Finite Group Databasebeta

← #33118956

Subgroups — #33118956

Order 18688.

1364 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988898C211yesmaximal, characteristic, complemented
2C22×#14556515C211yesmaximal, characteristic, complemented
3#22614239C211yesmaximal, characteristic, complemented
4#22611167C212yesmaximal, complemented
5#22611131C212yesmaximal, complemented
6#22610360C211yesmaximal, characteristic, complemented
7#22610224C211yesmaximal, characteristic, complemented
8#22610185C212yesmaximal, complemented
9#22609871C212yesmaximal, complemented
10#22609866C211yesmaximal, characteristic, complemented
11C2×#548×C73C211yesmaximal, characteristic
12C23×#10596831C2211yescharacteristic
13C2×#14556572—21no—
14C22×C4⋊Dic73C2211yescharacteristic
15#17988898—24no—
16C2×#14556524C2212yes—
17C2×#14556515—24no—
18C2×#14556515C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×#14556515C2212yes—
22C2×#14556512C2212yes—
23C2×#14556512C2212yes—
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C22⋊#10596831C2212yes—
26#17988517—21no—
27#17988512—21no—
28#17988485C2212yes—
29#17988448—21no—
30#17988447C2211yescharacteristic
31#17988447C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34C22×#41×C73C2211yescharacteristic
35C23×Q8×C73C2211yescharacteristic
36#548×C73—24no—
37#548×C73—24no—
38C2×#124×C73—21no—
39C2×#124×C73C2211yescharacteristic
40C2×C22⋊Q8×C73—21no—
41C2×C22⋊Q8×C73C2211yescharacteristic
42C2×#121×C73—21no—
43C2×#121×C73C2211yescharacteristic
44C2×C4×Q8×C73C2211yescharacteristic
45C2×C4×D4×C73—21no—
46C2×C4×D4×C73C2211yescharacteristic
47C23×Dic73—21no—
48C22×#10596831—22no—
49C22×#10596831—42no—
50C22×#10596831C2312yes—
51C22×#10596831C2312yes—
52#14556572—44no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×C4⋊Dic73—21no—
56C2×C4⋊Dic73C2312yes—
57C2×C4⋊Dic73C2312yes—
58C2×#11924007—22no—
59C2×C22⋊Dic73—21no—
60C2×C22⋊Dic73—22no—
61#14556524—24no—
62#14556515—24no—
63#14556515—24no—
64#14556515—24no—
65#14556515—42no—
66#14556512—24no—
67#14556512—42no—
68C22⋊#10596831—22no—
69C22⋊#10596831—42no—
70#14556498—22no—
71#14556498—21no—
72#14556498—22no—
73C2×#41×C73—22no—
74C2×#41×C73—22no—
75C2×#41×C73—24no—
76C2×#41×C73—42no—
77C22×Q8×C73—21no—
78C22×Q8×C73—22no—
79C22×Q8×C73—24no—
80C22×Q8×C73—42no—
81C22×Q8×C73D414yes—
82C22×Q8×C73C2311yescharacteristic
83C22×Q8×C73C2311yescharacteristic
84C22×D4×C73—21no—
85C22×D4×C73—21no—
86C22×D4×C73D411yescharacteristic
87C22×D4×C73C2311yescharacteristic
88C23×C4×C73—21no—
89C23×C4×C73—21no—
90C23×C4×C73C2311yescharacteristic
91C23×C4×C73C2311yescharacteristic
92#124×C73—28no—
93#124×C73—22no—
94#124×C73—22no—
95C22⋊Q8×C73—28no—
96C22⋊Q8×C73—22no—
97C22⋊Q8×C73—22no—
98#121×C73—28no—
99#121×C73—22no—
100#121×C73D414yes—
101C4×Q8×C73—22no—
102C4×Q8×C73—22no—
103C4×D4×C73—28no—
104C4×D4×C73—22no—
105C4×D4×C73—22no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—42no—
121C22×Dic73—42no—
122C22×Dic73—42no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—44no—
130C4⋊Dic73—22no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—22no—
134C4⋊Dic73—42no—
135#11924007—44no—
136C22⋊Dic73—44no—
137C22⋊Dic73—42no—
138C24×C73—21no—
139C24×C73C2×D411yescharacteristic
140C24×C73C2×D411yescharacteristic
141#41×C73—48no—
142#41×C73—44no—
143#41×C73—44no—
144C2×Q8×C73—24no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73—22no—
148C2×Q8×C73—28no—
149C2×Q8×C73—22no—
150C2×Q8×C73—22no—
151C2×Q8×C73—44no—
152C2×Q8×C73—44no—
153C2×Q8×C73—48no—
154C2×Q8×C73C2×D412yes—
155C2×Q8×C73C2×D412yes—
156C2×Q8×C73#4114yes—
157C2×D4×C73—24no—
158C2×D4×C73—22no—
159C2×D4×C73—24no—
160C2×D4×C73—22no—
161C2×D4×C73—22no—
162C2×D4×C73—21no—
163C2×D4×C73—22no—
164C2×D4×C73—22no—
165C2×D4×C73—44no—
166C2×D4×C73—42no—
167C2×D4×C73—42no—
168C2×D4×C73C2×D412yes—
169C22×C4×C73—21no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—24no—
