Finite Group Databasebeta

← C2×#24918798 #36277207

Subgroups — C2×#24918798 #36277207

Order 21248.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128256C211yesmaximal, characteristic, complemented
2C2×#19128238C211yesmaximal, characteristic
3#24918798C2116yesmaximal, direct product factor
4C2×#19128010C212yesmaximal, complemented
5C2×#19127986C211yesmaximal, characteristic, complemented
6C2×#19127982C211yesmaximal, characteristic, complemented
7C2×#19127939C212yesmaximal, complemented
8C2×#19127933C212yesmaximal
9C2×#19127925C212yesmaximal, complemented
10C2×#19127903C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D83C211yesmaximal, characteristic, complemented
12C2×#462×C83C211yesmaximal, characteristic, complemented
13C22×#13050577C2212yes—
14C22×C8×D83C2212yes—
15#19128256—24no—
16#19128256C2218yescomplemented
17C2×#14581792C2211yescharacteristic
18#19128238—24no—
19#19128238C2218yes—
20C2×#14581787—21no—
21C2×#14581783—21no—
22C2×#14581783C2211yescharacteristic
23C2×#14581781C2211yescharacteristic
24C2×#14581777—21no—
25C2×#14581772—21no—
26C2×#14581768C2212yes—
27C2×#14581768C2211yescharacteristic
28C2×#14581767C2211yescharacteristic
29C2×Q8×Dic83C2212yes—
30C2×#14581757—21no—
31C2×C4×D332—21no—
32C2×C4×D332C2211yescharacteristic
33C2×#14581750—21no—
34C2×#14581749—21no—
35C2×C42×D83C2211yescharacteristic
36#19128010C22116yescomplemented
37#19127986C2218yescomplemented
38#19127982C2218yescomplemented
39#19127939C22116yescomplemented
40#19127933C22116yes—
41#19127925C22116yescomplemented
42#19127903C2218yescomplemented
43C2×#14581726C2212yes—
44C2×#14581723C2212yes—
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46C2×Dic83⋊C8C2212yes—
47C4⋊C8×D83C2218yescomplemented
48C2×#14581716C2212yes—
49C2×#14581716C2212yes—
50C2×#14581714C2212yes—
51C22×Q16×C83C2212yes—
52C2×C4⋊Q8×C83C2211yescharacteristic
53C2×C4×Q8×C83C2211yescharacteristic
54#462×C83C2218yescomplemented
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×#103×C83C2212yes—
57C22×Dic166—21no—
58C22×D332—21no—
59C22×D332C2311yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83D411yescharacteristic
62C22×C4×D83D411yescharacteristic
63C22×C4×D83C2311yescharacteristic
64C2×#13050577—44no—
65C2×#13050577C2318yes—
66C22×C83⋊C8C2312yes—
67C2×C8×D83—44no—
68C2×C8×D83C2318yes—
69#14581792—22no—
70#14581792C2314yes—
71#14581787—28no—
72#14581783—28no—
73#14581783—22no—
74#14581783C2314yes—
75#14581781—22no—
76#14581781C2314yes—
77#14581777—28no—
78#14581772—28no—
79#14581768—22no—
80#14581768—24no—
81#14581768C2318yes—
82#14581768C2314yes—
83#14581767—22no—
84#14581767C2314yes—
85Q8×Dic83—24no—
86Q8×Dic83C2318yes—
87#14581757—28no—
88C4×D332—28no—
