Finite Group Databasebeta

← #33089045

Subgroups — #33089045

Order 18688.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2#22612395C211yesmaximal, characteristic
3#22610558C211yesmaximal, characteristic
4#22610551C211yesmaximal, characteristic
5#22610523C211yesmaximal, characteristic, complemented
6#22610502C211yesmaximal, characteristic, complemented
7#22610286C211yesmaximal, characteristic
8#22610286C211yesmaximal, characteristic
9#22610199C211yesmaximal, characteristic, complemented
10#22610185C211yesmaximal, characteristic, complemented
11#22609829C211yesmaximal, characteristic, complemented
12#22609761C211yesmaximal, characteristic, complemented
13#22609759C211yesmaximal, characteristic
14#22609680C211yesmaximal, characteristic, complemented
15C42×Q8×C73C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556548—21no—
19#17988685C418yescomplemented
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556502C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31#17988489C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988478—21no—
36#17988478—21no—
37C4×C4⋊Dic73C2211yescharacteristic
38C4×#11924005—21no—
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C4×C4⋊C4×C73—21no—
53C4×C4⋊C4×C73—21no—
54C4×C4⋊C4×C73—21no—
55C4×C4⋊C4×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C43×C73—21no—
59C43×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C2×#11924015—22no—
63C2×#11924015—22no—
64C2×#11924015C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C4×D292—22no—
70C4×D292C2×C414yes—
71#14556548—44no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924005—21no—
78C2×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×C4×Dic73—21no—
83#14556524—22no—
84#14556524C2×C414yes—
85#14556522—22no—
86#14556515—22no—
87#14556515—22no—
88#14556515C2×C414yes—
89#14556511—22no—
90#14556511—22no—
91#14556511C2×C414yes—
92#14556502—22no—
93#14556502C2×C414yes—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C4×Q8×C73—22no—
99C4×Q8×C73—22no—
100C4×Q8×C73—22no—
101C4×Q8×C73—22no—
102C4×Q8×C73—22no—
103C4×Q8×C73C2×C414yes—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C42×C73—21no—
111C2×C42×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73—21no—
114C2×C42×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73D411yescharacteristic
117C2×C42×C73D411yescharacteristic
118C2×C42×C73D411yescharacteristic
119C2×C42×C73D411yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22×C412yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C2×C4×D73—24no—
137C2×C4×D73—42no—
138C2×C4×D73—42no—
139#11924015—28no—
140C2×#10596831—21no—
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×#10596831C22×C412yes—
151C2×#10596831C22×C412yes—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73C22×C412yes—
158#11924007—42no—
159#11924007—42no—
160#11924005—24no—
161#11924005—42no—
162#11924005—42no—
163#11924005—42no—
164#11924005—42no—
165C4×Dic73—42no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73C22×C412yes—
170C2×Q8×C73#4112yes—
171C2×Q8×C73#4112yes—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C4⋊C4×C73—24no—
184C4⋊C4×C73—24no—
185C4⋊C4×C73—24no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73C22×C412yes—
190C4⋊C4×C73#4112yes—
191C4⋊C4×C73#4112yes—
192C42×C73—24no—
193C42×C73—24no—
194C42×C73—24no—
195C42×C73—22no—
196C42×C73—22no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—41no—
202C42×C73—41no—
203C42×C73—42no—
204C42×C73—42no—
205C42×C73—42no—
206C42×C73C22×C412yes—
207C42×C73#4112yes—
208C23×C73C22×D411yescharacteristic
209C22×D73—22no—
210C22×D73—41no—
211C22×D73—41no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—22no—
219Q8×C73#11714yescomplemented
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
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—22no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
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×C73C4×D412yes—
247C2×C4×C73C4×D412yes—
248C2×C4×C73AGL(1,ℤ/8)12yes—
249C2×C4×C73#13712yes—
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
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—
265D292—22no—
266D292—22no—
267D292—41no—
268D292—41no—
269C4×D73—48no—
270#10596831—22no—
271#10596831—22no—
272#10596831—41no—
273#10596831—41no—
274C22×C73C2×C4×D411yescharacteristic
275C22×C73#52511yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C73C2×#13711yescharacteristic
279C22×C73#57511yescharacteristic
280C22×C73#57511yescharacteristic
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—81no—
285C2×D73—81no—
286C2×D73—81no—
287C2×D73—81no—
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—24no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—42no—
297C4×C73—42no—
298C4×C73C2×#11711yescharacteristic
299C4×C73C4×#4111yescharacteristic
300C4×C73#52911yescharacteristic
301C4×C73#52911yescharacteristic
302Dic73—42no—
303Dic73—81no—
304Dic73—81no—
305#21124—731nomaximal, Sylow
306C2×C73#229011yescharacteristic
307C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
308C2×C73C4×#13711yescharacteristic
309C2×C73#311111yescharacteristic
310C2×C73#311211yescharacteristic
311C2×C73#311511yescharacteristic
312C2×C73#311711yescharacteristic
313D73—82no—
