Finite Group Databasebeta

← C2×#24913298 #35571470

Subgroups — C2×#24913298 #35571470

Order 21248.

2614 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581783C211yesmaximal, characteristic, complemented
2C22×#14581749C211yesmaximal, characteristic
3#24913298C2116yesmaximal, direct product factor
4C2×#19127635C212yesmaximal, complemented
5C2×#19127632C212yesmaximal, complemented
6C2×#19127625C212yesmaximal
7C2×#19127623C212yesmaximal
8C2×#19127599C211yesmaximal, characteristic, complemented
9C2×#19127570C211yesmaximal, characteristic, complemented
10C2×#19127558C211yesmaximal, characteristic
11C2×#19127558C211yesmaximal, characteristic
12C2×#389×C83C211yesmaximal, characteristic, complemented
13C23×C4×D83C2211yescharacteristic
14C2×#14581783—28no—
15C2×#14581783C2218yescomplemented
16C2×#14581749—28no—
17C2×#14581749C2218yes—
18C22×C4⋊Dic83C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20C22×C4⋊Dic83C2212yes—
21C22×Dic83⋊C4C2211yescharacteristic
22C22×Dic83⋊C4C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2212yes—
26C22×Dic83⋊C4C2212yes—
27C22×(C2×D83)⋊C4C2211yescharacteristic
28C22×(C2×D83)⋊C4C2211yescharacteristic
29C22×(C2×D83)⋊C4C2212yes—
30C22×(C2×D83)⋊C4C2212yes—
31C22×C4×Dic83C2211yescharacteristic
32#19127635C22116yescomplemented
33#19127632C22116yescomplemented
34#19127625C22116yes—
35#19127623C22116yes—
36#19127599C2218yescomplemented
37#19127570C2218yescomplemented
38#19127558C2218yes—
39#19127558C2218yes—
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2212yes—
42C2×C83⋊#95C2212yes—
43C22×C4⋊C4×C83C2211yescharacteristic
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2212yes—
46C22×C42×C83C2211yescharacteristic
47#389×C83C2218yescomplemented
48C2×#95×C83C2212yes—
49C24×D83C2311yescharacteristic
50C23×Dic83C2312yes—
51C23×Dic83C2312yes—
52C23×Dic83C2311yescharacteristic
53C23×Dic83C2311yescharacteristic
54C23×Dic83C2311yescharacteristic
55C22×C4×D83—24no—
56C22×C4×D83—24no—
57C22×C4×D83Q818yes—
58C22×C4×D83C2314yes—
59#14581783—232no—
60#14581749—232no—
61C2×C4⋊Dic83—28no—
62C2×C4⋊Dic83—24no—
63C2×C4⋊Dic83—24no—
64C2×C4⋊Dic83C2314yes—
65C2×C4⋊Dic83C2314yes—
66C2×C4⋊Dic83C2318yes—
67C2×Dic83⋊C4—24no—
68C2×Dic83⋊C4—24no—
69C2×Dic83⋊C4—28no—
70C2×Dic83⋊C4—44no—
71C2×Dic83⋊C4—44no—
72C2×Dic83⋊C4—44no—
73C2×Dic83⋊C4C2318yes—
74C2×Dic83⋊C4C2314yes—
75C2×Dic83⋊C4C2314yes—
76C2×Dic83⋊C4C2318yes—
77C2×Dic83⋊C4C2318yes—
78C2×Dic83⋊C4C2318yes—
79C2×(C2×D83)⋊C4—24no—
80C2×(C2×D83)⋊C4—24no—
81C2×(C2×D83)⋊C4—44no—
82C2×(C2×D83)⋊C4—44no—
83C2×(C2×D83)⋊C4C2314yes—
84C2×(C2×D83)⋊C4C2314yes—
85C2×(C2×D83)⋊C4C2318yes—
86C2×(C2×D83)⋊C4C2318yes—
87C2×C4×Dic83D418yes—
88C2×C4×Dic83C2314yes—
89C83⋊#95C2318yes—
90C83⋊#95C2318yes—
91C83⋊#95C2318yes—
92C23×C4×C83C2311yescharacteristic
93C23×C4×C83C2311yescharacteristic
94C23×C4×C83C2312yes—
95C23×C4×C83C2312yes—
96C23×C4×C83C2311yescharacteristic
97C2×C4⋊C4×C83—24no—
98C2×C4⋊C4×C83—24no—
99C2×C4⋊C4×C83—44no—
100C2×C4⋊C4×C83C2314yes—
101C2×C4⋊C4×C83C2314yes—
102C2×C4⋊C4×C83C2318yes—
103C2×C42×C83—24no—
104C2×C42×C83C2314yes—
105#95×C83C2318yes—
106C23×D83—22no—
107C23×D83—22no—
108C23×D83—22no—
109C23×D83—22no—