89C4×D332—22no—
90C4×D332C2314yes—
91#14581750—28no—
92#14581749—28no—
93C2×C4⋊Dic83—21no—
94C2×C4⋊Dic83—22no—
95C2×C4⋊Dic83—21no—
96C2×C4⋊Dic83C2311yescharacteristic
97C2×C4⋊Dic83C2312yes—
98C2×Dic83⋊C4—21no—
99C2×Dic83⋊C4—21no—
100C2×Dic83⋊C4—22no—
101C2×(C2×D83)⋊C4—22no—
102C2×(C2×D83)⋊C4—21no—
103C2×(C2×D83)⋊C4—21no—
104C2×(C2×D83)⋊C4—21no—
105C2×(C2×D83)⋊C4—21no—
106C2×(C2×D83)⋊C4—21no—
107C2×C4×Dic83—22no—
108C2×C4×Dic83D411yescharacteristic
109C2×C4×Dic83D411yescharacteristic
110C2×C4×Dic83C2311yescharacteristic
111C42×D83D414yes—
112C42×D83C2314yes—
113#14581726C2318yes—
114#14581723C2318yes—
115C4⋊(C83⋊C8)C2314yes—
116Dic83⋊C8C2318yes—
117#14581716C2318yes—
118#14581716C2318yes—
119#14581714C2318yes—
120C22×Q8×C83C2311yescharacteristic
121C22×Q8×C83C2312yes—
122C2×Q16×C83—44no—
123C2×Q16×C83C2318yes—
124C22×C8×C83C2312yes—
125C4⋊Q8×C83—22no—
126C4⋊Q8×C83C2314yes—
127C4×Q8×C83—22no—
128C4×Q8×C83C2314yes—
129C2×C4⋊C4×C83—21no—
130C2×C4⋊C4×C83—21no—
131C2×C4⋊C4×C83—21no—
132C2×C4⋊C4×C83C2311yescharacteristic
133C2×C42×C83—21no—
134C2×C42×C83C2311yescharacteristic
135C4⋊C8×C83C2314yes—
136#103×C83C2318yes—
137C23×D83—21no—
138C23×D83C2×D411yescharacteristic
139C2×Dic166—24no—
140C2×Dic166—44no—
141C2×D332—22no—
142C2×D332—22no—
143C2×D332—24no—
144C2×D332—21no—
145C2×D332—44no—
146C2×D332C2412yes—
147C22×Dic83—21no—
148C22×Dic83—22no—
149C22×Dic83C2×D411yescharacteristic
150C22×Dic83C2×D411yescharacteristic
151C22×Dic83C2×D411yescharacteristic
152C2×C4×D83—24no—
153C2×C4×D83—24no—
154C2×C4×D83—22no—
155C2×C4×D83—22no—
156C2×C4×D83—24no—
157C2×C4×D83—44no—
158C2×C4×D83C2×D412yes—
159C2×C4×D83C2×D414yes—
160C2×C4×D83C2×D414yes—
161C2×C4×D83C2412yes—
162#13050577—48no—
163C2×C83⋊C8—24no—
164C2×C83⋊C8C2414yes—
165C8×D83—48no—
166C4⋊Dic83—21no—
167C4⋊Dic83—22no—
168C4⋊Dic83—24no—
169C4⋊Dic83—28no—
170C4⋊Dic83—24no—
171C4⋊Dic83C2412yes—
172C4⋊Dic83C2414yes—
173Dic83⋊C4—28no—
174Dic83⋊C4—24no—
175Dic83⋊C4—24no—
176(C2×D83)⋊C4—28no—
177(C2×D83)⋊C4—24no—
178(C2×D83)⋊C4—24no—
179(C2×D83)⋊C4—24no—
180(C2×D83)⋊C4—24no—
181(C2×D83)⋊C4—24no—
182C4×Dic83—28no—
183C4×Dic83C2×D412yes—
184C4×Dic83C2×D414yes—
185C4×Dic83C2×D414yes—
186C4×Dic83C2412yes—
187C2×Q8×C83—21no—
188C2×Q8×C83—22no—
189C2×Q8×C83—24no—
190C2×Q8×C83—22no—
191C2×Q8×C83—22no—
192C2×Q8×C83—24no—
193C2×Q8×C83C2412yes—
194C2×Q8×C83C2414yes—
195C22×C4×C83—21no—
196C22×C4×C83—21no—
197C22×C4×C83—22no—
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83C2×D411yescharacteristic
200C22×C4×C83C2411yescharacteristic
201Q16×C83—48no—
202C2×C8×C83—24no—
203C2×C8×C83C2414yes—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—24no—
206C4⋊C4×C83—24no—
207C4⋊C4×C83—24no—
