Finite Group Databasebeta

← #33091639

Subgroups — #33091639

Order 18688.

1868 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic73C211yesmaximal, characteristic
2C2×#17988688C212yesmaximal, complemented
3C22⋊Q8×Dic73C211yesmaximal, characteristic, complemented
4#22610267C212yesmaximal, complemented
5#22610239C211yesmaximal, characteristic, complemented
6#22610180C212yesmaximal, complemented
7#22610121C211yesmaximal, characteristic, complemented
8#22609862C211yesmaximal, characteristic
9#22609758C212yesmaximal, complemented
10#22609695C211yesmaximal, characteristic, complemented
11C2×#450×C73C211yesmaximal, characteristic
12C24×Dic73—21no—
13C2×Q8×Dic73C2211yescharacteristic, complemented
14C2×D4×Dic73—28no—
15C2×D4×Dic73—21no—
16C2×D4×Dic73—21no—
17C2×D4×Dic73C2212yes—
18C2×D4×Dic73C2212yes—
19C22×C4⋊Dic73C2211yescharacteristic
20C22×C22⋊Dic73—21no—
21C22×C4×Dic73C2211yescharacteristic
22#17988688—28no—
23C2×#14556525C2212yescomplemented
24C2×#14556515C2212yescomplemented
25C2×#14556512C2212yes—
26C2×#14556512C2212yes—
27C2×C22⋊#10596831C2211yescharacteristic
28C2×#14556501C2212yescomplemented
29#17988494C2211yescharacteristic, complemented
30#17988486C2211yescharacteristic, complemented
31#17988473—21no—
32#17988462—21no—
33#17988447C2211yescharacteristic
34C4⋊C4×Dic73—21no—
35C4⋊C4×Dic73C2211yescharacteristic, complemented
36C22⋊C4×Dic73—21no—
37#17988350C2212yescomplemented
38#17988349C2212yescomplemented
39#17988341C2212yescomplemented
40C23×D4×C73C2211yescharacteristic
41C22×SD16×C73C2212yes—
42C2×C22⋊Q8×C73C2211yescharacteristic
43#450×C73C418yescomplemented
44C2×#102×C73C2212yes—
45C2×C22⋊C8×C73C2211yescharacteristic
46C23×Dic73—28no—
47C23×Dic73—22no—
48C23×Dic73—22no—
49C23×Dic73—21no—
50C23×Dic73—44no—
51C23×Dic73—44no—
52C23×Dic73D411yescharacteristic
53C23×Dic73D411yescharacteristic
54C22×#10596831C2312yes—
55Q8×Dic73—24no—
56D4×Dic73—28no—
57D4×Dic73—28no—
58D4×Dic73—24no—
59D4×Dic73—24no—
60D4×Dic73—24no—
61D4×Dic73D418yes—
62C2×C4⋊Dic73—24no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2312yes—
69C2×C22⋊Dic73—28no—
70C2×C22⋊Dic73—22no—
71C2×C22⋊Dic73—22no—
72C2×C22⋊Dic73—21no—
73C2×C4×Dic73—24no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73D411yescharacteristic
77C2×C4×Dic73D411yescharacteristic
78C2×C4×Dic73C2311yescharacteristic
79C2×C4×Dic73C2311yescharacteristic
80#14556525—24no—
81#14556515—24no—
82#14556512—24no—
83#14556512—24no—
84C22⋊#10596831—22no—
85#14556501—24no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C25×C73—21no—
92C22×Q8×C73C2311yescharacteristic
93C22×D4×C73—24no—
94C22×D4×C73—24no—
95C22×D4×C73—21no—
96C22×D4×C73—22no—
97C22×D4×C73C2×C414yes—
98C22×D4×C73C2312yes—
99C23×C4×C73C2311yescharacteristic
100C2×SD16×C73—24no—
101C2×SD16×C73—24no—
102C2×SD16×C73C2×C418yescomplemented
103C22×C8×C73C2312yes—
104C22⋊Q8×C73—22no—
105C22⋊Q8×C73C2×C414yescomplemented
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C22⋊C4×C73—21no—
109#102×C73C2×C418yescomplemented
110C22⋊C8×C73C2×C414yescomplemented
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—24no—
115C22×Dic73—28no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—28no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—41no—