110C23×D83—22no—
111C23×D83—22no—
112C23×D83C2×Q814yes—
113C23×D83C2412yes—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—24no—
121C22×Dic83—24no—
122C22×Dic83—24no—
123C22×Dic83—24no—
124C22×Dic83—24no—
125C22×Dic83—24no—
126C22×Dic83C2×D414yes—
127C22×Dic83C2×D414yes—
128C22×Dic83C2×Q814yes—
129C22×Dic83C2414yes—
130C22×Dic83C2414yes—
131C22×Dic83C2412yes—
132C22×Dic83C2412yes—
133C22×Dic83C2412yes—
134C2×C4×D83—232no—
135C2×C4×D83—28no—
136C2×C4×D83—28no—
137C2×C4×D83C2×Q8116yes—
138C4⋊Dic83—28no—
139C4⋊Dic83—28no—
140C4⋊Dic83—216no—
141Dic83⋊C4—216no—
142Dic83⋊C4—28no—
143Dic83⋊C4—28no—
144Dic83⋊C4—48no—
145Dic83⋊C4—48no—
146Dic83⋊C4—48no—
147(C2×D83)⋊C4—28no—
148(C2×D83)⋊C4—28no—
149(C2×D83)⋊C4—48no—
150(C2×D83)⋊C4—48no—
151C4×Dic83C2×D4116yes—
152C24×C83C2411yescharacteristic
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—24no—
156C22×C4×C83—24no—
157C22×C4×C83—22no—
158C22×C4×C83—24no—
159C22×C4×C83—24no—
160C22×C4×C83—24no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—24no—
164C22×C4×C83—22no—
165C22×C4×C83C2×D414yes—
166C22×C4×C83C2×Q814yes—
167C22×C4×C83#4114yes—
168C22×C4×C83C2412yes—
169C22×C4×C83C2414yes—
170C22×C4×C83C2414yes—
171C22×C4×C83C2412yes—
172C22×C4×C83C2412yes—
173C4⋊C4×C83—28no—
174C4⋊C4×C83—28no—
175C4⋊C4×C83—48no—
176C42×C83—28no—
177C23×C83C22×D412yes—
178C23×C83C22×Q812yes—
179C23×C83C2×#4112yes—
180C23×C83H(5,𝔽2)12yes—
181C23×C83H(5,𝔽2)12yes—
182C23×C832−1+412yes—
183C23×C832−1+412yes—
184C23×C83C2511yescharacteristic
185C22×D83—28no—
186C22×D83—28no—
187C22×D83—28no—
188C22×D83—22no—
189C22×D83—22no—
190C22×D83—22no—
191C22×D83—22no—
192C22×D83—22no—
193C22×D83—22no—
194C22×D83—44no—
195C22×D83—44no—
196C22×D83—44no—
197C22×D83—44no—
198C22×D83C22×Q814yes—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—22no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—22no—
208C2×C4×C83—24no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—44no—
212C2×C4×C83—48no—
213C2×C4×C83—44no—
214C2×C4×C83—48no—
215C2×C4×C83C22⋊Q8116yes—
216C2×C4×C83C22×D414yes—
217C2×C4×C83C22×Q814yes—
218C2×C4×C83C2×#4114yes—
219C2×Dic83—28no—
220C2×Dic83—28no—
221C2×Dic83—28no—
222C2×Dic83—22no—
223C2×Dic83—22no—
224C2×Dic83—22no—
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—48no—
235C2×Dic83—48no—
236C2×Dic83C22×D414yes—
237C2×Dic83C22×D414yes—
238C2×Dic83C22×Q814yes—
239C4×D83—232no—
240C22×C83C2×C22⋊Q814yes—
241C22×C83#53714yes—
242C22×C83#53714yes—
243C22×C83#54014yes—
244C22×C83#54114yes—
245C22×C83#54414yes—
246C22×C83#54414yes—
247C22×C83C23×D411yescharacteristic
248C22×C83C23×Q811yescharacteristic
249C22×C83C22×#4111yescharacteristic
250C22×C83C2×H(5,𝔽2)11yescharacteristic
251C22×C83C2×H(5,𝔽2)11yescharacteristic
252C22×C83C2×2−1+411yescharacteristic
253C22×C83C2×2−1+411yescharacteristic
254C2×D83—24no—
255C2×D83—24no—
256C2×D83—24no—
257C2×D83—416no—
258C2×D83—42no—
259C2×D83—42no—
260C2×D83—42no—
261C2×D83—42no—
262C4×C83—42no—
263C4×C83—44no—
264C4×C83—42no—
265C4×C83—44no—
266C4×C83C2×C22⋊Q818yes—
267Dic83—24no—
268Dic83—24no—
269Dic83—24no—
270Dic83—44no—
271Dic83—44no—
272C2×#2629—831nomaximal, Sylow
273C2×C83#262918yescomplemented
