Finite Group Databasebeta

← C2×#22624388 #33993756

Subgroups — C2×#22624388 #33993756

Order 18688.

6566 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624388C2132yesmaximal, direct product factor
2C2×#17989341C212yesmaximal, complemented
3C2×#17989335C211yesmaximal, characteristic, complemented
4C2×#17989293C214yesmaximal, complemented
5C2×#17989290C214yesmaximal, complemented
6C2×#17989275C214yesmaximal, complemented
7C2×#17989196C214yesmaximal, complemented
8C2×#17989185C214yesmaximal, complemented
9C2×#17989185C214yesmaximal, complemented
10C2×#17989122C212yesmaximal, complemented
11C2×#125×D73C211yesmaximal, characteristic, complemented
12C2×#564×C73C211yesmaximal, characteristic, complemented
13#17989341C22132yescomplemented
14#17989335C22116yescomplemented
15#17989293C22164yescomplemented
16#17989290C22164yescomplemented
17#17989275C22164yescomplemented
18#17989196C22164yescomplemented
19#17989185C22164yescomplemented
20#17989185C22164yescomplemented
21C2×#14556590C2214yes—
22#17989122C22132yescomplemented
23C2×#14556586C2214yes—
24C2×#14556586C2214yes—
25C2×#14556586C2214yes—
26C2×#14556586C2214yes—
27C2×#14556585C2214yes—
28C2×#14556583C2211yescharacteristic
29C2×#14556582C2214yes—
30C2×#14556582C2212yes—
31C2×#14556581C2214yes—
32C2×#14556581C2214yes—
33#125×D73C22116yescomplemented
34C2×#14556579C2212yes—
35C2×#14556576C2214yes—
36C2×#14556576C2214yes—
37C2×#14556576C2214yes—
38C2×#14556576C2214yes—
39C2×#14556576C2214yes—
40C2×#14556571C2214yes—
41C2×#14556571C2214yes—
42C2×#14556571C2212yes—
43C2×#14556569C2214yes—
44C2×#14556569C2214yes—
45C2×#14556569C2214yes—
46C2×#14556569C2214yes—
47C2×#14556566C2214yes—
48C2×#14556566C2214yes—
49C2×#14556566C2212yes—
50C2×C4×Dic146C2214yes—
51C2×#14556556C2212yes—
52C2×C4×D292C2214yes—
53C2×C4×D292C2214yes—
54C2×#14556553C2212yes—
55C2×#14556553C2214yes—
56C2×#14556553C2214yes—
57C2×#14556549C2214yes—
58C2×#14556549C2214yes—
59C2×C4⋊C4×D73C2212yes—
60C2×C4⋊C4×D73C2214yes—
61C2×C4⋊C4×D73C2214yes—
62C2×C42×D73C2211yescharacteristic
63#564×C73C22116yescomplemented
64C2×#125×C73C2214yes—
65C2×#125×C73C2214yes—
66C2×#125×C73C2211yescharacteristic
67C2×#125×C73C2214yes—
68C2×#125×C73C2212yes—
69C22×Dic146C2312yes—
70C22×Dic146C2314yes—
71C22×D292C2314yes—
72C22×D292C2312yes—
73C22×D292C2314yes—
74C22×C4×D73C2311yescharacteristic
75C22×C4×D73C2312yes—
76C22×C4×D73C2314yes—
77C22×C4×D73C2314yes—
78C22×C4×D73C2314yes—
79C22×C4×D73C2314yes—
80#14556590C23132yes—
81#14556586C23132yes—
82#14556586C23132yes—
83#14556586C23132yes—
84#14556586C23132yes—
85#14556585C23132yes—
86#14556583C2318yes—
87#14556582C23132yes—
88#14556582C23116yes—
89#14556581C23132yes—
90#14556581C23132yes—
91#14556579C23116yes—
