214yescomplemented
29C2×C31×#107—21no—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#103—22no—
33C2×C31×#103—21no—
34C2×C31×#102—21no—
35C2×C4×C8×C31C2211yescharacteristic
36C31×#333C2214yescomplemented
37C31×Q8⋊C8C2214yescomplemented
38C31×D4⋊C8C2214yes—
39C22×Q8×C31—21no—
40C22×Q8×C31D411yescharacteristic
41C22×Q8×C31D412yes—
42C22×D4×C31D411yescharacteristic
43C23×C4×C31—22no—
44C23×C4×C31—21no—
45C2×Q16×C31—24no—
46C2×Q16×C31—44no—
47C22×C8×C31—21no—
48C22×C8×C31—21no—
49C22×C8×C31—21no—
50C31×C4⋊Q8—21no—
51C31×C4⋊Q8—21no—
52C31×C4⋊Q8—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8D412yes—
56C31×C4⋊Q8C2312yes—
57C31×C4⋊Q8C2312yes—
58C31×#125—24no—
59C31×#125—24no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—28no—
62C31×C22⋊Q8—24no—
63C31×C22⋊Q8—28no—
64C4×Q8×C31—21no—
65C4×Q8×C31—21no—
66C4×Q8×C31D412yes—
67C4×Q8×C31C2312yes—
68C4×D4×C31—24no—
69C4×D4×C31—24no—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31D412yes—
73C4×D4×C31C2312yes—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—24no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—24no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31D411yescharacteristic
84C2×C4⋊C4×C31D411yescharacteristic
85C2×C22⋊C4×C31—22no—
86C2×C22⋊C4×C31—21no—
87C2×C42×C31—21no—
88C2×C42×C31C2311yescharacteristic
89C31×#107—24no—
90C31×C4⋊C8—21no—
91C31×C4⋊C8—21no—
92C31×C4⋊C8C2312yes—
93C31×C4⋊C8C2312yes—
94C31×#103—28no—
95C31×#103—24no—
96C31×#102—24no—
97C4×C8×C31—21no—
98C4×C8×C31C2312yes—
99C24×C31—21no—
100C2×Q8×C31—21no—
101C2×Q8×C31—22no—
102C2×Q8×C31—22no—
103C2×Q8×C31—22no—
104C2×Q8×C31—22no—
105C2×Q8×C31—44no—
106C2×Q8×C31—41no—
107C2×Q8×C31—42no—
108C2×Q8×C31—42no—
109C2×Q8×C31C2×D412yes—
110C2×Q8×C31C2×D414yes—
111C2×D4×C31—21no—
112C2×D4×C31—22no—
113C2×D4×C31Q1614yescomplemented
114C2×D4×C31C2×D412yes—
115C22×C4×C31—21no—
116C22×C4×C31—21no—
117C22×C4×C31—21no—
118C22×C4×C31—22no—
119C22×C4×C31—28no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—42no—
123C22×C4×C31—42no—
124C22×C4×C31—41no—
125C22×C4×C31—84no—
126C22×C4×C31C2×D411yescharacteristic
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2×D411yescharacteristic
129Q16×C31—48no—
130C2×C8×C31—22no—
131C2×C8×C31—22no—
132C2×C8×C31—22no—
133C2×C8×C31—42no—
134C2×C8×C31—42no—
135C2×C8×C31—42no—
136C4⋊C4×C31—28no—
137C4⋊C4×C31—28no—
138C4⋊C4×C31—21no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—22no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—21no—
143C4⋊C4×C31—22no—
144C4⋊C4×C31—22no—
145C4⋊C4×C31—22no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—41no—
149C4⋊C4×C31—41no—
150C4⋊C4×C31C2×D412yes—
151C4⋊C4×C31C2×D412yes—
152C22⋊C4×C31—28no—
153C22⋊C4×C31—22no—
154C22⋊C4×C31—22no—
155C42×C31—22no—
156C42×C31—22no—
157C42×C31C2×D411yescharacteristic
158C42×C31C2×D411yescharacteristic
159C42×C31C2×D411yescharacteristic
160C42×C31C2411yescharacteristic
161C2×#1644C3111yesmaximal, characteristic, direct product factor, Sylow
162C23×C31—22no—
163C23×C31—24no—
164C23×C31—42no—
165C23×C31—42no—
166C23×C31#12011yescharacteristic
167Q8×C31—22no—
168Q8×C31—22no—
169Q8×C31—44no—
170Q8×C31—42no—
171Q8×C31—42no—
172Q8×C31—42no—
173Q8×C31—82no—
174Q8×C31—82no—
175D4×C31—22no—