314C2×#440—731no—
315#2293—731no—
316C42×Q8D7311yescharacteristic, complemented
317#1926—731no—
318#1925—731no—
319#1921—731no—
320#1918—731no—
321#1880—731no—
322#1880—731no—
323#1865—731no—
324#1863—731no—
325#1776—731no—
326#1763—731no—
327#1762—731no—
328#1745—731no—
329C73#2112411yescharacteristic, complemented, Sylow
330C22×SD16—731no—
331C2×C4×Q8C2×D7311yescharacteristic
332C2×C4×Q8C2×D7311yescharacteristic
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4—731no—
335C2×#117—1461no—
336#440—738no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#106—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×#102—731no—
344C2×#102—731no—
345C2×#102—731no—
346C2×#97—731no—
347C2×#97—731no—
348#390—731no—
349#390—731no—
350#389—731no—
351#389—731no—
352#388—1461no—
353#388—1461no—
354C4×C4⋊C4—21no—
355C4×C4⋊C4—21no—
356C4×C4⋊C4—21no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4C2×D7311yescharacteristic
359C4×C4⋊C4C2×D7311yescharacteristic
360C4×C4⋊C4C2×D7311yescharacteristic
361C4×C22⋊C4—1461no—
362C43—21no—
363C43C2×D7311yescharacteristic
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374C22×Q8C22×D7311yescharacteristic
375C22×D4—731no—
376C23×C4—1461no—
377C2×SD16—734no—
378C2×SD16—1462no—
379C2×SD16—1462no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8—731no—
383C22×C8—731no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8—22no—
389C4×Q8C4×D7314yes—
390C4×D4—734no—
391C4×D4—1462no—
392#117—2924no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4C22×D7311yescharacteristic
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C22⋊C4—1461no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—1461no—
416C2×C42C22⋊D7311yescharacteristic
417C2×C42C22⋊D7311yescharacteristic
418C2×C42C22⋊D7311yescharacteristic
419C2×C42C22⋊D7311yescharacteristic
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423#107—734no—
424#107—1462no—
425#106—1462no—
426#103—734no—
427#103—1462no—
428#103—1462no—
429#102—734no—
430#102—1462no—
431#102—1462no—
432#97—734no—
433#97—1462no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440C24—1461no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D7312yes—
445C2×Q8#1192402812yes—
446C2×Q8#1192402812yes—
447C2×D4—732no—
448C2×D4—1461no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1464no—
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468C22×C4C2×C22⋊D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470SD16—1468no—
471C2×C8—732no—
472C2×C8—732no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—732no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4C2×C4×D7312yes—
498C4⋊C4#1192402812yes—
499C4⋊C4#1192402812yes—
500C22⋊C4—2922no—
501C22⋊C4—2922no—
502C22⋊C4—2922no—
503C22⋊C4—2922no—
504C22⋊C4—1464no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—2922no—
520C42C2×C4×D7312yes—
521C42#1192402812yes—
522C23—2921no—
523C23—2921no—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23C22×C22⋊D7311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1455654814yescomplemented
534D4—2921no—
535D4—2921no—
536D4—1462no—
537D4—1462no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—2928no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—1462no—
574C2×C4C4×C22⋊D7312yes—
575C2×C4C4×C22⋊D7312yes—
576C2×C4#1455661612yes—
577C2×C4#1455662212yes—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4C2×#1192402811yescharacteristic
580C2×C4C2×#1192402811yescharacteristic
581C2×C4C2×#1192402811yescharacteristic
582C2×C4C2×#1192402811yescharacteristic
583C2×C4C2×#1192402811yescharacteristic
584C8—2921no—
585C8—2921no—
586C8—1462no—
587C8—1462no—
588C22—2922no—
589C22—2922no—
590C22—2922no—
591C22—5841no—
592C22—5841no—
593C22—5841no—
594C22—5841no—
595C22C2×C4×C22⋊D7311yescharacteristic
596C22#1798909811yescharacteristic
597C22#1798909811yescharacteristic
598C22C2×#1455661611yescharacteristic
599C22C2×#1455662211yescharacteristic
600C22#1798938511yescharacteristic
601C22#1798938511yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—2922no—
613C4—5841no—
614C4—5841no—
615C4C2×#1455654811yescharacteristic
616C4C4×#1192402811yescharacteristic
617C4#1798912111yescharacteristic
618C4#1798912111yescharacteristic
619C2—5842no—
620C2#2261237411yescharacteristic
621C2C4×#1455661611yescharacteristic
622C2C4×#1455662211yescharacteristic
623C2#2262004011yescharacteristic
624C2#2262005311yescharacteristic
625C2#2262008211yescharacteristic
626C2#2262010211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC42×Q8×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC42×Q8D73
hypocenterC73#21124
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
5. #22610523C2semidirect
6. #22610502C2semidirect
9. #22610199C2semidirect
10. #22610185C2semidirect
11. #22609829C2semidirect
12. #22609761C2semidirect
14. #22609680C2semidirect
15. C42×Q8×C73C2semidirect
19. #17988685C4semidirect
219. Q8×C73#117semidirect
316. C42×Q8D73semidirect
329. C73#21124semidirect
533. Q8#14556548semidirect

Maximal subgroups

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

subgroupindex
C2×#179886852
#226123952
#226105582
#226105512
#226105232
#226105022
#226102862
#226101992
#226101852
#226098292
#226097612
#226097592
#226096802
C42×Q8×C732
#2112473