92#14556576C23132yes—
93#14556576C23132yes—
94#14556576C23132yes—
95#14556576C23132yes—
96#14556576C23132yes—
97#14556571C23116yes—
98#14556571C23132yes—
99#14556571C23132yes—
100#14556569C23132yes—
101#14556569C23132yes—
102#14556569C23132yes—
103#14556569C23132yes—
104#14556566C23116yes—
105#14556566C23132yes—
106#14556566C23132yes—
107C4×Dic146C23132yes—
108#14556556C23116yes—
109C4×D292C23132yes—
110C4×D292C23132yes—
111#14556553C23116yes—
112#14556553C23132yes—
113#14556553C23132yes—
114#14556549C23132yes—
115#14556549C23132yes—
116C2×C4⋊Dic73C2314yes—
117C2×C4⋊Dic73C2314yes—
118C2×C4⋊Dic73C2314yes—
119C2×C4⋊Dic73C2312yes—
120C2×#11924007C2314yes—
121C2×#11924007C2314yes—
122C2×#11924007C2314yes—
123C2×#11924007C2314yes—
124C2×#11924007C2314yes—
125C2×#11924007C2314yes—
126C4⋊C4×D73C23116yes—
127C4⋊C4×D73C23132yes—
128C4⋊C4×D73C23132yes—
129C2×#11924005C2314yes—
130C2×#11924005C2314yes—
131C2×#11924005C2314yes—
132C2×#11924005C2314yes—
133C2×#11924005C2314yes—
134C2×#11924005C2314yes—
135C2×#11924005C2314yes—
136C2×#11924005C2314yes—
137C2×#11924005C2314yes—
138C2×C4×Dic73C2314yes—
139C2×C4×Dic73C2311yescharacteristic
140C2×C4×Dic73C2312yes—
141C2×C4×Dic73C2314yes—
142C42×D73C2318yes—
143#125×C73C23116yes—
144#125×C73C23132yes—
145#125×C73C23132yes—
146#125×C73C23132yes—
147#125×C73C2318yes—
148C2×C4⋊C4×C73C2314yes—
149C2×C4⋊C4×C73C2314yes—
150C2×C4⋊C4×C73C2314yes—
151C2×C4⋊C4×C73C2312yes—
152C2×C4⋊C4×C73C2314yes—
153C2×C4⋊C4×C73C2314yes—
154C2×C4⋊C4×C73C2312yes—
155C2×C4⋊C4×C73C2314yes—
156C2×C4⋊C4×C73C2312yes—
157C2×C42×C73C2311yescharacteristic
158C2×C42×C73C2314yes—
159C23×D73C2414yes—
160C23×D73C2411yescharacteristic
161C2×Dic146—48no—
162C2×Dic146—44no—
163C2×Dic146C2418yes—
164C2×Dic146C24116yes—
165C2×D292—216no—
166C2×D292—48no—
167C2×D292—44no—
168C2×D292C24116yes—
169C2×D292C2418yes—
170C2×D292C24116yes—
171C22×Dic73C2412yes—
172C22×Dic73C2411yescharacteristic
173C22×Dic73C2414yes—
174C22×Dic73C2414yes—
175C2×C4×D73—24no—
176C2×C4×D73—48no—
177C2×C4×D73—44no—
178C2×C4×D73—48no—
179C2×C4×D73—48no—
180C2×C4×D73—48no—
181C2×C4×D73C2414yes—
182C2×C4×D73C2418yes—
183C2×C4×D73C24116yes—
184C2×C4×D73C24116yes—
185C2×C4×D73C24116yes—
186C2×C4×D73C24116yes—
187C4⋊Dic73C24116yes—
188C4⋊Dic73C24116yes—
189C4⋊Dic73C2418yes—
190C4⋊Dic73C24116yes—
191#11924007C24116yes—
192#11924007C24116yes—
193#11924007C24116yes—
194#11924007C24116yes—
195#11924007C24116yes—
196#11924007C24116yes—
197#11924005C24116yes—
198#11924005C24116yes—
199#11924005C24116yes—
200#11924005C24116yes—
201#11924005C24116yes—
202#11924005C24116yes—
203#11924005C24116yes—
204#11924005C24116yes—
205#11924005C24116yes—
206C4×Dic73C2414yes—
207C4×Dic73C2418yes—
208C4×Dic73C24116yes—