176D4×C31—24no—
177D4×C31C2×Q1614yescomplemented
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—22no—
181C2×C4×C31—22no—
182C2×C4×C31—24no—
183C2×C4×C31—42no—
184C2×C4×C31—42no—
185C2×C4×C31—42no—
186C2×C4×C31—44no—
187C2×C4×C31—42no—
188C2×C4×C31—42no—
189C2×C4×C31—44no—
190C2×C4×C31—42no—
191C2×C4×C31—88no—
192C2×C4×C31—82no—
193C2×C4×C31#12011yescharacteristic
194C2×C4×C31#12011yescharacteristic
195C2×C4×C31#12011yescharacteristic
196C2×C4×C31C2×Q1612yes—
197C2×C4×C31#13512yes—
198C2×C4×C31AGL(1,ℤ/8)12yes—
199C2×C4×C31#13712yes—
200C2×C4×C31C22×D411yescharacteristic
201C2×C4×C31C22×D411yescharacteristic
202C2×C4×C31C22×D411yescharacteristic
203C8×C31—42no—
204C8×C31—42no—
205C8×C31—42no—
206C2×#554C2×C3111yescharacteristic
207C2×#487C2×C3111yescharacteristic, complemented
208C2×#474C2×C3111yescharacteristic
209C2×#462C2×C3111yescharacteristic, complemented
210#1644C2×C3118yesdirect product factor
211C2×#333C2×C3111yescharacteristic, complemented
212C2×Q8⋊C8C2×C3111yescharacteristic, complemented
213C2×D4⋊C8C2×C3111yescharacteristic
214C22×C31—24no—
215C22×C31—48no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31#44912yes—
219C22×C31#45112yes—
220C22×C31#45512yes—
221C22×C31C2×#12011yescharacteristic
222C4×C31—21no—
223C4×C31—21no—
224C4×C31—42no—
225C4×C31—42no—
226C4×C31—42no—
227C4×C31—44no—
228C4×C31—82no—
229C4×C31#44912yes—
230C4×C31#45112yes—
231C4×C31C22×Q1611yescharacteristic
232C4×C31C2×#13511yescharacteristic
233C4×C31C2×AGL(1,ℤ/8)11yescharacteristic
234C4×C31C2×#13711yescharacteristic
235C22×Q16—21no—
236#554—22no—
237#554—22no—
238#554—22no—
239#554C22×C3114yes—
240C2×C4⋊Q8C22×C3111yescharacteristic
241C2×C4⋊Q8C22×C3111yescharacteristic
242C2×#125—21no—
243C2×C22⋊Q8—21no—
244C2×C22⋊Q8—22no—
245C2×C4×Q8C22×C3111yescharacteristic
246C2×C4×D4—21no—
247C2×C4×D4C22×C3111yescharacteristic
248C22×C4⋊C4—21no—
249#487—22no—
250#487C22×C3114yescomplemented
251#474—22no—
252#474C22×C3114yes—
253#462—22no—
254#462C22×C3114yescomplemented
255C2×#107—21no—
256C2×C4⋊C8C22×C3111yescharacteristic
257C2×C4⋊C8C22×C3111yescharacteristic
258C2×#103—22no—
259C2×#103—21no—
260C2×#102—21no—
261C2×C4×C8C22×C3111yescharacteristic
262#333C22×C3114yescomplemented
263Q8⋊C8C22×C3114yescomplemented
264D4⋊C8C22×C3114yes—
265C2×C31—44no—
266C2×C31#164414yesdirect product factor
267C2×C31C2×#45111yescharacteristic
268C2×C31C2×#44911yescharacteristic
269C2×C31C2×#45511yescharacteristic
270C22×Q8—21no—
271C22×Q8D4×C3111yescharacteristic
272C22×Q8D4×C3112yes—
273C22×D4D4×C3111yescharacteristic
274C23×C4—22no—
275C23×C4—21no—
276C2×Q16—24no—
277C2×Q16—44no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8D4×C3112yes—
287C4⋊Q8C23×C3112yes—
288C4⋊Q8C23×C3112yes—
289#125—24no—
290#125—24no—
291C22⋊Q8—24no—
292C22⋊Q8—28no—
293C22⋊Q8—24no—
294C22⋊Q8—28no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8D4×C3112yes—
298C4×Q8C23×C3112yes—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4D4×C3112yes—
304C4×D4C23×C3112yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4D4×C3111yescharacteristic
315C2×C4⋊C4D4×C3111yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C42—21no—
319C2×C42C23×C3111yescharacteristic
