Finite Group Databasebeta

← #25127692

Subgroups — #25127692

Order 11008.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10621885C211yesmaximal, characteristic, complemented
2C2×D4×Dic86C211yesmaximal, characteristic
3C2×#14583055C211yesmaximal, characteristic, complemented
4C2×#14583054C212yesmaximal, complemented
5#19135694C212yesmaximal, complemented
6#19135603C212yesmaximal, complemented
7#19135016C211yesmaximal, characteristic, complemented
8#19134939C212yesmaximal, complemented
9#19134888C211yesmaximal, characteristic
10#19134529C211yesmaximal, characteristic, complemented
11C43×#2409C211yesmaximal, characteristic
12C23×Dic86C2211yescharacteristic
13C23×Dic86C2211yescharacteristic
14C22×#10621885—24no—
15C22×#10621885—28no—
16C22×#10621885C2212yes—
17C22×#10621885C2212yes—
18C22×#10621885C2212yes—
19C23×C43⋊C8C2211yescharacteristic
20D4×Dic86—24no—
21D4×Dic86—24no—
22C2×#13063325C2211yescharacteristic
23C2×#13063324—21no—
24C2×#13063304C2211yescharacteristic
25C2×#13063304C2211yescharacteristic
26C2×#13063302—21no—
27C2×#13063302—21no—
28C2×C4×Dic86C2211yescharacteristic
29C2×D4×Dic43—21no—
30#14583055—24no—
31#14583054—28no—
32C2×#13063249C2212yes—
33C2×#13063248C2212yes—
34C2×#13063248C2212yes—
35C2×#13063248C2212yes—
36C2×C22⋊(C43⋊C8)C2211yescharacteristic
37C2×C22⋊(C43⋊C8)C2212yes—
38#14582661C2212yes—
39#14582654C2212yes—
40C23×Q8×C43C2211yescharacteristic
41C2×C22⋊Q8×C43C2212yes—
42C2×C4×D4×C43C2211yescharacteristic
43C43×#395—21no—
44C43×#393—21no—
45C43×#391C2212yes—
46C43×#386—21no—
47C43×#385—21no—
48C43×#380C2211yescharacteristic
49C22×Dic86—24no—
50C22×Dic86—22no—
51C22×Dic86—21no—
52C22×Dic86—21no—
53C22×Dic86—22no—
54C22×Dic86—22no—
55C22×Dic86—24no—
56C22×Dic86—42no—
57C22×Dic86—42no—
58C22×Dic86D414yes—
59C22×Dic86C2311yescharacteristic
60C22×Dic86C2311yescharacteristic
61C22×Dic86C2311yescharacteristic
62C23×Dic43—21no—
63C23×Dic43—21no—
64C2×#10621885—232no—
65C2×#10621885—24no—
66C2×#10621885—24no—
67C2×#10621885—24no—
68C2×#10621885—24no—
69C2×#10621885—24no—
70C2×#10621885—24no—
71C2×#10621885—24no—
72C2×#10621885—42no—
73C2×#10621885D418yescomplemented
74C22×C43⋊C8—22no—
75C22×C43⋊C8—24no—
76C22×C43⋊C8C2312yes—
77C22×C43⋊C8C2312yes—
78#13063325—22no—
79#13063325—22no—
80#13063324—28no—
81#13063304—22no—
82#13063304—22no—
83#13063304—22no—
84#13063304D414yes—
85#13063302—28no—
86#13063302—28no—
87C4×Dic86—22no—
88C4×Dic86—22no—
89D4×Dic43—28no—
90C2×C4⋊Dic43—21no—
91C2×C4⋊Dic43—21no—
92C2×C4⋊Dic43C2311yescharacteristic
93C2×Dic43⋊C4—21no—
94C2×Dic43⋊C4—21no—
95C2×Dic43⋊C4—21no—
96C2×C22⋊Dic43—21no—
97C2×C22⋊Dic43—21no—
98C2×C4×Dic43—21no—
99#13063249—24no—
100#13063248—24no—
101#13063248—24no—
102#13063248—42no—