209C4×Dic73C24116yes—
210C22×C4×C73C2414yes—
211C22×C4×C73C2411yescharacteristic
212C22×C4×C73C2412yes—
213C22×C4×C73C2414yes—
214C22×C4×C73C2414yes—
215C4⋊C4×C73C24116yes—
216C4⋊C4×C73C24116yes—
217C4⋊C4×C73C2418yes—
218C4⋊C4×C73C24116yes—
219C4⋊C4×C73C24116yes—
220C4⋊C4×C73C2418yes—
221C4⋊C4×C73C24116yes—
222C4⋊C4×C73C24116yes—
223C4⋊C4×C73C2418yes—
224C42×C73C2414yes—
225C42×C73C24116yes—
226C23×C73C2511yescharacteristic
227C22×D73—24no—
228C22×D73—28no—
229C22×D73—28no—
230C22×D73—28no—
231C22×D732−1+414yes—
232C22×D73C2518yes—
233C22×D73C2512yes—
234C2×C4×C73—22no—
235C2×C4×C73—28no—
236C2×C4×C73—28no—
237C2×C4×C73—28no—
238C2×C4×C73—24no—
239C2×C4×C73C2518yes—
240C2×C4×C73C2512yes—
241C2×C4×C73C2514yes—
242C2×C4×C73C2518yes—
243C2×C4×C73C2518yes—
244C2×Dic73—22no—
245C2×Dic73—24no—
246C2×Dic73—28no—
247C2×Dic73—28no—
248C2×Dic73C2514yes—
249C2×Dic73C2512yes—
250C2×Dic73C2518yes—
251C2×Dic73C2518yes—
252D292—232no—
253D292—48no—
254D292—416no—
255C4×D73—28no—
256C4×D73—416no—
257C4×D73—416no—
258C4×D73—416no—
259C4×D73—416no—
260C4×D73—48no—
261Dic146—48no—
262Dic146—416no—
263C22×C73C2×2−1+412yes—
264C22×C73#58514yes—
265C22×C73C2611yescharacteristic
266C2×D73—24no—
267C2×D73—28no—
268C2×D73—28no—
269C2×D73—28no—
270C2×D73—28no—
271C2×D73—416no—
272C2×D73C2×2−1+414yes—
273C4×C73—22no—
274C4×C73—28no—
275C4×C73—28no—
276C4×C73—24no—
277C4×C73—28no—
278Dic73—22no—
279Dic73—28no—
280Dic73—28no—
281Dic73—24no—
282C2×#3553—731nomaximal, Sylow
283C2×C73#355314yescomplemented
284C2×C73C22×2−1+411yescharacteristic
285C2×C73C2×#58512yes—
286D73—24no—
287D73—48no—
288#3553—7332no—
289C2×#564D7311yescharacteristic, complemented
290C2×#563—732no—
291C2×#562—731no—
292C2×#554—734no—
293C2×#553—734no—
294C2×#552—734no—
295C2×#541—734no—
296C2×#539—734no—
297C2×#539—734no—
298C2×#529—732no—
299C22×#125—731no—
300C73C2×#355311yescharacteristic, complemented, Sylow
301#564C2×D73116yescomplemented
302#563—7332no—
303#562—7316no—
304#554—7364no—
305#553—7364no—
306#552—7364no—
307#541—7364no—
308#539—7364no—
309#539—7364no—
310C2×C4⋊Q8—734no—
311#529—7332no—
312C2×#126—734no—
313C2×#126—734no—
314C2×#126—734no—
315C2×#126—734no—
316C2×#126—734no—
317C2×#125—7316no—
318C2×#125—734no—
319C2×#125—732no—
320C2×#125—734no—
321C2×#125—734no—
322C2×#125—731no—
323C2×#125C2×D7311yescharacteristic
324C2×#125C2×D7312yes—
325C2×#125C2×D7314yes—
326C2×#125C2×D7314yes—
327C2×#125C2×D7314yes—
328C2×#124—732no—
329C2×#123—734no—
330C2×#123—734no—
331C2×#123—734no—
332C2×#123—734no—
333C2×#123—734no—
334C2×C22⋊Q8—732no—
335C2×C22⋊Q8—734no—
336C2×C22⋊Q8—734no—
337C2×C22⋊Q8—734no—
338C2×C22⋊Q8—734no—
339C2×C22⋊Q8—734no—
340C2×C22⋊Q8—734no—
341C2×#121—732no—
342C2×#121—734no—
343C2×#121—734no—
