Finite Group Databasebeta

← #33016042

Subgroups — #33016042

Order 18688.

1966 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988480C212yesmaximal, complemented
2C2×#17988467C212yesmaximal
3#22608199C211yesmaximal, characteristic, complemented
4#22608188C212yesmaximal, complemented
5#22608176C212yesmaximal, complemented
6#22608163C211yesmaximal, characteristic, complemented
7#22608153C212yesmaximal, complemented
8#22608153C212yesmaximal, complemented
9C2×#394×C73C211yesmaximal, characteristic
10C24×Dic73C2212yes—
11C22×C4⋊Dic73C2212yes—
12C22×C22⋊Dic73C2212yes—
13C22×C22⋊Dic73C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C4×Dic73C2211yescharacteristic
16#17988480—28no—
17#17988467—28no—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2211yescharacteristic
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C23×D4×C73C2211yescharacteristic
29C22×C4⋊C4×C73C2211yescharacteristic
30C22×C22⋊C4×C73C2212yes—
31C22×C22⋊C4×C73C2212yes—
32#394×C73C418yescomplemented
33C2×#95×C73C2211yescharacteristic
34C23×Dic73—24no—
35C23×Dic73—24no—
36C23×Dic73—24no—
37C23×Dic73—42no—
38C23×Dic73—42no—
39C23×Dic73—42no—
40C23×Dic73—42no—
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43C23×Dic73C2312yes—
44C23×Dic73C2312yes—
45C2×C4⋊Dic73—24no—
46C2×C4⋊Dic73—42no—
47C2×C4⋊Dic73—42no—
48C2×C22⋊Dic73—24no—
49C2×C22⋊Dic73—24no—
50C2×C22⋊Dic73—24no—
51C2×C22⋊Dic73—24no—
52C2×C22⋊Dic73—42no—
53C2×C22⋊Dic73—42no—
54C2×C22⋊Dic73—42no—
55C2×C22⋊Dic73—42no—
56C2×C22⋊Dic73D418yes—
57C2×C4×Dic73—24no—
58C2×C4×Dic73—41no—
59#14556498—24no—
60#14556498—24no—
61#14556498—24no—
62#14556498—24no—
63#14556498—24no—
64#14556498—42no—
65#14556498—41no—
66#14556498—41no—
67#14556498—41no—
68#14556498—42no—
69C25×C73C2312yes—
70C22×D4×C73—24no—
71C22×D4×C73—24no—
72C22×D4×C73—41no—
73C22×D4×C73—41no—
74C22×D4×C73C2×C414yes—
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2311yescharacteristic
77C23×C4×C73C2311yescharacteristic
78C23×C4×C73C2312yes—
79C2×C4⋊C4×C73—41no—
80C2×C4⋊C4×C73—41no—
81C2×C4⋊C4×C73C2×C414yescomplemented
82C2×C22⋊C4×C73—24no—
83C2×C22⋊C4×C73—24no—
84C2×C22⋊C4×C73—42no—
85C2×C22⋊C4×C73—42no—
86C2×C22⋊C4×C73C2×C418yescomplemented
87C2×C22⋊C4×C73C2×C418yescomplemented
88#95×C73C2×C414yescomplemented
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—48no—
115C22×Dic73—48no—
116C22×Dic73—48no—
117C22×Dic73—48no—
118C22×Dic73—48no—
119C22×Dic73—48no—
120C22×Dic73—48no—
121C22×Dic73C2×D414yes—
122C22×Dic73C2×D414yes—
123C22×Dic73#4114yes—
124C4⋊Dic73—48no—
125C22⋊Dic73—48no—
126C22⋊Dic73—48no—
127C22⋊Dic73—48no—
128C4×Dic73—44no—
129C24×C73—22no—
130C24×C73—22no—
131C24×C73—22no—
132C24×C73—22no—
133C24×C73—22no—
134C24×C73—22no—
135C24×C73—22no—
136C24×C73—22no—
137C24×C73—22no—
138C24×C73—22no—
139C24×C73—22no—
140C24×C73—22no—
141C24×C73C22×C414yes—
142C24×C73C2×D414yes—
143C24×C73#4114yes—
144C24×C73C2411yescharacteristic
145C2×D4×C73—216no—
146C2×D4×C73—44no—
147C2×D4×C73—44no—
148C2×D4×C73—48no—
149C2×D4×C73—44no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73C22×C414yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C4⋊C4×C73—44no—
173C22⋊C4×C73—216no—
174C22⋊C4×C73—48no—
175C23×C73—24no—
176C23×C73—24no—
177C23×C73—24no—
178C23×C73—24no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—24no—
183C23×C73—24no—
184C23×C73—24no—
185C23×C73—24no—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—24no—