103C22⋊(C43⋊C8)—22no—
104C22⋊(C43⋊C8)—24no—
105C22×Q8×C43—21no—
106C22×Q8×C43—24no—
107C22×Q8×C43—22no—
108C22×Q8×C43—22no—
109C22×Q8×C43—42no—
110C22×Q8×C43C2312yes—
111C22×D4×C43D411yescharacteristic
112C23×C4×C43—21no—
113C23×C4×C43C2311yescharacteristic
114C23×C4×C43C2311yescharacteristic
115C22⋊Q8×C43—24no—
116C22⋊Q8×C43—42no—
117C4×D4×C43—22no—
118C4×D4×C43—22no—
119C2×C4⋊C4×C43—22no—
120C2×C4⋊C4×C43D411yescharacteristic
121C2×C4⋊C4×C43C2312yes—
122C2×C22⋊C4×C43—22no—
123C2×C22⋊C4×C43—21no—
124C2×C22⋊C4×C43D411yescharacteristic
125C2×C22⋊C4×C43D411yescharacteristic
126C2×C42×C43C2311yescharacteristic
127#95×C43—21no—
128#95×C43—22no—
129#95×C43—22no—
130C24×C43C2×D411yescharacteristic
131C24×C43C2×D411yescharacteristic
132C2×Q8×C43—22no—
133C2×Q8×C43—28no—
134C2×Q8×C43—22no—
135C2×Q8×C43—22no—
136C2×Q8×C43—22no—
137C2×Q8×C43—22no—
138C2×Q8×C43—44no—
139C2×Q8×C43—48no—
140C2×Q8×C43—44no—
141C2×Q8×C43—42no—
142C2×Q8×C43C2×D414yes—
143C2×D4×C43—22no—
144C2×D4×C43—22no—
145C2×D4×C43Q1614yescomplemented
146C22×C4×C43—22no—
147C22×C4×C43—22no—
148C22×C4×C43—21no—
149C22×C4×C43—21no—
150C22×C4×C43—22no—
151C22×C4×C43—22no—
152C22×C4×C43—22no—
153C22×C4×C43—22no—
154C22×C4×C43—44no—
155C22×C4×C43—42no—
156C22×C4×C43C2×D412yes—
157C22×C4×C43C2×D412yes—
158C22×C4×C43C2×D411yescharacteristic
159C22×C4×C43C2×D411yescharacteristic
160C22×C4×C43C2×D411yescharacteristic
161C22×C4×C43C2×D411yescharacteristic
162C22×C4×C43C2411yescharacteristic
163C22×Dic43—24no—
164C22×Dic43—24no—
165C22×Dic43—21no—
166C22×Dic43—21no—
167C22×Dic43—21no—
168C22×Dic43—21no—
169C22×Dic43—44no—
170C22×Dic43—44no—
171C2×Dic86—21no—
172C2×Dic86—22no—
173C2×Dic86—22no—
174C2×Dic86—22no—
175C2×Dic86—22no—
176C2×Dic86—22no—
177C2×Dic86—28no—
178C2×Dic86—24no—
179C2×Dic86—24no—
180C2×Dic86—21no—
181C2×Dic86—21no—
182C2×Dic86—44no—
183C2×Dic86—48no—
184C2×Dic86—48no—
185C2×Dic86—44no—
186C2×Dic86—44no—
187C2×Dic86—44no—
188C2×Dic86—44no—
189C2×Dic86C2×D412yes—
190C2×Dic86C2×D412yes—
191C2×Dic86C2×D414yes—
192C2×Dic86C2×D412yes—
193C4⋊C4×C43—22no—
194C4⋊C4×C43—22no—
195C4⋊C4×C43—22no—
196C4⋊C4×C43—44no—
197C22⋊C4×C43—22no—
198C22⋊C4×C43—22no—
199C22⋊C4×C43—44no—
200C22⋊C4×C43—42no—
201C42×C43—21no—
202C42×C43—21no—
203C22⋊Dic43—24no—
204C22⋊Dic43—24no—
205#10621885—216no—
206#10621885—216no—
207#10621885—48no—
208C4⋊Dic43—24no—
209C4⋊Dic43—24no—
210C4⋊Dic43—21no—
211C4⋊Dic43C2×D412yes—
212Dic43⋊C4—24no—
213Dic43⋊C4—24no—
214Dic43⋊C4—24no—
215C4×Dic43—24no—
216C2×C43⋊C8—28no—
217C2×C43⋊C8—22no—
218C2×C43⋊C8—22no—
219C2×C43⋊C8—22no—
220C2×C43⋊C8—22no—
221C2×C43⋊C8—22no—
222C2×C43⋊C8C2×D414yes—
223C23×C43—22no—
224C23×C43—22no—
225C23×C43—21no—
226C23×C43—22no—
227C23×C43—21no—