344C2×C4×Q8—732no—
345C2×C4×Q8—734no—
346C2×C4×D4—732no—
347C2×C4×D4—734no—
348C2×C4×D4—734no—
349C2×C4×D4—734no—
350C2×C4×D4—734no—
351C2×#117—734no—
352C2×#117—734no—
353C22×C4⋊C4—732no—
354C22×C4⋊C4—734no—
355C22×C4⋊C4—734no—
356C22×C42—731no—
357C22×Q8—734no—
358C22×Q8—732no—
359C22×D4—734no—
360C22×D4—732no—
361C22×D4—734no—
362C23×C4—734no—
363C23×C4—732no—
364C23×C4—734no—
365C23×C4—734no—
366C23×C4—734no—
367C23×C4—731no—
368C4⋊Q8—7332no—
369#126—7332no—
370#126—7332no—
371#126—7332no—
372#126—7332no—
373#126—7332no—
374#125—7332no—
375#125—7316no—
376#125—7332no—
377#125—7332no—
378#125—738no—
379#125C22×D7318yes—
380#125C22×D73116yes—
381#125C22×D73132yes—
382#125C22×D73132yes—
383#125C22×D73132yes—
384#124—7316no—
385#123—7332no—
386#123—7332no—
387#123—7332no—
388#123—7332no—
389#123—7332no—
390C22⋊Q8—7316no—
391C22⋊Q8—7332no—
392C22⋊Q8—7332no—
393C22⋊Q8—7332no—
394C22⋊Q8—7332no—
395C22⋊Q8—7332no—
396C22⋊Q8—7332no—
397#121—7316no—
398#121—7332no—
399#121—7332no—
400C4×Q8—7316no—
401C4×Q8—7332no—
402C4×D4—7316no—
403C4×D4—7332no—
404C4×D4—7332no—
405C4×D4—7332no—
406C4×D4—7332no—
407#117—7332no—
408#117—7332no—
409C2×C4⋊C4—734no—
410C2×C4⋊C4—734no—
411C2×C4⋊C4—734no—
412C2×C4⋊C4—732no—
413C2×C4⋊C4—734no—
414C2×C4⋊C4—734no—
415C2×C4⋊C4—734no—
416C2×C4⋊C4—734no—
417C2×C4⋊C4—734no—
418C2×C4⋊C4—734no—
419C2×C4⋊C4—7316no—
420C2×C4⋊C4—7332no—
421C2×C4⋊C4—7332no—
422C2×C4⋊C4C22×D7314yes—
423C2×C4⋊C4C22×D7314yes—
424C2×C4⋊C4C22×D7312yes—
425C2×C4⋊C4C22×D7314yes—
426C2×C4⋊C4C22×D7314yes—
427C2×C4⋊C4C22×D7312yes—
428C2×C4⋊C4C22×D7314yes—
429C2×C4⋊C4C22×D7314yes—
430C2×C4⋊C4C22×D7312yes—
431C2×C22⋊C4—734no—
432C2×C22⋊C4—734no—
433C2×C22⋊C4—734no—
434C2×C22⋊C4—734no—
435C2×C22⋊C4—734no—
436C2×C22⋊C4—734no—
437C2×C22⋊C4—734no—
438C2×C22⋊C4—734no—
439C2×C22⋊C4—734no—
440C2×C42—731no—
441C2×C42—734no—
442C2×C42—734no—
443C2×C42—732no—
444C2×C42—738no—
445C2×C42C22×D7314yes—
446C2×C42C22×D7311yescharacteristic
447C24—731no—
448C24—734no—
449C2×Q8—2928no—
450C2×Q8—2924no—
451C2×Q8—7316no—
452C2×Q8—738no—
453C2×D4—2928no—
454C2×D4—2924no—
455C2×D4—7316no—
456C2×D4—738no—
457C2×D4—7316no—
458C2×D4—14616no—
459C22×C4—2928no—
460C22×C4—2928no—
461C22×C4—2928no—
462C22×C4—2928no—
463C22×C4—2924no—
464C22×C4—738no—
465C22×C4—7316no—
466C22×C4—7316no—
467C22×C4—7316no—
468C22×C4—7316no—
469C22×C4—732no—
470C22×C4—731no—
471C22×C4—734no—
472C22×C4—734no—
473C22×C4—734no—
474C22×C4—1464no—
475C22×C4C23×D7314yes—
476C22×C4C23×D7311yescharacteristic
477C22×C4C23×D7312yes—
478C22×C4C23×D7314yes—
479C22×C4C23×D7314yes—
480C4⋊C4—7316no—
481C4⋊C4—7316no—
482C4⋊C4—7316no—
483C4⋊C4—7316no—
484C4⋊C4—7316no—
485C4⋊C4—7316no—
486C4⋊C4—7316no—
487C4⋊C4—7316no—
488C4⋊C4—738no—
