Finite Group Databasebeta

← #33159177

Subgroups — #33159177

Order 18688.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612987C211yesmaximal, characteristic, complemented
2#22611475C211yesmaximal, characteristic, complemented
3#22611248C211yesmaximal, characteristic
4#22611177C211yesmaximal, characteristic, complemented
5#22611013C211yesmaximal, characteristic
6#22610784C211yesmaximal, characteristic
7#22610565C211yesmaximal, characteristic, complemented
8#22610555C211yesmaximal, characteristic
9#22610506C211yesmaximal, characteristic, complemented
10#22610490C211yesmaximal, characteristic, complemented
11#22610289C211yesmaximal, characteristic
12#22610289C211yesmaximal, characteristic
13#22609933C211yesmaximal, characteristic, complemented
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2512C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556591C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988493C2211yescharacteristic
35#17988483—21no—
36#17988470—21no—
37#17988470—21no—
38C4⋊C4×Dic73—21no—
39C4×C4⋊Dic73C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#126×C73—21no—
49#397×C73—21no—
50#396×C73—21no—
51#390×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#390×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#388×C73—21no—
56#384×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C22×Dic146C2311yescharacteristic
61C2×#11924016—22no—
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016—41no—
65C2×#11924016—41no—
66C2×#11924016—41no—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556591—22no—
72#14556591—22no—
73C4×Dic146—22no—
74C4×Dic146—22no—
75C4×Dic146—41no—
76C4×Dic146—41no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73—21no—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86#14556522—22no—
87#14556522—41no—
88C4⋊#10596831—41no—
89#14556514—22no—
90#14556514—22no—
91#14556514—22no—
92#14556514—41no—
93#14556514—41no—
94#14556514—41no—
95#14556512—22no—
96#14556512—41no—
97C4×#10596831—41no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73C2311yescharacteristic
104#126×C73—44no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73D411yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146#4112yes—
138C2×Dic146#4112yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143#11924016—44no—
144#11924016—44no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831#4112yes—
157C4⋊Dic73—24no—
158C4⋊Dic73—24no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73#4112yes—
163#11924007—42no—
164#11924007—42no—
165C4×Dic73—42no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C24×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—41no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73—42no—
194C4⋊C4×C73—81no—
195C4⋊C4×C73SD1614yescomplemented
196C22⋊C4×C73—42no—
197C22⋊C4×C73—42no—
198C22⋊C4×C73—42no—
199C22⋊C4×C73—42no—
200C22⋊C4×C73—81no—
201C22⋊C4×C73—81no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—42no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73C22×D411yescharacteristic
217D4×C73—41no—
218D4×C73—41no—
219D4×C73—41no—
220D4×C73—41no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
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—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
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—41no—
244C2×C4×C73—41no—
245C2×C4×C73C2×SD1612yes—
246C2×C4×C73C2×SD1612yes—
247C2×C4×C73C2×SD1612yes—
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×C4×C73H(5,𝔽2)11yescharacteristic
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266Dic146—22no—
267Dic146—22no—
268Dic146—22no—
269Dic146—22no—
270Dic146—41no—
271Dic146—41no—
272Dic146—41no—
273Dic146—41no—
274#10596831—22no—
275#10596831—41no—
276#10596831—41no—
277#10596831—41no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—81no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73#52511yescharacteristic
286C22×C73#54611yescharacteristic
287C22×C73#55011yescharacteristic
288C22×C73C22×SD1611yescharacteristic
289C22×C73#57511yescharacteristic
290C22×C73#57711yescharacteristic
291C22×C73#57811yescharacteristic
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—22no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73—81no—
301C4×C73#52911yescharacteristic
302C4×C73#53811yescharacteristic
303C4×C73#55111yescharacteristic
304C4×C73#55311yescharacteristic
305Dic73—42no—
306Dic73—42no—
307Dic73—42no—
308Dic73—81no—
309#27066—731nomaximal, Sylow
310C2×C73—161no—
311C2×C73#246711yescharacteristic
312C2×C73#310311yescharacteristic
313C2×C73#312111yescharacteristic
314C2×C73#329811yescharacteristic
315C2×C73#330711yescharacteristic
316C2×C73#339211yescharacteristic
317C2×C73#340211yescharacteristic
318#2512D7311yescharacteristic