123C22×Dic73—432no—
124C22×Dic73—44no—
125C22×Dic73—44no—
126C22×Dic73—44no—
127C22×Dic73—42no—
128C22×Dic73—42no—
129C22×Dic73—44no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C4⋊Dic73—24no—
138C4⋊Dic73—22no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141C4⋊Dic73—42no—
142C4⋊Dic73—42no—
143C4⋊Dic73C2×D414yes—
144C22⋊Dic73—28no—
145C22⋊Dic73—28no—
146C22⋊Dic73—42no—
147C4×Dic73—22no—
148C4×Dic73—22no—
149C4×Dic73—22no—
150C4×Dic73—42no—
151C4×Dic73—42no—
152C4×Dic73C2×D414yes—
153C24×C73—24no—
154C24×C73—24no—
155C24×C73—24no—
156C24×C73—22no—
157C24×C73—42no—
158C24×C73—42no—
159C24×C73—42no—
160C24×C73—42no—
161C24×C73C2×D411yescharacteristic
162C2×Q8×C73—21no—
163C2×Q8×C73—22no—
164C2×Q8×C73—22no—
165C2×Q8×C73C22×C412yescomplemented
166C2×D4×C73—24no—
167C2×D4×C73—216no—
168C2×D4×C73—24no—
169C2×D4×C73—216no—
170C2×D4×C73—24no—
171C2×D4×C73—22no—
172C2×D4×C73—22no—
173C2×D4×C73—22no—
174C2×D4×C73C22×C414yes—
175C2×D4×C73C2×D414yes—
176C2×D4×C73#4114yes—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73C22×C412yes—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73#4112yes—
184C22×C4×C73C2411yescharacteristic
185SD16×C73—216no—
186C2×C8×C73—22no—
187C2×C8×C73—22no—
188C2×C8×C73C22×C414yescomplemented
189C4⋊C4×C73—24no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73C22×C412yescomplemented
192C22⋊C4×C73—24no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—216no—
198C23×C73—24no—
199C23×C73—24no—
200C23×C73—24no—
201C23×C73—48no—
202C23×C73—48no—
203C23×C73—48no—
204C23×C73—48no—
205C23×C73—48no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73—48no—
211C23×C73C4×D412yes—
212C23×C73AGL(1,ℤ/8)12yes—
213C23×C73#13712yes—
214C23×C73C22×D411yescharacteristic
215Q8×C73—24no—
216Q8×C73—42no—
217D4×C73—216no—
218D4×C73—24no—
219D4×C73—24no—
220D4×C73—24no—
221D4×C73C4×D418yes—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73C4×D412yes—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C22×D412yes—
235C2×C4×C73C2×#4112yes—
236C2×C4×C73C2×#4111yescharacteristic
237C2×Dic73—28no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—24no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—44no—
249C2×Dic73—416no—
250C2×Dic73—416no—
251C2×Dic73—48no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73#12012yes—
261C2×Dic73#12012yes—
262C2×Dic73AGL(1,ℤ/8)12yes—
263C8×C73—24no—
264#10596831—42no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—28no—
272C22×C73—432no—
273C22×C73—44no—
274C22×C73—44no—
275C22×C73—44no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73#44014yescomplemented
280C22×C73C2×C4×D411yescharacteristic
281C22×C73C2×#12011yescharacteristic
282C22×C73#52511yescharacteristic
283C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73C2×#13711yescharacteristic
286C22×C73#57511yescharacteristic
287C4×C73—22no—
288C4×C73—42no—
289C4×C73—42no—
290C4×C73C2×C4×D412yes—
291C4×C73#52512yes—
292Dic73—22no—
293Dic73—22no—
294Dic73—22no—
295Dic73—44no—
296Dic73—41no—
297Dic73—81no—
298Dic73—81no—
299#21453—731nomaximal, Sylow
300C2×C73—24no—
301C2×C73—48no—
302C2×C73C4×#12011yescharacteristic
303C2×C73C2×#44011yescharacteristic
304C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
305C2×C73#300311yescharacteristic