320#107—24no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8C23×C3112yes—
324C4⋊C8C23×C3112yes—
325#103—28no—
326#103—24no—
327#102—24no—
328C4×C8—21no—
329C4×C8C23×C3112yes—
330C31C2×#164411yescharacteristic, direct product factor, Sylow
331C24—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—44no—
338C2×Q8—41no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8C2×D4×C3112yes—
342C2×Q8C2×D4×C3114yes—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4Q16×C3114yescomplemented
346C2×D4C2×D4×C3112yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—84no—
358C22×C4C2×D4×C3111yescharacteristic
359C22×C4C2×D4×C3111yescharacteristic
360C22×C4C2×D4×C3111yescharacteristic
361Q16—48no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4C2×D4×C3112yes—
383C4⋊C4C2×D4×C3112yes—
384C22⋊C4—28no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—22no—
388C42—22no—
389C42C2×D4×C3111yescharacteristic
390C42C2×D4×C3111yescharacteristic
391C42C2×D4×C3111yescharacteristic
392C42C24×C3111yescharacteristic
393C23—24no—
394C23—22no—
395C23—42no—
396C23—42no—
397C23C31×#12011yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—44no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8—82no—
405Q8—82no—
406D4—24no—
407D4—22no—
408D4C2×Q16×C3114yescomplemented
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—88no—
423C2×C4—82no—
424C2×C4C31×#12011yescharacteristic
425C2×C4C31×#12011yescharacteristic
426C2×C4C31×#12011yescharacteristic
427C2×C4C2×Q16×C3112yes—
428C2×C4C31×#13512yes—
429C2×C4C31×AGL(1,ℤ/8)12yes—
430C2×C4C31×#13712yes—
431C2×C4C22×D4×C3111yescharacteristic
432C2×C4C22×D4×C3111yescharacteristic
433C2×C4C22×D4×C3111yescharacteristic
434C8—42no—
435C8—42no—
436C8—42no—
437C22—24no—
438C22—48no—
439C22—42no—
440C22—42no—
441C22C31×#44912yes—
442C22C31×#45112yes—
443C22C31×#45512yes—
444C22C2×C31×#12011yescharacteristic
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—44no—
451C4—82no—
452C4C31×#44912yes—
453C4C31×#45112yes—
454C4C22×Q16×C3111yescharacteristic
455C4C2×C31×#13511yescharacteristic
456C4C2×C31×AGL(1,ℤ/8)11yescharacteristic
457C4C2×C31×#13711yescharacteristic
458C2—44no—
459C2C31×#164414yesdirect product factor
460C2C2×C31×#45111yescharacteristic
461C2C2×C31×#44911yescharacteristic
462C2C2×C31×#45511yescharacteristic
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×#487C2semidirect
4. C2×C31×#462C2semidirect
5. C31×#1644C2direct (N × H)
6. C2×C31×#333C2semidirect
7. C2×C31×Q8⋊C8C2semidirect
24. C31×#487C22semidirect
28. C31×#462C22semidirect
36. C31×#333C22semidirect
37. C31×Q8⋊C8C22semidirect
113. C2×D4×C31Q16semidirect
161. C2×#1644C31direct (N × H)
177. D4×C31C2×Q16semidirect
207. C2×#487C2×C31semidirect
209. C2×#462C2×C31semidirect
210. #1644C2×C31direct (N × H)
211. C2×#333C2×C31semidirect
212. C2×Q8⋊C8C2×C31semidirect
250. #487C22×C31semidirect
254. #462C22×C31semidirect
262. #333C22×C31semidirect
263. Q8⋊C8C22×C31semidirect
266. C2×C31#1644direct (N × H)
330. C31C2×#1644direct (N × H)
345. C2×D4Q16×C31semidirect
408. D4C2×Q16×C31semidirect
459. C2C31×#1644direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5542
C2×C31×#4872
C2×C31×#4742
C2×C31×#4622
C31×#16442
C2×C31×#3332
C2×C31×Q8⋊C82
C2×C31×D4⋊C82
C2×#164431