Finite Group Databasebeta

← #33149085

Subgroups — #33149085

Order 18688.

848 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
1C2×#17988887C211yesmaximal, characteristic
2C2×C4×#11924016C211yesmaximal, characteristic
3#22613792C211yesmaximal, characteristic, complemented
4#22611482C211yesmaximal, characteristic, complemented
5#22611413C211yesmaximal, characteristic, complemented
6#22611309C211yesmaximal, characteristic, complemented
7#22611269C211yesmaximal, characteristic, complemented
8#22610656C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610265C211yesmaximal, characteristic
11#22610265C211yesmaximal, characteristic
12#22610192C211yesmaximal, characteristic
13#22610184C211yesmaximal, characteristic, complemented
14#22610178C211yesmaximal, characteristic
15C73×#2349C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19#17988887—24no—
20C4×#11924016—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988528—21no—
35#17988518—21no—
36#17988493C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988482—21no—
41#17988469—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#385×C73—21no—
56#385×C73—21no—
57#382×C73C2211yescharacteristic
58#380×C73—21no—
59C4×C4⋊C4×C73C2211yescharacteristic
60C2×#95×C73—21no—
61C22×Dic146C2311yescharacteristic
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016Q814yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69#14556583—22no—
70#14556583—41no—
71C4×Dic146—22no—
72C4×Dic146—41no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C4×Dic73—21no—
82#14556524—22no—
83#14556524—41no—
84#14556522—22no—
85#14556522—22no—
86#14556522—22no—
87C4⋊#10596831—22no—
88#14556514—22no—
89#14556514—22no—
90#14556514—41no—
91#14556512—22no—
92#14556512—22no—
93#14556512D414yes—
94C4×#10596831—22no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73C2311yescharacteristic
102C23×C4×C73—21no—
103C23×C4×C73—21no—
104C23×C4×C73—21no—
105C4×D4×C73—22no—
106C4×D4×C73—22no—
107C4×D4×C73—22no—
108C4×D4×C73—22no—
109C4×D4×C73—41no—
110C4×D4×C73—41no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146C2×Q812yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140#11924016—28no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C2×D412yes—
152C2×#10596831C2×Q812yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73C2×D412yes—
159#11924007—42no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—42no—
164C4×Dic73—42no—
165C24×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73C2×D412yes—
170C2×D4×C73C2×Q812yes—
171C2×D4×C73#4112yes—
172C22×C4×C73—21no—
173C22×C4×C73—24no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73—42no—
180C22×C4×C73—81no—
181C22×C4×C73—81no—
182C22×C4×C73—81no—
183C22×C4×C73—81no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C4⋊C4×C73—22no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—41no—
198C4⋊C4×C73—81no—
199C4⋊C4×C73—81no—
200C4⋊C4×C73#4112yes—
201C22⋊C4×C73—24no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C23×C73—22no—
211C23×C73—22no—
212C23×C73—22no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73C22×D411yescharacteristic
218D4×C73—22no—
219D4×C73—22no—
220D4×C73—22no—
221D4×C73C22⋊Q814yes—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
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—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—84no—
249C2×C4×C73—84no—
250C2×C4×C73—84no—
251C2×C4×C73#13512yes—
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C732−1+411yescharacteristic
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270Dic146—22no—
271Dic146—22no—
272Dic146—41no—
273Dic146—41no—
274#10596831—22no—
275#10596831—22no—
276#10596831—41no—
277#10596831—41no—
278C22×C73—24no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73#52511yescharacteristic
286C22×C73D4211yescharacteristic
287C22×C73D4×Q811yescharacteristic
288C22×C73C2×#13511yescharacteristic
289C22×C73#57511yescharacteristic
290C22×C73#57711yescharacteristic
291C22×C73#57711yescharacteristic
292C4×C73—22no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—81no—
300C4×C73—81no—
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#53911yescharacteristic
303C4×C73#54711yescharacteristic
304C4×C73#55411yescharacteristic
305Dic73—42no—
306Dic73—42no—
307Dic73—81no—
308Dic73—81no—
309#26378—731nomaximal, Sylow
310C2×C73—44no—
311C2×C73#244911yescharacteristic
312C2×C73#310211yescharacteristic
313C2×C73#312311yescharacteristic
314C2×C73#326711yescharacteristic
315C2×C73#328211yescharacteristic
316C2×C73#338111yescharacteristic
317C2×C73#338811yescharacteristic
318C2×#478—731no—
319C2×C4×SD16—731no—
320#2453—731no—
321#2349D7311yescharacteristic
322#2097—731no—
323#2085—731no—
324#2066—731no—