274C2×C83C22×C22⋊Q811yescharacteristic
275C2×C83C2×#53711yescharacteristic
276C2×C83C2×#53711yescharacteristic
277C2×C83C2×#54011yescharacteristic
278C2×C83C2×#54111yescharacteristic
279C2×C83C2×#54411yescharacteristic
280C2×C83C2×#54411yescharacteristic
281D83—44no—
282C22×#125—831no—
283C22×#117—831no—
284#2629—8316no—
285C2×#398—832no—
286C2×#398—832no—
287C2×#396—832no—
288C2×#396—832no—
289C2×#389—831no—
290C2×#389D8311yescharacteristic, complemented
291C2×#384—831no—
292C2×#380—831no—
293C2×#380—831no—
294C83C2×#262911yescharacteristic, complemented, Sylow
295C24×C4—831no—
296C2×#125—838no—
297C2×#125—1668no—
298C2×#117—838no—
299C2×#117—1668no—
300C22×C4⋊C4—832no—
301C22×C4⋊C4—832no—
302C22×C4⋊C4—832no—
303C22×C4⋊C4—832no—
304C22×C4⋊C4—832no—
305C22×C4⋊C4—831no—
306C22×C4⋊C4—831no—
307C22×C4⋊C4—831no—
308C22×C4⋊C4—831no—
309C22×C4⋊C4C2×D8311yescharacteristic
310C22×C4⋊C4C2×D8311yescharacteristic
311C22×C4⋊C4C2×D8312yes—
312C22×C22⋊C4—832no—
313C22×C22⋊C4—832no—
314C22×C22⋊C4—831no—
315C22×C22⋊C4—831no—
316C22×C42—831no—
317C22×C42C2×D8311yescharacteristic
318#398—8316no—
319#398—8316no—
320#396—8316no—
321#396—8316no—
322#389—838no—
323#389C2×D8318yescomplemented
324#384—838no—
325#380—838no—
326#380—838no—
327C2×#95—832no—
328C2×#95—832no—
329C2×#95—832no—
330C2×#95C2×D8312yes—
331C25—831no—
332C23×C4—831no—
333C23×C4—831no—
334C23×C4—831no—
335C23×C4—838no—
336C23×C4—834no—
337C23×C4—832no—
338C23×C4—832no—
339C23×C4—1664no—
340C23×C4—1664no—
341C23×C4C22×D8311yescharacteristic
342C23×C4C22×D8312yes—
343C23×C4C22×D8312yes—
344C23×C4C22×D8311yescharacteristic
345C23×C4C22×D8311yescharacteristic
346#125—16632no—
347#117—16632no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—3324no—
352C2×C4⋊C4—3324no—
353C2×C4⋊C4—3324no—
354C2×C4⋊C4—838no—
355C2×C4⋊C4—838no—
356C2×C4⋊C4—838no—
357C2×C4⋊C4—834no—
358C2×C4⋊C4—834no—
359C2×C4⋊C4—834no—
360C2×C4⋊C4—838no—
361C2×C4⋊C4—838no—
362C2×C4⋊C4—834no—
363C2×C4⋊C4—1664no—
364C2×C4⋊C4—1664no—
365C2×C4⋊C4—1668no—
366C2×C4⋊C4—1668no—
367C2×C4⋊C4—1664no—
368C2×C4⋊C4—1664no—
369C2×C4⋊C4C22×D8314yes—
370C2×C4⋊C4C22×D8314yes—
371C2×C4⋊C4C22×D8318yes—
372C2×C22⋊C4—3324no—
373C2×C22⋊C4—3324no—
374C2×C22⋊C4—838no—
375C2×C22⋊C4—838no—
376C2×C22⋊C4—834no—
377C2×C22⋊C4—834no—
378C2×C22⋊C4—1664no—
379C2×C22⋊C4—1664no—
380C2×C42—24no—
381C2×C42—838no—
382C2×C42—834no—
383C2×C42C22×D8314yes—
384#95—838no—
385#95—838no—
386#95—838no—
387#95C22×D8318yes—
388C24—834no—
389C24—832no—
390C24—1662no—
391C24—1662no—
392C24—1662no—
393C24—1662no—
394C24—1662no—
395C24—1662no—
396C24C23×D8311yescharacteristic
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—834no—
410C22×C4—834no—
411C22×C4—834no—
412C22×C4—832no—
413C22×C4—832no—
414C22×C4—832no—
415C22×C4—8316no—
416C22×C4—834no—
417C22×C4—834no—
418C22×C4—1664no—
419C22×C4—1664no—
420C22×C4—1664no—
421C22×C4—1664no—
422C22×C4—1664no—
423C22×C4—1664no—
424C22×C4—1662no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4—16632no—
431C22×C4—1668no—
432C22×C4—1668no—
433C22×C4D4×D8314yes—
434C22×C4Q8×D8314yes—
435C22×C4#1305059014yes—
436C22×C4C23×D8312yes—
437C22×C4C23×D8314yes—
438C22×C4C23×D8314yes—