190C23×C73—24no—
191C23×C73—24no—
192C23×C73—24no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73C4×D418yes—
210C23×C73C23×C411yescharacteristic
211C23×C73C22×D412yes—
212C23×C73C2×#4112yes—
213C23×C73C2×#4112yes—
214C23×C73H(5,𝔽2)12yes—
215C23×C73H(5,𝔽2)11yescharacteristic
216C23×C73H(5,𝔽2)11yescharacteristic
217C23×C73H(5,𝔽2)12yes—
218C23×C732−1+412yes—
219D4×C73—416no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—24no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—432no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C22×C73—28no—
272C22×C73—28no—
273C22×C73—28no—
274C22×C73—44no—
275C22×C73—44no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73—44no—
280C22×C73—44no—
281C22×C73—44no—
282C22×C73—44no—
283C22×C73—44no—
284C22×C73—44no—
285C22×C73—44no—
286C22×C73C2×C4×D412yes—
287C22×C73C4×#4111yescharacteristic
288C22×C73#51811yescharacteristic
289C22×C73#51811yescharacteristic
290C22×C73#52012yes—
291C22×C73#53412yes—
292C22×C73#53712yes—
293C22×C73#53812yes—
294C22×C73#53812yes—
295C22×C73#53912yes—
296C22×C73#54012yes—
297C22×C73D4211yescharacteristic
298C22×C73#54612yes—
299C22×C73#54612yes—
300C22×C73#54712yes—
301C22×C73#55112yes—
302C22×C73#55111yescharacteristic
303C22×C73#55212yes—
304C22×C73H(3,𝔽4)11yescharacteristic
305C22×C73#56212yes—
306C22×C73#56311yescharacteristic
307C4×C73—44no—
308C4×C73—42no—
309C4×C73—42no—
310C4×C73—42no—
311Dic73—44no—
312Dic73—44no—
313Dic73—44no—
314Dic73—82no—
315#14571—731nomaximal, Sylow
316C2×C73—48no—
317C2×C73#232512yes—
318C2×C73#233311yescharacteristic
319C2×C73#235811yescharacteristic
320C2×C73#235912yes—
321C2×C73#237911yescharacteristic
322C2×C73#267112yes—
323C2×C73#268412yes—
324C2×C73#269512yes—
325C2×C73#274312yes—
326C2×#394D7311yescharacteristic
327C2×#388—732no—
328C2×#386—732no—
329#1445—731no—
330#1442—732no—
331#1439—732no—
332#1436—731no—
333C22⋊#95—732no—
334C22⋊#95—732no—
335C73#1457111yescharacteristic, complemented, Sylow
336C23×D4C2×D7311yescharacteristic
337C24×C4—732no—
338C22×C4⋊C4—732no—
339C22×C4⋊C4C2×D7311yescharacteristic
340C22×C22⋊C4—732no—
341C22×C22⋊C4—732no—
342C22×C22⋊C4—732no—
343C22×C22⋊C4C2×D7312yes—
344C22×C22⋊C4C2×D7312yes—
345C22×C42—731no—
346#394Dic7318yescomplemented
347#388—1468no—
348#386—1468no—
349C2×#95—731no—
350C2×#95—732no—
351C2×#95—731no—
352C2×#95—732no—
353C2×#95—731no—
354C2×#95—732no—
355C2×#95—732no—
356C2×#95—732no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95C2×D7311yescharacteristic
360C25C22×D7312yes—
361C22×D4—24no—
362C22×D4—24no—
363C22×D4—41no—
364C22×D4—41no—
365C22×D4C2×Dic7314yes—
366C23×C4—2922no—
367C23×C4—2922no—
368C23×C4—2922no—
369C23×C4—2922no—
370C23×C4—732no—
371C23×C4—732no—
372C23×C4—732no—
373C23×C4—732no—
374C23×C4—1464no—
375C23×C4—1464no—
376C23×C4—1464no—
377C23×C4C22×D7312yes—
378C23×C4C22×D7311yescharacteristic
379C23×C4C22×D7311yescharacteristic
380C23×C4C22×D7311yescharacteristic
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—2922no—
384C2×C4⋊C4—2922no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4C2×Dic7314yescomplemented
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—2922no—
392C2×C22⋊C4—2922no—
393C2×C22⋊C4—2922no—
394C2×C22⋊C4—2922no—
395C2×C22⋊C4—738no—
396C2×C22⋊C4—1464no—
397C2×C22⋊C4—1464no—
398C2×C22⋊C4—1464no—
399C2×C22⋊C4—1464no—