306C2×C73#300711yescharacteristic
307C2×C73#311011yescharacteristic
308C2×C73#311411yescharacteristic
309C22×C4×D4—731no—
310C2×#450D7311yescharacteristic
311C2×#440—732no—
312C4×C22⋊Q8—731no—
313#1877—732no—
314#1872—731no—
315#1862—732no—
316#1852—731no—
317#1786—731no—
318#1762—732no—
319#1749—731no—
320C73#2145311yescharacteristic, complemented, Sylow
321C23×D4C2×D7311yescharacteristic
322C24×C4—1461no—
323C22×SD16C2×D7312yes—
324C2×C22⋊Q8C2×D7311yescharacteristic
325C2×C4×Q8—731no—
326C2×C4×D4—732no—
327C2×C4×D4—732no—
328C2×C4×D4—1461no—
329C2×C4×D4—1461no—
330C2×C4×D4—1468no—
331C22×C4⋊C4—731no—
332C22×C22⋊C4—1461no—
333C22×C42—731no—
334#450Dic7318yescomplemented
335#440—1468no—
336C2×#107—732no—
337C2×#103—732no—
338C2×#102—732no—
339C2×#102—732no—
340C2×#102C2×D7312yes—
341C2×C22⋊C8—731no—
342C2×C22⋊C8C2×D7311yescharacteristic
343C2×#97—732no—
344#391—731no—
345#389—731no—
346#387—1461no—
347#385—1461no—
348#380—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4—1461no—
351C4×C22⋊C4—1461no—
352#340—732no—
353#340—732no—
354#336—732no—
355C25—21no—
356C22×Q8C22×D7311yescharacteristic
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4C2×Dic7314yes—
362C22×D4C22×D7312yes—
363C23×C4—2924no—
364C23×C4—2924no—
365C23×C4—731no—
366C23×C4—731no—
367C23×C4—1462no—
368C23×C4—1462no—
369C23×C4—1468no—
370C23×C4—1461no—
371C23×C4C22×D7311yescharacteristic
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16C2×Dic7318yescomplemented
375C22×C8—732no—
376C22×C8C22×D7312yes—
377C22⋊Q8—22no—
378C22⋊Q8C2×Dic7314yescomplemented
379C4×Q8—1464no—
380C4×D4—738no—
381C4×D4—1464no—
382C4×D4—1464no—
383C4×D4—1468no—
384C4×D4—1468no—
385C4×D4—1464no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—732no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1464no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1461no—
397C2×C22⋊C4—1462no—
398C2×C22⋊C4—1462no—
399C2×C22⋊C4—1468no—
400C2×C42—731no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—1461no—
405C2×C42—1461no—
406C2×C42—1464no—
407#107—1464no—
408#103—1464no—
409#102—1464no—
410#102—1464no—
411#102C2×Dic7318yescomplemented
412C22⋊C8—1462no—
413C22⋊C8C2×Dic7314yescomplemented
414#97—1464no—
415#95—1461no—
416#95—1461no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24D4×D7311yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C22×Dic7312yescomplemented
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4C22×Dic7314yes—
442C2×D4D4×D7314yes—
443C2×D4#1192402714yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2922no—
449C22×C4—2922no—
450C22×C4—2924no—
451C22×C4—29232no—
452C22×C4—2924no—
453C22×C4—2924no—
454C22×C4—2924no—
455C22×C4—2921no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1462no—
463C22×C4—1468no—
464C22×C4—1464no—
465C22×C4—1468no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4C22×Dic7312yes—
472C22×C4D4×D7311yescharacteristic
473C22×C4#1192402712yes—
474C22×C4C23×D7311yescharacteristic
475SD16—216no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—1462no—
479C2×C8—1462no—
480C2×C8—1462no—
481C2×C8C22×Dic7314yescomplemented
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—2922no—
485C4⋊C4—2922no—
486C4⋊C4—734no—
487C4⋊C4—1464no—