208C4⋊C4×C83C2412yes—
209C42×C83—24no—
210C42×C83C2×D412yes—
211C42×C83C2412yes—
212C23×C83C22×D411yescharacteristic
213C22×D83—22no—
214C22×D83—22no—
215C22×D83—22no—
216C22×D83—42no—
217C22×D83—42no—
218C22×D83—42no—
219C22×D83#13714yes—
220C22×D83C22×D412yes—
221Q8×C83—22no—
222Q8×C83—24no—
223Q8×C83—22no—
224Q8×C83—24no—
225Q8×C83—42no—
226Q8×C83—44no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—24no—
231C2×C4×C83—42no—
232C2×C4×C83—44no—
233C2×C4×C83—42no—
234C2×C4×C83#13514yes—
235C2×C4×C83C22×D412yes—
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C22×D412yes—
238C2×C4×C83H(5,𝔽2)12yes—
239C2×C4×C832−1+412yes—
240C2×C4×C83C2511yescharacteristic
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—24no—
245C2×Dic83—22no—
246C2×Dic83—42no—
247C2×Dic83—44no—
248C2×Dic83C22×D412yes—
249C2×Dic83C22×D412yes—
250C2×Dic83C22×D412yes—
251D332—22no—
252D332—22no—
253D332—48no—
254D332—42no—
255C4×D83—28no—
256C4×D83—24no—
257C4×D83—28no—
258C4×D83—22no—
259C4×D83—22no—
260C4×D83—48no—
261Dic166—48no—
262C8×C83—24no—
263C83⋊C8—24no—
264C22×C83#53712yes—
265C22×C83C2×#13512yes—
266C22×C83C2×#13712yes—
267C22×C83C23×D411yescharacteristic
268C2×D83—22no—
269C2×D83—28no—
270C2×D83—22no—
271C2×D83—42no—
272C2×D83—42no—
273C2×D83—42no—
274C2×D83—84no—
275C2×D83C2×#13714yes—
276C4×C83—22no—
277C4×C83—42no—
278C4×C83—44no—
279C4×C83—42no—
280C4×C83#53712yes—
281C4×C83C2×#13514yes—
282C4×C83C2×H(5,𝔽2)11yescharacteristic
283C4×C83C2×2−1+411yescharacteristic
284Dic83—22no—
285Dic83—22no—
286Dic83—22no—
287Dic83—44no—
288Dic83—42no—
289C2×#3248—831nomaximal, Sylow
290C2×C83#324814yescomplemented
291C2×C83C2×#53711yescharacteristic
292C2×C83C22×#13511yescharacteristic
293C2×C83C22×#13711yescharacteristic
294D83—24no—
295D83—82no—
296C2×#533—831no—
297C2×#529—831no—
298#3248—8316no—
299C2×#479—832no—
300C2×#475—831no—
301C2×#474—831no—
302C2×#467—832no—
303C2×#465—832no—
304C2×#464—832no—
305C2×#462D8311yescharacteristic, complemented
306C2×#461—831no—
307C22×C4⋊C8—831no—
308C83C2×#324811yescharacteristic, complemented, Sylow
309C22×Q16C2×D8312yes—
310C22×SD16—832no—
311C23×C8—832no—
312#533—838no—
313#533—1664no—
314C2×C4⋊Q8—831no—
315C2×C4⋊Q8C2×D8311yescharacteristic
316#529—838no—
317#529—1664no—
318C2×#126—1661no—
319C2×#125—831no—
320C2×#125—1661no—
321C2×#124—831no—
322C2×#123—1661no—
323C2×C22⋊Q8—831no—
324C2×C22⋊Q8—832no—
325C2×C22⋊Q8—1661no—
326C2×#121—831no—
327C2×C4×Q8—832no—
328C2×C4×Q8—1661no—
329C2×C4×Q8C2×D8311yescharacteristic
330C2×C4×D4—831no—
331C2×C4×D4—1661no—
332C2×#117—1661no—
333C2×#117—1661no—
334C22×C42—831no—
335#479—8316no—
336#475—838no—
337#474—838no—
338#467—8316no—
339#465—8316no—
340#464—8316no—