319#2365—731no—
320#2096—731no—
321#2055—731no—
322#2043—731no—
323#2013—731no—
324#1968—731no—
325#1927—731no—
326#1926—731no—
327#1919—731no—
328#1916—731no—
329#1880—731no—
330#1880—731no—
331#1803—731no—
332C4×#103—731no—
333C73#2706611yescharacteristic, complemented, Sylow
334C22×SD16—731no—
335C22×SD16—731no—
336C2×C4⋊Q8—731no—
337C2×#126—21no—
338C2×C4×Q8—731no—
339C2×C4×Q8—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×C4⋊C8—731no—
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103—731no—
348C2×#103—731no—
349C2×#102—731no—
350C2×#102—731no—
351C2×C4×C8—731no—
352#397—21no—
353#396—21no—
354#391—731no—
355#391—731no—
356#390C2×D7311yescharacteristic
357#390C2×D7311yescharacteristic
358#390C2×D7311yescharacteristic
359#389—1461no—
360#389C2×D7311yescharacteristic
361#388—21no—
362#387—1461no—
363#387—1461no—
364#384C2×D7311yescharacteristic
365C4×C4⋊C4—731no—
366C4×C4⋊C4—1461no—
367C4×C4⋊C4C2×D7311yescharacteristic
368C4×C4⋊C4C2×D7311yescharacteristic
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377C22×Q8—731no—
378C22×Q8—731no—
379C22×D4C22×D7311yescharacteristic
380C2×SD16—2921no—
381C2×SD16—2921no—
382C2×SD16—2921no—
383C2×SD16—1462no—
384C2×SD16—1462no—
385C2×SD16—1462no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389C22×C8—731no—
390C4⋊Q8—1462no—
391C4⋊Q8—1462no—
392#126—44no—
393C4×Q8—2921no—
394C4×Q8—2921no—
395C4×Q8—1462no—
396C4×Q8—1462no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4C22⋊D7311yescharacteristic
407C2×C4⋊C4C22⋊D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C4⋊C4C22×D7311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C42—21no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423C2×C42C22×D7311yescharacteristic
424#106—2921no—
425#106—1462no—
426C4⋊C8—2921no—
427#103—2921no—
428#103—2921no—
429#103—2921no—
430#103—1462no—
431#103—1462no—
432#103—1462no—
433#102—2921no—
434#102—1462no—
435C4×C8—2921no—
436#95—21no—
437#95—21no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95C22⋊D7311yescharacteristic
444#95C22⋊D7311yescharacteristic
445C24—21no—
446C2×Q8—732no—
447C2×Q8—732no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×Q8—1461no—
455C2×Q8—1461no—
456C2×Q8—1461no—
457C2×Q8—1461no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478SD16—2924no—
479SD16—2924no—
480C2×C8—732no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—2922no—
503C4⋊C4—2922no—
504C4⋊C4—732no—
505C4⋊C4—1464no—
506C4⋊C4—1464no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4#1192401614yescomplemented
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—2922no—
525C42—2922no—
526C42—2922no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C22×C22⋊D7311yescharacteristic
535Q8—2921no—
536Q8—2921no—
537Q8—2921no—
538Q8—2921no—
539Q8—1462no—
540Q8—1462no—
541Q8—1462no—
542Q8—1462no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1462no—
583C2×C4C2×#1192401612yes—
584C2×C4C2×#1192401612yes—
585C2×C4C2×#1192401612yes—
586C2×C4#41×D7311yescharacteristic
587C2×C4#41×D7311yescharacteristic
588C2×C4#41×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#1455663711yescharacteristic
591C2×C4#1455663711yescharacteristic
592C8—2921no—
593C8—2921no—
594C8—2921no—
595C8—1462no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1798909411yescharacteristic
604C22#1798922511yescharacteristic
605C22#1798925611yescharacteristic
606C22C22×#1192401611yescharacteristic
607C22#1798938511yescharacteristic
608C22#1798939811yescharacteristic
609C22#1798940211yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—2922no—
620C4—2922no—
621C4—2922no—
622C4—5841no—
623C4#1798912011yescharacteristic
624C4#1798917611yescharacteristic
625C4#1798926211yescharacteristic
626C4#1798928811yescharacteristic
627C2—161no—
628C2#2261397811yescharacteristic
629C2#2261995811yescharacteristic
630C2#2262015411yescharacteristic
631C2#2262179011yescharacteristic
632C2#2262190111yescharacteristic
633C2#2262291211yescharacteristic
634C2#2262300611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2512C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2512D73
hypocenterC73#27066
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2512

Complements

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

NH ≅ G/Ntype
1. #22612987C2semidirect
2. #22611475C2semidirect
4. #22611177C2semidirect
7. #22610565C2semidirect
9. #22610506C2semidirect
10. #22610490C2semidirect
13. #22609933C2semidirect
14. C4×#14556514C2semidirect
195. C4⋊C4×C73SD16semidirect
333. C73#27066semidirect
510. C4⋊C4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226129872
#226114752
#226112482
#226111772
#226110132
#226107842
#226105652
#226105552
#226105062
#226104902
#226102892
#226099332
C4×#145565142
C73×#25122
#2706673