488C4⋊C4—1462no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C4⋊C4C22×Dic7312yescomplemented
492C22⋊C4—24no—
493C22⋊C4—2922no—
494C22⋊C4—1468no—
495C22⋊C4—1468no—
496C42—2922no—
497C42—2922no—
498C42—734no—
499C42—1462no—
500C42—1462no—
501C42—1462no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23D4×Dic7312yes—
521C23#1455661812yes—
522C23#1455662012yes—
523C23C2×D4×D7311yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4D4×Dic7318yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—2928no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—2924no—
545C2×C4—29216no—
546C2×C4—29216no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—732no—
554C2×C4—732no—
555C2×C4—732no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1464no—
560C2×C4—1468no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4D4×Dic7312yes—
568C2×C4C23×Dic7311yescharacteristic
569C2×C4C2×D4×D7312yes—
570C2×C4C2×#1192402712yes—
571C2×C4C2×#1192402711yescharacteristic
572C8—24no—
573C8—2922no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22#1798868814yescomplemented
589C22C2×D4×Dic7311yescharacteristic
590C22#120×D7311yescharacteristic
591C22#1798909311yescharacteristic
592C22AGL(1,ℤ/8)×D7311yescharacteristic
593C22C2×#1455661811yescharacteristic
594C22C2×#1455662011yescharacteristic
595C22#1798938311yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—2924no—
600C4—2921no—
601C4—5841no—
602C4—5841no—
603C4—1462no—
604C4—1462no—
605C4—1462no—
606C4C2×D4×Dic7312yes—
607C4#1798909312yes—
608C2—24no—
609C2—48no—
610C2#120×Dic7311yescharacteristic
611C2C2×#1798868811yescharacteristic
612C2AGL(1,ℤ/8)×Dic7311yescharacteristic
613C2#2261915911yescharacteristic
614C2#2261918611yescharacteristic
615C2#2262002511yescharacteristic
616C2#2262007011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#450×C73C2
socleC23×C73C22×D4
prime radicalC23×D4×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#450D73
hypocenterC73#21453
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. C2×#17988688C2semidirect
3. C22⋊Q8×Dic73C2semidirect
4. #22610267C2semidirect
5. #22610239C2semidirect
6. #22610180C2semidirect
7. #22610121C2semidirect
9. #22609758C2semidirect
10. #22609695C2semidirect
13. C2×Q8×Dic73C22semidirect
23. C2×#14556525C22semidirect
24. C2×#14556515C22semidirect
28. C2×#14556501C22semidirect
29. #17988494C22semidirect
30. #17988486C22semidirect
35. C4⋊C4×Dic73C22semidirect
37. #17988350C22semidirect
38. #17988349C22semidirect
39. #17988341C22semidirect
43. #450×C73C4semidirect
102. C2×SD16×C73C2×C4semidirect
105. C22⋊Q8×C73C2×C4semidirect
109. #102×C73C2×C4semidirect
110. C22⋊C8×C73C2×C4semidirect
165. C2×Q8×C73C22×C4semidirect
188. C2×C8×C73C22×C4semidirect
191. C4⋊C4×C73C22×C4semidirect
279. C22×C73#440semidirect
320. C73#21453semidirect
334. #450Dic73semidirect
374. C2×SD16C2×Dic73semidirect
378. C22⋊Q8C2×Dic73semidirect
411. #102C2×Dic73semidirect
413. C22⋊C8C2×Dic73semidirect
432. C2×Q8C22×Dic73semidirect
481. C2×C8C22×Dic73semidirect
491. C4⋊C4C22×Dic73semidirect
588. C22#17988688semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic732
C2×#179886882
C22⋊Q8×Dic732
#226102672
#226102392
#226101802
#226101212
#226098622
#226097582
#226096952
C2×#450×C732
#2145373