173C22×C4×C73—24no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—44no—
181C22×C4×C73—44no—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D412yes—
187C22×C4×C73C2×D412yes—
188C22×C4×C73C2411yescharacteristic
189C4⋊C4×C73—24no—
190C4⋊C4×C73—24no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73C2×D412yes—
194C22⋊C4×C73—24no—
195C22⋊C4×C73—24no—
196C22⋊C4×C73—24no—
197C22⋊C4×C73—22no—
198C22⋊C4×C73—24no—
199C22⋊C4×C73—22no—
200C42×C73—24no—
201C42×C73—21no—
202C42×C73—21no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—21no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—22no—
211C23×C73—42no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—42no—
215C23×C73#12012yes—
216C23×C73#13512yes—
217C23×C73AGL(1,ℤ/8)12yes—
218C23×C73C22×D411yescharacteristic
219Q8×C73—22no—
220Q8×C73—24no—
221Q8×C73—22no—
222Q8×C73—22no—
223Q8×C73—48no—
224Q8×C73—42no—
225Q8×C73#12314yescomplemented
226D4×C73—24no—
227D4×C73—48no—
228D4×C73—44no—
229D4×C73—48no—
230D4×C73—41no—
231D4×C73—41no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—24no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—24no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—48no—
248C2×C4×C73—48no—
249C2×C4×C73—42no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—42no—
255C2×C4×C73#12012yes—
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C2×#4112yes—
259C2×C4×C73H(5,𝔽2)12yes—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—84no—
267#10596831—42no—
268#10596831—42no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—24no—
272C22×C73—24no—
273C22×C73—42no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—42no—
277C22×C73—41no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—82no—
282C22×C73—82no—
283C22×C73#44914yes—
284C22×C73C2×#12011yescharacteristic
285C22×C73#54612yes—
286C22×C73C2×#13511yescharacteristic
287C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C73#57812yes—
289C4×C73—22no—
290C4×C73—24no—
291C4×C73—41no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73C2×#12311yescharacteristic
297C4×C73#53411yescharacteristic
298C4×C73#54612yes—
299Dic73—82no—
300Dic73—82no—
301#24035—731nomaximal, Sylow
302C2×C73—24no—
303C2×C73—81no—
304C2×C73—82no—
305C2×C73#240511yescharacteristic
306C2×C73C2×#44911yescharacteristic
307C2×C73#301111yescharacteristic
308C2×C73#330012yes—
309C2×C73#330812yes—
310C2×#548D7311yescharacteristic
311C2×#480—731no—
312C22×#103—731no—
313#2486—731no—
314#2042—732no—
315#2036—732no—
316#1893—731no—
317#1869—731no—
318#1863—732no—
319#1788—732no—
320#1787—731no—
321C73#2403511yescharacteristic, complemented, Sylow
322C22×#41C2×D7311yescharacteristic
323C23×Q8C2×D7311yescharacteristic
324C23×C8—731no—
325#548—24no—
326#548—24no—
327C2×#124—21no—
328C2×#124C2×D7311yescharacteristic
329C2×#123—1461no—
330C2×C22⋊Q8—21no—
331C2×C22⋊Q8C2×D7311yescharacteristic
332C2×#121—21no—
333C2×#121C2×D7311yescharacteristic
334C2×C4×Q8C2×D7311yescharacteristic
335C2×C4×D4—21no—
336C2×C4×D4C2×D7311yescharacteristic
337C22×C4⋊C4—731no—
338#480—1464no—
339C2×#107—732no—
340C2×#103—732no—
341C2×#103—732no—
342C2×#103—732no—
343C2×#103—732no—
344C2×#103—1464no—
345C2×#102—732no—
346C2×#102—732no—
347C2×C22⋊C8—732no—
348C2×C22⋊C8—731no—
349#398—1461no—
350#397—1461no—
351#389—732no—
352#381—1461no—
353#380—731no—
354#380—732no—
355#340—732no—
356#336—732no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—24no—
360C2×#41—42no—
361C22×Q8—21no—
362C22×Q8—22no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8C22⋊D7314yes—
366C22×Q8C22×D7311yescharacteristic
367C22×Q8C22×D7311yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22⋊D7311yescharacteristic
371C22×D4C22×D7311yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—1461no—
375C23×C4C22×D7311yescharacteristic