325#2059—731no—
326#1941—731no—
327#1916—731no—
328#1877—731no—
329#1877—731no—
330#1864—731no—
331#1863—731no—
332#1862—731no—
333C73#2637811yescharacteristic, complemented, Sylow
334C22×SD16—731no—
335C2×#125—731no—
336C2×C4×Q8—731no—
337C2×C4×D4C2×D7311yescharacteristic
338C2×C4×D4C2×D7311yescharacteristic
339C2×C4×D4C2×D7311yescharacteristic
340#478—1464no—
341C4×SD16—1464no—
342C2×#107—731no—
343C2×#107—731no—
344C2×#106—731no—
345C2×#106—731no—
346C2×#106—731no—
347C2×C4⋊C8—731no—
348C2×#103—731no—
349C2×#103—731no—
350C2×#103—731no—
351C2×#102—731no—
352C2×#102—731no—
353C2×#102—731no—
354C2×C4×C8—731no—
355#400—1461no—
356#398—1461no—
357#391—731no—
358#391—731no—
359#389—731no—
360#389—731no—
361#389—1461no—
362#389C2×D7311yescharacteristic
363#389C2×D7311yescharacteristic
364#387—1461no—
365#385—21no—
366#385—21no—
367#382C2×D7311yescharacteristic
368#380—21no—
369C4×C4⋊C4C2×D7311yescharacteristic
370C2×#95—21no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#336—731no—
376#336—731no—
377#336—731no—
378#336—731no—
379C22×Q8—731no—
380C22×D4C22×D7311yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4—21no—
384C2×SD16—734no—
385C2×SD16—1462no—
386C2×SD16—1462no—
387C22×C8—731no—
388C22×C8—731no—
389C22×C8—731no—
390C22×C8—731no—
391#125—2921no—
392#125—1462no—
393C4×Q8—2921no—
394C4×Q8—1462no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—41no—
400C4×D4—41no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4C22⋊D7311yescharacteristic
412C2×C4⋊C4C22⋊D7311yescharacteristic
413C2×C4⋊C4C22×D7311yescharacteristic
414C2×C4⋊C4C22×D7311yescharacteristic
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423#107—2921no—
424#107—1462no—
425#106—1462no—
426#106—1462no—
427#106—1462no—
428C4⋊C8—1462no—
429#103—2921no—
430#103—1462no—
431#103—1462no—
432#102—734no—
433#102—1462no—
434#102—1462no—
435C4×C8—1462no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95C22⋊D7311yescharacteristic
447#95C22⋊D7311yescharacteristic
448C24—21no—
449C2×Q8—732no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×Q8—1461no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4D4×D7312yes—
459C2×D4Q8×D7312yes—
460C2×D4#1192402812yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484SD16—1468no—
485C2×C8—732no—
486C2×C8—732no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—2922no—
507C4⋊C4—2922no—
508C4⋊C4—2922no—
509C4⋊C4—2922no—
510C4⋊C4—2922no—
511C4⋊C4—732no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4#1192402812yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—2922no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C22×C22⋊D7311yescharacteristic
536Q8—2921no—
537Q8—2921no—
538Q8—1462no—
539Q8—1462no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4#1455656914yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
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—22no—
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—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—1462no—
584C2×C4—1462no—
585C2×C4#1455661112yes—
586C2×C4C2×D4×D7311yescharacteristic
587C2×C4C2×Q8×D7311yescharacteristic
588C2×C4C2×#1192402811yescharacteristic
589C2×C4C2×#1192402811yescharacteristic
590C2×C4#1455663611yescharacteristic
591C2×C4#1455663911yescharacteristic
592C8—2921no—
593C8—2921no—
594C8—1462no—
595C8—1462no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22#1798909811yescharacteristic
604C22D4×C22⋊D7311yescharacteristic
605C22Q8×C22⋊D7311yescharacteristic
606C22C2×#1455661111yescharacteristic
607C22#1798938511yescharacteristic
608C22#1798939811yescharacteristic
609C22#1798939811yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—2922no—
620C4—2922no—
621C4—5841no—
622C4—5841no—
623C4C2×#1455656911yescharacteristic
624C4#1798918411yescharacteristic
625C4#1798923811yescharacteristic
626C4#1798929211yescharacteristic
627C2—44no—
628C2#2261374011yescharacteristic
629C2#2261994911yescharacteristic
630C2#2262018311yescharacteristic
631C2#2262145211yescharacteristic
632C2#2262163411yescharacteristic
633C2#2262280611yescharacteristic
634C2#2262287611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2349C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2349D73
hypocenterC73#26378
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #22613792C2semidirect
4. #22611482C2semidirect
5. #22611413C2semidirect
6. #22611309C2semidirect
7. #22611269C2semidirect
8. #22610656C2semidirect
9. #22610490C2semidirect
13. #22610184C2semidirect
333. C73#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#179888872
C2×C4×#119240162
#226137922
#226114822
#226114132
#226113092
#226112692
#226106562
#226104902
#226102652
#226101922
#226101842
#226101782
C73×#23492
#2637873