228C23×C43—22no—
229C23×C43#12012yes—
230C23×C43C2×Q1612yes—
231C23×C43C2×Q1612yescomplemented
232C23×C43#13712yes—
233C23×C43C22×D411yescharacteristic
234Q8×C43—24no—
235Q8×C43—24no—
236Q8×C43—48no—
237Q8×C43—42no—
238Q8×C43—42no—
239D4×C43—28no—
240C2×C4×C43—21no—
241C2×C4×C43—22no—
242C2×C4×C43—21no—
243C2×C4×C43—22no—
244C2×C4×C43—22no—
245C2×C4×C43—24no—
246C2×C4×C43—21no—
247C2×C4×C43—21no—
248C2×C4×C43—24no—
249C2×C4×C43—22no—
250C2×C4×C43—22no—
251C2×C4×C43—48no—
252C2×C4×C43—42no—
253C2×C4×C43—42no—
254C2×C4×C43—41no—
255C2×C4×C43—41no—
256C2×C4×C43—42no—
257C2×C4×C43—42no—
258C2×C4×C43—42no—
259C2×C4×C43—42no—
260C2×C4×C43#12012yes—
261C2×C4×C43C2×Q1612yescomplemented
262C2×C4×C43C22×D411yescharacteristic
263C2×C4×C43C22×D412yes—
264C2×C4×C43C22×D411yescharacteristic
265C2×C4×C43H(5,𝔽2)12yes—
266C2×Dic43—22no—
267C2×Dic43—24no—
268C2×Dic43—22no—
269C2×Dic43—22no—
270C2×Dic43—22no—
271C2×Dic43—48no—
272C2×Dic43—48no—
273C2×Dic43—42no—
274C2×Dic43—42no—
275C2×Dic43—42no—
276Dic86—22no—
277Dic86—24no—
278Dic86—22no—
279Dic86—22no—
280Dic86—24no—
281Dic86—48no—
282Dic86—48no—
283Dic86—41no—
284Dic86—41no—
285Dic86—42no—
286Dic86#12014yes—
287C43⋊C8—24no—
288C43⋊C8—42no—
289#23856—431nomaximal, Sylow
290C22×C43—22no—
291C22×C43—22no—
292C22×C43—24no—
293C22×C43—24no—
294C22×C43—22no—
295C22×C43—24no—
296C22×C43—41no—
297C22×C43—41no—
298C22×C43—42no—
299C22×C43#45114yescomplemented
300C22×C43C2×#12011yescharacteristic
301C22×C43D4212yes—
302C22×C43C22×Q1611yescharacteristic
303C22×C43C2×#13711yescharacteristic
304C22×C43#57812yes—
305C4×C43—22no—
306C4×C43—22no—
307C4×C43—41no—
308C4×C43—44no—
309C4×C43—41no—
310C4×C43—82no—
311C4×C43C2×#12011yescharacteristic
312C4×C43#53411yescharacteristic
313C4×C43D4212yes—
314Dic43—24no—
315Dic43—42no—
316Dic43—42no—
317Dic43—42no—
318C23×Q16—431no—
319C2×D4×Q8—431no—
320C2×#451—431no—
321C2×#451—432no—
322#2409D4311yescharacteristic
323#2016—432no—
324#2001—432no—
325#1892—431no—
326#1878—432no—
327#1869—431no—
328#1787—431no—
329C2×C43—24no—
330C2×C43—42no—
331C2×C43#240311yescharacteristic
332C2×C43C2×#45111yescharacteristic
333C2×C43#301111yescharacteristic
334C2×C43#327012yes—
335C2×C43D4×Q1612yes—
336C23×Q8—431no—
337C23×Q8—431no—
338C23×Q8C2×D4311yescharacteristic
339C22×Q16—432no—
340C22×Q16—432no—
341C22×Q16—432no—
342C22×Q16—868no—
343C22×Q16—864no—
344C23×C8—431no—
345D4×Q8—864no—
346D4×Q8—864no—
347C2×C4⋊Q8—431no—
348C2×C4⋊Q8—861no—
349C2×C22⋊Q8—431no—
350C2×C22⋊Q8—431no—
351C2×C22⋊Q8—861no—
352C2×C22⋊Q8—861no—
353C2×C22⋊Q8C2×D4312yes—
354C2×C4×Q8—431no—
355C2×C4×D4—861no—
356C2×C4×D4C2×D4311yescharacteristic
357#451—868no—
358#451—864no—
359C2×#103—432no—
360C2×#103—432no—
361C2×#103—432no—