341#462C2×D8318yescomplemented
342#461—838no—
343C2×#107—832no—
344C2×#106—832no—
345C2×C4⋊C8—832no—
346C2×C4⋊C8—831no—
347C2×C4⋊C8—838no—
348C2×C4⋊C8C2×D8311yescharacteristic
349C2×#103—832no—
350C2×#103—832no—
351C2×#103C2×D8312yes—
352C2×#102—832no—
353C22×Q8—1661no—
354C22×Q8C22×D8312yes—
355C22×Q8C22×D8311yescharacteristic
356C22×D4—831no—
357C22×D4—1661no—
358C23×C4—831no—
359C23×C4—831no—
360C23×C4—831no—
361C23×C4—1661no—
362C2×Q16—44no—
363C2×Q16C22×D8318yes—
364C2×SD16—3324no—
365C2×SD16—838no—
366C22×C8—3324no—
367C22×C8—838no—
368C22×C8—832no—
369C22×C8C22×D8312yes—
370C4⋊Q8—22no—
371C4⋊Q8—834no—
372C4⋊Q8—1662no—
373C4⋊Q8C22×D8314yes—
374#126—1668no—
375#125—834no—
376#125—1662no—
377#125—1668no—
378#124—834no—
379#124—1662no—
380#123—1668no—
381C22⋊Q8—834no—
382C22⋊Q8—838no—
383C22⋊Q8—1668no—
384C22⋊Q8—1662no—
385C22⋊Q8—1664no—
386#121—834no—
387#121—1662no—
388C4×Q8—22no—
389C4×Q8—838no—
390C4×Q8—1668no—
391C4×Q8—1664no—
392C4×Q8C22×D8314yes—
393C4×D4—834no—
394C4×D4—1668no—
395C4×D4—1662no—
396#117—1668no—
397#117—1668no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—832no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1662no—
407C2×C4⋊C4—1662no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4—1662no—
414C2×C22⋊C4—1661no—
415C2×C22⋊C4—1661no—
416C2×C42—21no—
417C2×C42—831no—
418C2×C42—831no—
419C2×C42—831no—
420C2×C42—834no—
421C2×C42—834no—
422C2×C42—1662no—
423C2×C42C22×D8311yescharacteristic
424#107—838no—
425#106—838no—
426C4⋊C8—838no—
427C4⋊C8—834no—
428C4⋊C8C22×D8314yes—
429#103—838no—
430#103—838no—
431#103C22×D8318yes—
432#102—838no—
433C24—831no—
434C24—1661no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—3324no—
442C2×Q8—1664no—
443C2×Q8C23×D8312yes—
444C2×Q8C23×D8314yes—
445C2×D4—3324no—
446C2×D4—832no—
447C2×D4—1664no—
448C2×D4—1661no—
449C2×D4—1662no—
450C2×D4—1662no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—3324no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—834no—
459C22×C4—834no—
460C22×C4—832no—
461C22×C4—832no—
462C22×C4—1664no—
463C22×C4—1662no—
464C22×C4—1661no—
465C22×C4—1664no—
466C22×C4—1664no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4D4×D8311yescharacteristic
470C22×C4D4×D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472Q16—48no—
473SD16—3328no—
474C2×C8—24no—
475C2×C8—3328no—
476C2×C8—834no—
477C2×C8—1664no—
478C2×C8C23×D8314yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—834no—
484C4⋊C4—832no—
485C4⋊C4—1664no—
486C4⋊C4—1664no—
487C4⋊C4—1664no—
488C4⋊C4—1668no—
489C4⋊C4—1668no—
490C4⋊C4—1661no—
491C4⋊C4—1662no—
492C4⋊C4—1664no—
493C4⋊C4C23×D8312yes—
494C22⋊C4—1664no—
495C22⋊C4—1664no—
496C22⋊C4—1664no—
497C22⋊C4—1668no—
