Finite Group Databasebeta

← #33097185

Subgroups — #33097185

Order 18688.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988720C211yesmaximal, characteristic
2C2×#17988489C211yesmaximal, characteristic, complemented
3#22610806C212yesmaximal, complemented
4#22610627C212yesmaximal, complemented
5#22610558C212yesmaximal, complemented
6#22610279C212yesmaximal, complemented
7#22610249C211yesmaximal, characteristic, complemented
8#22610123C211yesmaximal, characteristic, complemented
9#22609861C211yesmaximal, characteristic, complemented
10#22609693C211yesmaximal, characteristic, complemented
11C4×#121×C73C211yesmaximal, characteristic, complemented
12C23×D292C2211yescharacteristic
13C22×#11924013C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924005—21no—
16C22×#11924005—21no—
17#17988720C418yescomplemented
18C2×#14556522C2212yescomplemented
19C2×#14556512C2212yescomplemented
20C2×#14556511C2212yes—
21C2×#14556511C2212yes—
22C2×#14556511C2212yes—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×#14556502C2212yescomplemented
26#17988489—21no—
27#17988489—21no—
28#17988489C2212yes—
29#17988489C2212yes—
30#17988349C2212yescomplemented
31#17988349C2212yescomplemented
32#17988342C2212yescomplemented
33C2×#121×C73C2211yescharacteristic
34C2×C4×D4×C73—21no—
35C2×C4×D4×C73C2211yescharacteristic, complemented
36C22×C42×C73C2211yescharacteristic
37#390×C73C2211yescharacteristic, complemented
38#389×C73C2211yescharacteristic, complemented
39#388×C73—21no—
40#386×C73—21no—
41#380×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic, complemented
43C4×C22⋊C4×C73—21no—
44C24×D73—21no—
45C22×D292—21no—
46C22×D292—22no—
47C22×D292—42no—
48C22×D292—42no—
49C22×D292C2×C414yescomplemented
50C22×D292C2312yes—
51C23×Dic73—21no—
52C2×#11924013—42no—
53C2×#11924013—42no—
54C2×#11924013C2×C418yescomplemented
55C22×#10596831C2312yes—
56C22×#10596831C2312yes—
57C2×C4⋊Dic73—21no—
58C2×C4⋊Dic73—42no—
59C2×C4⋊Dic73C2312yes—
60C2×#11924005—22no—
61C2×#11924005—22no—
62C2×#11924005—22no—
63C2×#11924005—22no—
64C2×#11924005—44no—
65C2×#11924005—44no—
66#14556522—42no—
67#14556512—42no—
68#14556511—42no—
69#14556511—42no—
70#14556511C2×C418yescomplemented
71C22⋊#10596831—22no—
72C22⋊#10596831C2×C414yescomplemented
73#14556502—42no—
74C22×D4×C73—21no—
75C22×D4×C73C2311yescharacteristic
76C23×C4×C73—21no—
77C23×C4×C73D411yescharacteristic
78C23×C4×C73D411yescharacteristic
79C23×C4×C73C2311yescharacteristic
80#121×C73—22no—
81#121×C73C2×C414yes—
82C4×D4×C73—42no—
83C4×D4×C73—44no—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C22⋊C4×C73—21no—
89C2×C22⋊C4×C73—21no—
90C2×C22⋊C4×C73—21no—
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—21no—
93C2×C42×C73—21no—
94C2×C42×C73—42no—
95C2×C42×C73D411yescharacteristic
96C2×C42×C73D411yescharacteristic
97C2×C42×C73C2311yescharacteristic
98C2×C42×C73C2311yescharacteristic
99#95×C73—21no—
100#95×C73—21no—
101#95×C73—21no—
102C23×D73—24no—
103C23×D73—22no—
104C23×D73—42no—
105C23×D73—42no—
106C23×D73—42no—
107C23×D73—42no—
108C23×D73—42no—
109C23×D73—42no—
110C2×D292—22no—
111C2×D292—22no—
112C2×D292—22no—
113C2×D292—22no—
114C2×D292—24no—
115C2×D292—22no—
116C2×D292—48no—
117C2×D292—42no—
118C2×D292—42no—
119C2×D292—48no—
120C2×D292C22×C414yes—
121C22×Dic73—22no—
122C22×Dic73—42no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125#11924013—48no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—22no—
131C2×#10596831C22×C414yes—
132C4⋊Dic73—22no—
133C4⋊Dic73—22no—
134C4⋊Dic73—42no—
135#11924005—44no—
136#11924005—44no—
137#11924005—44no—
138#11924005—44no—
139C24×C73—21no—
140C24×C73C2×D411yescharacteristic
141C2×D4×C73—24no—
142C2×D4×C73—21no—
143C2×D4×C73—22no—
144C2×D4×C73—22no—
145C2×D4×C73—42no—
146C2×D4×C73—42no—
147C2×D4×C73C22×C412yes—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—41no—
157C22×C4×C73—41no—
158C22×C4×C73—41no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73—42no—
162C22×C4×C73C22×C412yes—