489C4⋊C4—7316no—
490C4⋊C4C23×D73116yes—
491C4⋊C4C23×D73116yes—
492C4⋊C4C23×D7318yes—
493C4⋊C4C23×D73116yes—
494C4⋊C4C23×D73116yes—
495C4⋊C4C23×D7318yes—
496C4⋊C4C23×D73116yes—
497C4⋊C4C23×D73116yes—
498C4⋊C4C23×D7318yes—
499C22⋊C4—7316no—
500C22⋊C4—7316no—
501C22⋊C4—7316no—
502C22⋊C4—7316no—
503C22⋊C4—7316no—
504C22⋊C4—7316no—
505C22⋊C4—7316no—
506C22⋊C4—7316no—
507C22⋊C4—7316no—
508C42—734no—
509C42—738no—
510C42—7316no—
511C42—7316no—
512C42C23×D7314yes—
513C42C23×D73116yes—
514C23—734no—
515C23—732no—
516C23—738no—
517C23—1468no—
518C23—1468no—
519C23—1468no—
520C23—1464no—
521C23C24×D7311yescharacteristic
522Q8—29216no—
523Q8—2928no—
524D4—29216no—
525D4—2928no—
526D4—14632no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—29216no—
533C2×C4—29216no—
534C2×C4—29216no—
535C2×C4—29216no—
536C2×C4—2928no—
537C2×C4—734no—
538C2×C4—732no—
539C2×C4—738no—
540C2×C4—738no—
541C2×C4—1468no—
542C2×C4—1462no—
543C2×C4—1464no—
544C2×C4—1468no—
545C2×C4—1468no—
546C2×C4C24×D7318yes—
547C2×C4C24×D7312yes—
548C2×C4C24×D7314yes—
549C2×C4C24×D7318yes—
550C2×C4C24×D7318yes—
551C22—29216no—
552C22—734no—
553C22—1468no—
554C22—1468no—
555C22—1468no—
556C22—1468no—
557C22—1464no—
558C222−1+4×D7312yes—
559C22#1798942414yes—
560C22C25×D7311yescharacteristic
561C4—22no—
562C4—28no—
563C4—28no—
564C4—28no—
565C4—24no—
566C4—1462no—
567C4—1464no—
568C4—1468no—
569C4—1468no—
570C2—2928no—
571C2—1464no—
572C2#2262438814yesdirect product factor
573C2C2×2−1+4×D7311yescharacteristic
574C2C2×#1798942412yes—
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#564×C73C2
socleC23×C73C25
hypercenterC2×#564D73
hypocenterC73C2×#3553
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
1. #22624388C2direct (N × H)
2. C2×#17989341C2semidirect
3. C2×#17989335C2semidirect
4. C2×#17989293C2semidirect
5. C2×#17989290C2semidirect
6. C2×#17989275C2semidirect
7. C2×#17989196C2semidirect
8. C2×#17989185C2semidirect
9. C2×#17989185C2semidirect
10. C2×#17989122C2semidirect
11. C2×#125×D73C2semidirect
12. C2×#564×C73C2semidirect
13. #17989341C22semidirect
14. #17989335C22semidirect
15. #17989293C22semidirect
16. #17989290C22semidirect
17. #17989275C22semidirect
18. #17989196C22semidirect
19. #17989185C22semidirect
20. #17989185C22semidirect
22. #17989122C22semidirect
33. #125×D73C22semidirect
63. #564×C73C22semidirect
283. C2×C73#3553semidirect
289. C2×#564D73semidirect
300. C73C2×#3553semidirect
301. #564C2×D73semidirect
572. C2#22624388direct (N × H)

Maximal subgroups

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

subgroupindex
#226243882
C2×#179893412
C2×#179893352
C2×#179892932
C2×#179892902
C2×#179892752
C2×#179891962
C2×#179891852
C2×#179891222
C2×#125×D732
C2×#564×C732
C2×#355373