498C22⋊C4—1664no—
499C22⋊C4—1664no—
500C42—24no—
501C42—834no—
502C42—832no—
503C42—834no—
504C42—832no—
505C42—1668no—
506C42D4×D8312yes—
507C42C23×D8312yes—
508C23—3322no—
509C23—3322no—
510C23—3322no—
511C23—834no—
512C23—832no—
513C23—1662no—
514C23—1662no—
515C23—1662no—
516C23C2×D4×D8311yescharacteristic
517Q8—24no—
518Q8—22no—
519Q8—22no—
520Q8—24no—
521Q8—42no—
522Q8—44no—
523Q8—3328no—
524D4—3328no—
525D4—3322no—
526D4—1662no—
527D4—1662no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—3322no—
536C2×C4—3324no—
537C2×C4—3328no—
538C2×C4—832no—
539C2×C4—832no—
540C2×C4—832no—
541C2×C4—1668no—
542C2×C4—1664no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4—1662no—
547C2×C4—1668no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1664no—
551C2×C4#1458181514yes—
552C2×C4C2×D4×D8312yes—
553C2×C4C2×D4×D8312yes—
554C2×C4C2×D4×D8311yescharacteristic
555C2×C4#1458183712yes—
556C2×C4#1458183912yes—
557C2×C4C24×D8311yescharacteristic
558C8—24no—
559C8—1664no—
560C22—3322no—
561C22—3322no—
562C22—3322no—
563C22—834no—
564C22—6644no—
565C22—1668no—
566C22—1662no—
567C22—1662no—
568C22#1912828012yes—
569C22C2×#1458181512yes—
570C22#137×D8312yes—
571C22C22×D4×D8311yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—3322no—
577C4—3324no—
578C4—1662no—
579C4—1662no—
580C4—1662no—
581C4#1912828012yes—
582C4C2×#1458181514yes—
583C4C2×#1458183711yescharacteristic
584C4C2×#1458183911yescharacteristic
585C2—6642no—
586C2—1664no—
587C2#2491879814yesdirect product factor
588C2C2×#1912828011yescharacteristic
589C2C22×#1458181511yescharacteristic
590C2C2×#137×D8311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC2×#462×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581767C22
hypercenterC2×#462D83
hypocenterC83C2×#3248
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×#19128256C2semidirect
3. #24918798C2direct (N × H)
4. C2×#19128010C2semidirect
5. C2×#19127986C2semidirect
6. C2×#19127982C2semidirect
7. C2×#19127939C2semidirect
9. C2×#19127925C2semidirect
10. C2×#19127903C2semidirect
11. C2×C4⋊C8×D83C2semidirect
12. C2×#462×C83C2semidirect
16. #19128256C22semidirect
36. #19128010C22semidirect
37. #19127986C22semidirect
38. #19127982C22semidirect
39. #19127939C22semidirect
41. #19127925C22semidirect
42. #19127903C22semidirect
47. C4⋊C8×D83C22semidirect
54. #462×C83C22semidirect
290. C2×C83#3248semidirect
305. C2×#462D83semidirect
308. C83C2×#3248semidirect
341. #462C2×D83semidirect
587. C2#24918798direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191282562
C2×#191282382
#249187982
C2×#191280102
C2×#191279862
C2×#191279822
C2×#191279392
C2×#191279332
C2×#191279252
C2×#191279032
C2×C4⋊C8×D832
C2×#462×C832
C2×#324883