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73#4112yes—
169C22×C4×C73C2411yescharacteristic
170C4⋊C4×C73—21no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—22no—
173C4⋊C4×C73—42no—
174C4⋊C4×C73C22×C412yes—
175C22⋊C4×C73—24no—
176C22⋊C4×C73—42no—
177C22⋊C4×C73—42no—
178C22⋊C4×C73—42no—
179C22⋊C4×C73—42no—
180C42×C73—22no—
181C42×C73—22no—
182C42×C73—41no—
183C42×C73—41no—
184C42×C73—42no—
185C23×C73—21no—
186C23×C73—21no—
187C23×C73—21no—
188C23×C73—21no—
189C23×C73—22no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73C4×D412yes—
197C23×C73AGL(1,ℤ/8)12yes—
198C23×C73AGL(1,ℤ/8)12yes—
199C23×C73C22×D411yescharacteristic
200C22×D73—24no—
201C22×D73—48no—
202C22×D73—48no—
203C22×D73—42no—
204C22×D73—42no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73—48no—
210C22×D73—84no—
211C22×D73—84no—
212C22×D73—84no—
213C22×D73—84no—
214D4×C73—24no—
215D4×C73—48no—
216D4×C73—42no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—44no—
234C2×C4×C73—44no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—84no—
244C2×C4×C73C4×D412yes—
245C2×C4×C73AGL(1,ℤ/8)12yes—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)12yes—
249C2×C4×C73H(5,𝔽2)12yes—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—84no—
254D292—24no—
255D292—24no—
256D292—48no—
257D292—42no—
258D292—42no—
259#10596831—24no—
260#10596831—42no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—81no—
271C22×C73—81no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73#44214yes—
275C22×C73C2×C4×D411yescharacteristic
276C22×C73#53411yescharacteristic
277C22×C73#53511yescharacteristic
278C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73#57511yescharacteristic
282C2×D73—44no—
283C2×D73—44no—
284C2×D73—44no—
285C2×D73—816no—
286C2×D73—82no—
287C2×D73—82no—
288C2×D73—82no—
289C2×D73—82no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73#51812yes—
299C4×C73#53812yes—
300Dic73—82no—
301#22046—731nomaximal, Sylow
302C2×C73—24no—
303C2×C73—82no—
304C2×C73#232111yescharacteristic
305C2×C73C2×#44211yescharacteristic
306C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C73#319211yescharacteristic
308C2×C73#319411yescharacteristic
309C2×C73#321311yescharacteristic
310C2×C73#321711yescharacteristic
311D73—84no—
312C2×#447—731no—
313C4×#121D7311yescharacteristic, complemented
314C2×#390—731no—
315#1972—732no—
316#1936—732no—
317#1926—732no—
318#1879—732no—
319#1874—731no—
320#1853—731no—
321#1786—731no—
322#1749—731no—
323C73#2204611yescharacteristic, complemented, Sylow
324C23×D4—731no—
325C22×D8—732no—
326C2×#121C2×D7311yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C2×D7311yescharacteristic, complemented
329C22×C4⋊C4—731no—
330C22×C22⋊C4—1461no—
331C22×C22⋊C4—1461no—
332C22×C42C2×D7311yescharacteristic
333#447—738no—
334C2×#106—732no—
335C2×#102—732no—
336C2×#102—732no—
337C2×#102—732no—
338C2×#102—732no—
339C2×C22⋊C8—731no—
340C2×C22⋊C8—731no—
341C2×#97—732no—
342#390—732no—
343#390—732no—
344#390—1461no—
345#390—1461no—
346#390C2×D7311yescharacteristic, complemented
347#389C2×D7311yescharacteristic, complemented
348#388—21no—
349#386—21no—
350#380C2×D7311yescharacteristic
351C4×C4⋊C4C2×D7311yescharacteristic, complemented
352C4×C22⋊C4—21no—
353#340—732no—
354#340—732no—
355#336—732no—
356C25—1461no—
357C22×D4—21no—
358C22×D4—2922no—
359C22×D4—2922no—
360C22×D4—734no—
361C22×D4—732no—
362C22×D4—1461no—
363C22×D4—1462no—
364C22×D4C22×D7311yescharacteristic
365C23×C4—21no—
366C23×C4—1461no—
367C23×C4C22⋊D7311yescharacteristic
368C23×C4C22⋊D7311yescharacteristic
369C23×C4C22×D7311yescharacteristic
370C2×D8—2922no—
371C2×D8—2922no—
372C2×D8—738no—
373C22×C8—732no—
374C22×C8—732no—
375#121—22no—
376#121C4×D7314yes—
377C4×D4—42no—
378C4×D4—44no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—2922no—
381C2×C4⋊C4—732no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—2924no—
392C2×C22⋊C4—2924no—
393C2×C22⋊C4—1462no—
394C2×C22⋊C4—1462no—
395C2×C22⋊C4—1462no—
396C2×C22⋊C4—1462no—