400C2×C22⋊C4C2×Dic7318yescomplemented
401C2×C22⋊C4C2×Dic7318yescomplemented
402C2×C42—2921no—
403C2×C42—1464no—
404#95—2922no—
405#95—2921no—
406#95—2921no—
407#95—2921no—
408#95—2922no—
409#95—1464no—
410#95—1464no—
411#95—1464no—
412#95—1464no—
413#95—1464no—
414#95C2×Dic7314yescomplemented
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24C22×Dic7314yes—
428C24D4×D7314yes—
429C24#1192402714yes—
430C24C23×D7311yescharacteristic
431C2×D4—216no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—44no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—2928no—
455C22×C4—2928no—
456C22×C4—2928no—
457C22×C4—2928no—
458C22×C4—2928no—
459C22×C4—2928no—
460C22×C4—2928no—
461C22×C4—734no—
462C22×C4—734no—
463C22×C4—734no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4C22×Dic7314yes—
490C22×C4C22×Dic7312yes—
491C22×C4C22×Dic7312yes—
492C22×C4C22×Dic7312yes—
493C4⋊C4—44no—
494C4⋊C4—2928no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—2928no—
498C22⋊C4—2928no—
499C22⋊C4—2928no—
500C42—2924no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23D4×Dic7318yes—
536C23C23×Dic7311yescharacteristic
537C23C2×D4×D7312yes—
538C23#41×D7312yes—
539C23C2×#1192402712yes—
540C23#1455663612yes—
541C23#1455663611yescharacteristic
542C23#1455663611yescharacteristic
543C23#1455663712yes—
544C23#1455663812yes—
545D4—416no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—29232no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2922no—
576C2×C4—2922no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—1464no—
589C2×C4—1464no—
590C2×C4—1464no—
591C2×C4—1464no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22C2×D4×Dic7312yes—
613C22#41×Dic7311yescharacteristic
614C22#1798906211yescharacteristic
615C22#1798906211yescharacteristic
616C22#1798907012yes—
617C22#1798914512yes—
618C22#1798916412yes—
619C22#1798916912yes—
620C22#1798917412yes—
621C22#1798918212yes—
622C22#1798918712yes—
623C22#1798921911yescharacteristic
624C22#1798922412yes—
625C22#1798922912yes—
626C22#1798923512yes—
627C22#1798926012yes—
628C22#1798926811yescharacteristic
629C22#1798927412yes—
630C22#1798932911yescharacteristic
631C22#1798933112yes—
632C22#1798933711yescharacteristic
633C4—44no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—2924no—
638C4—2924no—
639C4—2924no—
640C4—5842no—
641C2—48no—
642C2#2261263612yes—
643C2#2261269811yescharacteristic
644C2#2261291511yescharacteristic
645C2#2261292712yes—
646C2#2261310511yescharacteristic
647C2#2261616012yes—
648C2#2261630612yes—
649C2#2261646512yes—
650C2#2261707212yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#394×C73C2
socle, cosocleC24×C73C24
prime radicalC23×D4×C73C22
hypercenterC2×#394D73
hypocenterC73#14571
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#17988480C2semidirect
3. #22608199C2semidirect
4. #22608188C2semidirect
5. #22608176C2semidirect
6. #22608163C2semidirect
7. #22608153C2semidirect
8. #22608153C2semidirect
32. #394×C73C4semidirect
81. C2×C4⋊C4×C73C2×C4semidirect
86. C2×C22⋊C4×C73C2×C4semidirect
87. C2×C22⋊C4×C73C2×C4semidirect
88. #95×C73C2×C4semidirect
335. C73#14571semidirect
346. #394Dic73semidirect
386. C2×C4⋊C4C2×Dic73semidirect
400. C2×C22⋊C4C2×Dic73semidirect
401. C2×C22⋊C4C2×Dic73semidirect
414. #95C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
C2×#179884672
#226081992
#226081882
#226081762
#226081632
#226081532
C2×#394×C732
#1457173