376C23×C4C22×D7311yescharacteristic
377C22×C8—2922no—
378C22×C8—732no—
379C22×C8—732no—
380C22×C8—1462no—
381#124—28no—
382#124—22no—
383#124—22no—
384#123—2924no—
385C22⋊Q8—28no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388#121—28no—
389#121—22no—
390#121C22⋊D7314yes—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×D4—28no—
394C4×D4—22no—
395C4×D4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—732no—
399C2×C4⋊C4—732no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1462no—
402C2×C4⋊C4—1462no—
403C2×C4⋊C4—1462no—
404C2×C4⋊C4C22⋊D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1462no—
412C2×C22⋊C4C22⋊D7311yescharacteristic
413C2×C22⋊C4C22⋊D7311yescharacteristic
414C2×C42—21no—
415C2×C42C22×D7311yescharacteristic
416#107—1464no—
417#103—2922no—
418#103—1464no—
419#103—1464no—
420#103—1464no—
421#102—2922no—
422#102—1464no—
423C22⋊C8—2922no—
424C22⋊C8—1462no—
425#95—1462no—
426#95—1461no—
427#95—1462no—
428C24—21no—
429C24C2×C22⋊D7311yescharacteristic
430C24C2×C22⋊D7311yescharacteristic
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—24no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—48no—
444C2×Q8C2×C22⋊D7312yes—
445C2×Q8C2×C22⋊D7312yes—
446C2×Q8#1192402714yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C22⋊D7312yes—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—2922no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7312yes—
482C22×C4C2×C22⋊D7312yes—
483C22×C4C23×D7311yescharacteristic
484C2×C8—2924no—
485C2×C8—1462no—
486C2×C8—1462no—
487C2×C8—1462no—
488C2×C8—1462no—
489C2×C8—1462no—
490C2×C8—1462no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—2924no—
496C4⋊C4—2922no—
497C4⋊C4—1462no—
498C4⋊C4—1462no—
499C4⋊C4—1462no—
500C4⋊C4—1462no—
501C4⋊C4C2×C22⋊D7312yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—2924no—
509C22⋊C4—2922no—
510C42—24no—
511C42—21no—
512C42—21no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23#1455656012yes—
526C23#1455661112yes—
527C23#1455661612yes—
528C23C22×C22⋊D7311yescharacteristic
529Q8—22no—
530Q8—24no—
531Q8—22no—
532Q8—22no—
533Q8—48no—
534Q8—42no—
535Q8#1455657214yescomplemented
536D4—24no—
537D4—48no—
538D4—44no—
539D4—48no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—2922no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2922no—
571C2×C4—5844no—
572C2×C4#1455656012yes—
573C2×C4C22×C22⋊D7311yescharacteristic
574C2×C4C22×C22⋊D7311yescharacteristic
575C2×C4C2×#1192402712yes—
576C2×C4#1455663612yes—
577C8—2922no—
578C8—2922no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22#1798873214yes—
594C22C2×#1455656011yescharacteristic
595C22#1798923112yes—
596C22C2×#1455661111yescharacteristic
597C22C2×#1455661611yescharacteristic
598C22#1798940212yes—
599C4—22no—
600C4—24no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—5842no—
607C4—5842no—
608C4C2×#1455657211yescharacteristic
609C4#1798914611yescharacteristic
610C4#1798923112yes—
611C2—24no—
612C2—81no—
613C2—82no—
614C2#2261325411yescharacteristic
615C2C2×#1798873211yescharacteristic
616C2#2261922611yescharacteristic
617C2#2262181812yes—
618C2#2262192212yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#548×C73C2
socleC23×C73C22×D4
hypercenterC2×#548D73
hypocenterC73#24035
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#17988898C2semidirect
2. C22×#14556515C2semidirect
3. #22614239C2semidirect
4. #22611167C2semidirect
5. #22611131C2semidirect
6. #22610360C2semidirect
7. #22610224C2semidirect
8. #22610185C2semidirect
9. #22609871C2semidirect
10. #22609866C2semidirect
225. Q8×C73#123semidirect
321. C73#24035semidirect
535. Q8#14556572semidirect

Maximal subgroups

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

subgroupindex
C2×#179888982
C22×#145565152
#226142392
#226111672
#226111312
#226103602
#226102242
#226101852
#226098712
#226098662
C2×#548×C732
#2403573