397C2×C42—21no—
398C2×C42—42no—
399C2×C42C22⋊D7311yescharacteristic
400C2×C42C22⋊D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403#106—2922no—
404#102—2922no—
405#102—2922no—
406#102—2922no—
407#102—738no—
408C22⋊C8—734no—
409C22⋊C8—1462no—
410#97—2922no—
411#95—21no—
412#95—21no—
413#95—21no—
414C24—21no—
415C24—2922no—
416C24—2922no—
417C24—2922no—
418C24—2922no—
419C24—2922no—
420C24—2922no—
421C24—1464no—
422C24—1462no—
423C24C2×C22⋊D7311yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—2922no—
431C2×D4—2922no—
432C2×D4—2928no—
433C2×D4—2928no—
434C2×D4—734no—
435C2×D4—1464no—
436C2×D4—1462no—
437C2×D4—1462no—
438C2×D4—1462no—
439C2×D4—1462no—
440C2×D4—1462no—
441C2×D4C2×C4×D7312yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—1462no—
460C22×C4C2×C4×D7312yes—
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4#1192402812yes—
467C22×C4C23×D7311yescharacteristic
468D8—2928no—
469C2×C8—734no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—2922no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4C2×C4×D7312yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—2924no—
489C22⋊C4—2924no—
490C22⋊C4—2924no—
491C22⋊C4—2924no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—2928no—
509C23—2928no—
510C23—2922no—
511C23—2922no—
512C23—2922no—
513C23—2928no—
514C23—2922no—
515C23—2922no—
516C23—2922no—
517C23—5844no—
518C23—5844no—
519C23—5844no—
520C23—5844no—
521C23—1464no—
522C23C4×C22⋊D7312yes—
523C23#1455661612yes—
524C23#1455661712yes—
525C23C22×C22⋊D7311yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—2928no—
530D4—2922no—
531D4—2922no—
532D4—1464no—
533D4—1464no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—5844no—
565C2×C4C4×C22⋊D7312yes—
566C2×C4#1455661612yes—
567C2×C4C22×C4×D7311yescharacteristic
568C2×C4C2×#1192402811yescharacteristic
569C2×C4#1455663612yes—
570C2×C4#1455663712yes—
571C8—2922no—
572C8—1464no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—2924no—
587C22—2924no—
588C22—2924no—
589C22—58416no—
590C22—5842no—
591C22—5842no—
592C22—5842no—
593C22—5842no—
594C22#1798869714yes—
595C22C2×C4×C22⋊D7311yescharacteristic
596C22#1798914611yescharacteristic
597C22#1798915111yescharacteristic
598C22C2×#1455661611yescharacteristic
599C22C2×#1455661611yescharacteristic
600C22C2×#1455661711yescharacteristic
601C22#1798938511yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—5842no—
611C4#1798905912yes—
612C4#1798917012yes—
613C2—24no—
614C2—82no—
615C2—5844no—
616C2#2261258511yescharacteristic
617C2C2×#1798869711yescharacteristic
618C2C4×#1455661611yescharacteristic
619C2#2262072811yescharacteristic
620C2#2262074111yescharacteristic
621C2#2262091411yescharacteristic
622C2#2262094611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#121×C73C2
socleC23×C73C22×D4
prime radicalC2×#17988720C2
cosocleC22×C4×C73C24
hypercenterC4×#121D73
hypocenterC73#22046
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#17988489C2semidirect
3. #22610806C2semidirect
4. #22610627C2semidirect
5. #22610558C2semidirect
6. #22610279C2semidirect
7. #22610249C2semidirect
8. #22610123C2semidirect
9. #22609861C2semidirect
10. #22609693C2semidirect
11. C4×#121×C73C2semidirect
17. #17988720C4semidirect
18. C2×#14556522C22semidirect
19. C2×#14556512C22semidirect
25. C2×#14556502C22semidirect
30. #17988349C22semidirect
31. #17988349C22semidirect
32. #17988342C22semidirect
35. C2×C4×D4×C73C22semidirect
37. #390×C73C22semidirect
38. #389×C73C22semidirect
42. C4×C4⋊C4×C73C22semidirect
49. C22×D292C2×C4semidirect
54. C2×#11924013C2×C4semidirect
70. #14556511C2×C4semidirect
72. C22⋊#10596831C2×C4semidirect
313. C4×#121D73semidirect
323. C73#22046semidirect
328. C2×C4×D4C2×D73semidirect
346. #390C2×D73semidirect
347. #389C2×D73semidirect
351. C4×C4⋊C4C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887202
C2×#179884892
#226108062
#226106272
#226105582
#226102792
#226102492
#226101232
#226098612
#226096932
C4×#121×C732
#2204673