Finite Group Databasebeta

← #33090324

Subgroups — #33090324

Order 18688.

874 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×#17988688C211yesmaximal, characteristic
2#22612406C211yesmaximal, characteristic, complemented
3#22612393C211yesmaximal, characteristic
4#22610814C211yesmaximal, characteristic
5#22610667C211yesmaximal, characteristic, complemented
6#22610664C211yesmaximal, characteristic, complemented
7#22610625C211yesmaximal, characteristic
8#22610618C211yesmaximal, characteristic, complemented
9#22610617C211yesmaximal, characteristic
10#22610610C211yesmaximal, characteristic, complemented
11#22609831C211yesmaximal, characteristic, complemented
12#22609763C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic
14C4×#14556501C211yesmaximal, characteristic, complemented
15C73×#1968C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556547—21no—
19#17988688C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988494C2211yescharacteristic
32#17988490C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988483C2211yescharacteristic
36#17988483C2211yescharacteristic
37#17988481—21no—
38#17988475—21no—
39#17988472—21no—
40#17988470—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43#17988438—21no—
44C22⋊C4×Dic73—21no—
45#17988432—21no—
46C42×Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53C22×SD16×C73C2211yescharacteristic
54C2×#106×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×C4×C8×C73C2211yescharacteristic
58#391×C73C2211yescharacteristic
59#390×C73C2211yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63Q8×Dic73—22no—
64Q8×Dic73C2×C414yes—
65D4×Dic73—22no—
66D4×Dic73C2×C414yes—
67#14556547—44no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73D411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89#14556525—22no—
90#14556525C2×C414yes—
91#14556515—22no—
92#14556515C2×C414yes—
93#14556512—22no—
94#14556512C2×C414yes—
95#14556501—22no—
96#14556501—22no—
97#14556501—22no—
98#14556501C2×C414yes—
99C4×#10596831—22no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×Q8×C73C2311yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C2×SD16×C73—22no—
111C2×SD16×C73—22no—
112C2×SD16×C73C2×C414yes—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C42×C73C2311yescharacteristic
120#106×C73—22no—
121#103×C73—22no—
122#102×C73—22no—
123C4×C8×C73—22no—
124#95×C73—21no—
125#95×C73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—24no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—42no—
132C22×Dic73—42no—
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831C22×C412yes—
142C2×#10596831#4112yes—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—24no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73C22×C412yes—
151C4⋊Dic73C22×C412yes—
152#11924007—42no—
153#11924007—42no—
154#11924007—42no—
155#11924007—42no—
156C22⋊Dic73—24no—
157C22⋊Dic73—42no—
158C22⋊Dic73—42no—
159C4×Dic73—22no—
160C4×Dic73—22no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—24no—
164C4×Dic73—41no—
165C4×Dic73—41no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73C22×C412yes—
169C4×Dic73#4112yes—
170C24×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73C22×C412yes—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73C22×C412yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189SD16×C73—28no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73C22×C412yes—
193C2×C8×C73#4112yes—
194C2×C8×C73#4112yes—
195C2×C8×C73#4112yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—42no—
202C42×C73—21no—
203C42×C73—21no—
204C23×C73—22no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73C22×D411yescharacteristic
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73—41no—
215Q8×C73—41no—
216D4×C73—22no—
217D4×C73—22no—
218D4×C73—41no—
219D4×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73AGL(1,ℤ/8)12yes—
234C2×C4×C73#13712yes—
235C2×C4×C73C23×C411yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—22no—
246C2×Dic73—22no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—48no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73C4×D412yes—
265C2×Dic73C4×D412yes—
266C8×C73—22no—
267C8×C73#11714yescomplemented
268#10596831—22no—
269#10596831—22no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73C2×C4×D411yescharacteristic
278C22×C73#53211yescharacteristic
279C22×C73#53211yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73C2×#13711yescharacteristic
282C22×C73#57511yescharacteristic
283C22×C73#57511yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—42no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73C2×#11711yescharacteristic
290C4×C73C4×#4111yescharacteristic
291C4×C73#52911yescharacteristic
292C4×C73#52911yescharacteristic
293Dic73—22no—
294Dic73—22no—
295Dic73—41no—
296Dic73—41no—
297Dic73—42no—
298Dic73—42no—
299Dic73—81no—
300Dic73—81no—
301#21288—731nomaximal, Sylow
302C2×C73—82no—
303C2×C73#229311yescharacteristic
304C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
305C2×C73C4×#13711yescharacteristic