362C2×#103—432no—
363C2×C22⋊C8—432no—
364C2×C22⋊C8—431no—
365#395—21no—
366#393—21no—
367#391C2×D4312yes—
368#386—21no—
369#385—21no—
370#380C2×D4311yescharacteristic
371#340—432no—
372#336—432no—
373C43#2385611yescharacteristic, complemented, Sylow
374C22×Q8—21no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8—22no—
378C22×Q8—42no—
379C22×Q8—434no—
380C22×Q8—431no—
381C22×Q8—431no—
382C22×Q8—431no—
383C22×Q8—862no—
384C22×Q8—862no—
385C22×Q8—864no—
386C22×Q8—864no—
387C22×Q8—862no—
388C22×Q8—861no—
389C22×Q8—861no—
390C22×Q8—1722no—
391C22×Q8—1722no—
392C22×Q8C22×D4312yes—
393C22×D4C22⋊D4311yescharacteristic
394C23×C4—21no—
395C23×C4—861no—
396C23×C4—861no—
397C23×C4C22×D4311yescharacteristic
398C23×C4C22×D4311yescharacteristic
399C2×Q16—438no—
400C2×Q16—864no—
401C2×Q16—864no—
402C2×Q16—864no—
403C2×Q16—864no—
404C2×Q16—8632no—
405C2×Q16—864no—
406C2×Q16—864no—
407C2×Q16—864no—
408C2×Q16—1722no—
409C22×C8—432no—
410C22×C8—432no—
411C22×C8—864no—
412C22×C8—862no—
413C4⋊Q8—862no—
414C4⋊Q8—862no—
415C4⋊Q8—868no—
416C22⋊Q8—24no—
417C22⋊Q8—42no—
418C22⋊Q8—434no—
419C22⋊Q8—868no—
420C22⋊Q8—868no—
421C22⋊Q8—862no—
422C22⋊Q8—862no—
423C22⋊Q8—862no—
424C4×Q8—862no—
425C4×Q8—862no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—868no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—431no—
431C2×C4⋊C4—861no—
432C2×C4⋊C4—861no—
433C2×C4⋊C4—861no—
434C2×C4⋊C4—861no—
435C2×C4⋊C4—861no—
436C2×C4⋊C4C22⋊D4311yescharacteristic
437C2×C4⋊C4C22×D4312yes—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—861no—
441C2×C22⋊C4—861no—
442C2×C22⋊C4C22⋊D4311yescharacteristic
443C2×C22⋊C4C22⋊D4311yescharacteristic
444C2×C42—861no—
445C2×C42C22×D4311yescharacteristic
446#103—864no—
447#103—864no—
448#103—864no—
449#103—1722no—
450C22⋊C8—864no—
451C22⋊C8—862no—
452#95—21no—
453#95—22no—
454#95—22no—
455C24C2×C22⋊D4311yescharacteristic
456C24C2×C22⋊D4311yescharacteristic
457C2×Q8—22no—
458C2×Q8—28no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—48no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—434no—
468C2×Q8—432no—
469C2×Q8—432no—
470C2×Q8—432no—
471C2×Q8—864no—
472C2×Q8—868no—
473C2×Q8—864no—
474C2×Q8—861no—
475C2×Q8—861no—
476C2×Q8—861no—
477C2×Q8—862no—
478C2×Q8—862no—
479C2×Q8—862no—
480C2×Q8—862no—
481C2×Q8—862no—
482C2×Q8—1724no—
483C2×Q8—1724no—
484C2×Q8—1724no—
485C2×Q8—1728no—
486C2×Q8—1724no—
487C2×Q8—1728no—
488C2×Q8—1724no—
489C2×Q8D4×D4314yes—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4#1062188514yescomplemented
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—861no—
504C22×C4—864no—
505C22×C4—864no—
506C22×C4—861no—
507C22×C4—861no—
508C22×C4—861no—
509C22×C4—1724no—
510C22×C4—1724no—
511C22×C4D4×D4312yes—