439C22×C4C23×D8312yes—
440C22×C4C23×D8312yes—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—48no—
444C4⋊C4—3328no—
445C4⋊C4—3328no—
446C4⋊C4—3328no—
447C4⋊C4—1668no—
448C4⋊C4—1668no—
449C4⋊C4—16616no—
450C4⋊C4—16616no—
451C4⋊C4—1668no—
452C4⋊C4—1668no—
453C22⋊C4—3328no—
454C22⋊C4—3328no—
455C22⋊C4—1668no—
456C22⋊C4—1668no—
457C42—28no—
458C42—8316no—
459C23—3324no—
460C23—3324no—
461C23—3324no—
462C23—3324no—
463C23—834no—
464C23—1668no—
465C23—1668no—
466C23—1668no—
467C23—1662no—
468C23—1662no—
469C23—1662no—
470C23—1662no—
471C23—1662no—
472C23—1662no—
473C23C2×D4×D8312yes—
474C23C2×Q8×D8312yes—
475C23C2×#1305059012yes—
476C23#1458183712yes—
477C23#1458183812yes—
478C23#1458183912yes—
479C23#1458184012yes—
480C23C24×D8311yescharacteristic
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—3328no—
498C2×C4—3328no—
499C2×C4—834no—
500C2×C4—834no—
501C2×C4—834no—
502C2×C4—1664no—
503C2×C4—1664no—
504C2×C4—1664no—
505C2×C4—1664no—
506C2×C4—1664no—
507C2×C4—1664no—
508C2×C4—1668no—
509C2×C4—1662no—
510C2×C4—1662no—
511C2×C4—1662no—
512C2×C4—1662no—
513C2×C4—1662no—
514C2×C4—1662no—
515C2×C4—1668no—
516C2×C4—1668no—
517C2×C4—16632no—
518C2×C4#14581770116yes—
519C2×C4C2×D4×D8314yes—
520C2×C4C2×Q8×D8314yes—
521C2×C4C2×#1305059014yes—
522C22—33216no—
523C22—3322no—
524C22—3322no—
525C22—3322no—
526C22—3322no—
527C22—1664no—
528C22—1664no—
529C22—1664no—
530C22C2×#1458177014yes—
531C22#1912828014yes—
532C22#1912828114yes—
533C22#1912830214yes—
534C22#1912830914yes—
535C22#1912832814yes—
536C22#1912833114yes—
537C22C22×D4×D8311yescharacteristic
538C22C22×Q8×D8311yescharacteristic
539C22C22×#1305059011yescharacteristic
540C22C2×#1458183711yescharacteristic
541C22C2×#1458183811yescharacteristic
542C22C2×#1458183911yescharacteristic
543C22C2×#1458184011yescharacteristic
544C4—42no—
545C4—44no—
546C4—42no—
547C4—44no—
548C4—3324no—
549C4—3324no—
550C4—1664no—
551C4—1664no—
552C4—1664no—
553C4C2×#1458177018yes—
554C2—3324no—
555C2#2491329818yesdirect product factor
556C2C22×#1458177011yescharacteristic
557C2C2×#1912828011yescharacteristic
558C2C2×#1912828111yescharacteristic
559C2C2×#1912830211yescharacteristic
560C2C2×#1912830911yescharacteristic
561C2C2×#1912832811yescharacteristic
562C2C2×#1912833111yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#389×C83C2
socleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×#389D83
hypocenterC83C2×#2629
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14581783C2semidirect
3. #24913298C2direct (N × H)
4. C2×#19127635C2semidirect
5. C2×#19127632C2semidirect
8. C2×#19127599C2semidirect
9. C2×#19127570C2semidirect
12. C2×#389×C83C2semidirect
15. C2×#14581783C22semidirect
32. #19127635C22semidirect
33. #19127632C22semidirect
36. #19127599C22semidirect
37. #19127570C22semidirect
47. #389×C83C22semidirect
273. C2×C83#2629semidirect
290. C2×#389D83semidirect
294. C83C2×#2629semidirect
323. #389C2×D83semidirect
555. C2#24913298direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145817832
C22×#145817492
#249132982
C2×#191276352
C2×#191276322
C2×#191276252
C2×#191276232
C2×#191275992
C2×#191275702
C2×#191275582
C2×#389×C832
C2×#262983