306C2×C73#318011yescharacteristic
307C2×C73#318111yescharacteristic
308C2×C73#318211yescharacteristic
309C2×C73#318311yescharacteristic
310C2×#440—731no—
311#2295—731no—
312#2293—731no—
313#1973—731no—
314#1968D7311yescharacteristic
315#1943—731no—
316#1942—731no—
317#1936—731no—
318#1935—731no—
319#1934—731no—
320#1933—731no—
321#1776—731no—
322#1763—731no—
323#1762—731no—
324C4×#97—731no—
325C73#2128811yescharacteristic, complemented, Sylow
326C22×SD16C2×D7311yescharacteristic
327C2×C4×Q8—731no—
328C2×C4×D4—731no—
329C2×#117—1461no—
330#440—738no—
331C2×#107—731no—
332C2×#107—731no—
333C2×#106C2×D7311yescharacteristic
334C2×#103—731no—
335C2×#103—731no—
336C2×#103C2×D7311yescharacteristic
337C2×#102—731no—
338C2×#102—731no—
339C2×#102C2×D7311yescharacteristic
340C2×#97—731no—
341C2×#97—731no—
342C2×#97—731no—
343C2×#97—731no—
344C2×C4×C8—731no—
345C2×C4×C8C2×D7311yescharacteristic
346#391—731no—
347#391C2×D7311yescharacteristic
348#390—731no—
349#390C2×D7311yescharacteristic
350#389—731no—
351#389—731no—
352#389—731no—
353#389—731no—
354#388—1461no—
355#388—1461no—
356#387—1461no—
357#387—1461no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362#376—1461no—
363C43—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370C22×Q8C22×D7311yescharacteristic
371C22×D4C22×D7311yescharacteristic
372C23×C4—1461no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C4×D7314yes—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C4×Q8—734no—
381C4×Q8—1462no—
382C4×D4—734no—
383C4×D4—1462no—
384#117—2924no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22×D7311yescharacteristic
411#107—734no—
412#107—1462no—
413#106—22no—
414#103—22no—
415#103—734no—
416#103—1462no—
417#102—22no—
418#102—734no—
419#102—1462no—
420#97—734no—
421#97—1462no—
422#97—1462no—
423#97—1462no—
424C4×C8—22no—
425C4×C8—1462no—
426#95—21no—
427#95—21no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D7312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1464no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4D4×D7311yescharacteristic
464C22×C4D4×D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—732no—
470C2×C8—732no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C2×C4×D7312yes—
476C2×C8#1192402712yes—
477C2×C8#1192402712yes—
478C2×C8#1192402712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—2922no—
484C4⋊C4—2922no—
485C4⋊C4—2922no—
486C4⋊C4—2922no—
487C4⋊C4—732no—
488C4⋊C4—732no—
489C4⋊C4—1464no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—2922no—
499C22⋊C4—2922no—
500C22⋊C4—1464no—
501C42—21no—
502C42—21no—
503C42—2921no—
504C42—2921no—
505C42—2922no—
506C42—2922no—
507C42—732no—
508C42—732no—
509C42—1464no—
510C42—1462no—
511C42—1462no—
512C42—1461no—
513C42—1461no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D7311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2928no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—732no—
555C2×C4—732no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1462no—
562C2×C4—1462no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4#1455661812yes—
569C2×C4#1455662012yes—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4#41×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C2×C4C2×#1192402711yescharacteristic
574C2×C4C2×#1192402711yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C8—22no—
577C8—1462no—
578C8—1462no—
579C8#1455654714yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D7311yescharacteristic
588C22#1798913211yescharacteristic
589C22#1798913511yescharacteristic
590C22C2×#1455661811yescharacteristic
591C22C2×#1455662011yescharacteristic
592C22#1798938411yescharacteristic
593C22#1798938411yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2922no—
600C4—2922no—
601C4—2921no—
602C4—2921no—
603C4—5841no—
604C4—5841no—
605C4—1462no—
606C4—1462no—
607C4C2×#1455654711yescharacteristic
608C4C4×#1192402711yescharacteristic
609C4#1798911611yescharacteristic
610C4#1798911811yescharacteristic
611C2—82no—
612C2#2261239311yescharacteristic
613C2C4×#1455661811yescharacteristic
614C2C4×#1455662011yescharacteristic
615C2#2262062211yescharacteristic
616C2#2262063311yescharacteristic
617C2#2262064711yescharacteristic
618C2#2262065711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1968C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1968D73
hypocenterC73#21288
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #22612406C2semidirect
5. #22610667C2semidirect
6. #22610664C2semidirect
8. #22610618C2semidirect
10. #22610610C2semidirect
11. #22609831C2semidirect
12. #22609763C2semidirect
14. C4×#14556501C2semidirect
19. #17988688C4semidirect
267. C8×C73#117semidirect
325. C73#21288semidirect
579. C8#14556547semidirect

Maximal subgroups

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

subgroupindex
C2×#179886882
#226124062
#226123932
#226108142
#226106672
#226106642
#226106252
#226106182
#226106172
#226106102
#226098312
#226097632
#226097582
C4×#145565012
C73×#19682
#2128873