512C22×C4D4×D4312yes—
513C22×C4C2×C22⋊D4311yescharacteristic
514C22×C4C2×C22⋊D4311yescharacteristic
515C22×C4C2×C22⋊D4311yescharacteristic
516C22×C4C2×C22⋊D4311yescharacteristic
517C22×C4C23×D4311yescharacteristic
518Q16—8616no—
519Q16—8616no—
520Q16—1728no—
521C2×C8—434no—
522C2×C8—868no—
523C2×C8—862no—
524C2×C8—862no—
525C2×C8—862no—
526C2×C8—862no—
527C2×C8—862no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—432no—
533C4⋊C4—864no—
534C4⋊C4—864no—
535C4⋊C4—864no—
536C4⋊C4—864no—
537C4⋊C4—864no—
538C4⋊C4—861no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—864no—
544C22⋊C4—864no—
545C42—21no—
546C42—21no—
547C42—864no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23#1306329312yes—
555C23C2×#1062188512yes—
556C23C2×#1062188512yescomplemented
557C23#1306335512yes—
558C23C22×C22⋊D4311yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—434no—
565Q8—862no—
566Q8—864no—
567Q8—862no—
568Q8—862no—
569Q8—864no—
570Q8—1728no—
571Q8—1728no—
572Q8—1721no—
573Q8—1721no—
574Q8—1722no—
575D4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—862no—
597C2×C4—864no—
598C2×C4—862no—
599C2×C4—862no—
600C2×C4—862no—
601C2×C4—1728no—
602C2×C4—1728no—
603C2×C4—1722no—
604C2×C4—1722no—
605C2×C4—1722no—
606C2×C4#1306329312yes—
607C2×C4C2×#1062188512yescomplemented
608C2×C4C2×D4×D4311yescharacteristic
609C2×C4C2×D4×D4312yes—
610C2×C4C2×D4×D4311yescharacteristic
611C2×C4#1306337012yes—
612C8—864no—
613C8—1722no—
614C22—24no—
615C22—22no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22#1458305414yescomplemented
624C22C2×#1306329311yescharacteristic
625C22D4×C22⋊D4312yes—
626C22C22×#1062188511yescharacteristic
627C22C2×#1306335511yescharacteristic
628C22#1458371412yes—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—44no—
633C4—41no—
634C4—82no—
635C4—864no—
636C4—1722no—
637C4—1722no—
638C4—1722no—
639C4#120×D4311yescharacteristic
640C4#1458345911yescharacteristic
641C4#1458353112yes—
642C2—24no—
643C2—42no—
644C2#1913790611yescharacteristic
645C2C2×#1458305411yescharacteristic
646C2#1914388811yescharacteristic
647C2#1914616112yes—
648C2D4×#1062188512yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2409C2
socleC23×C43C22×D4
prime radicalC22×D4×C43D4
hypercenter#2409D43
hypocenterC43#23856
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#10621885C2semidirect
3. C2×#14583055C2semidirect
4. C2×#14583054C2semidirect
5. #19135694C2semidirect
6. #19135603C2semidirect
7. #19135016C2semidirect
8. #19134939C2semidirect
10. #19134529C2semidirect
73. C2×#10621885D4semidirect
145. C2×D4×C43Q16semidirect
231. C23×C43C2×Q16semidirect
261. C2×C4×C43C2×Q16semidirect
299. C22×C43#451semidirect
373. C43#23856semidirect
492. C2×D4#10621885semidirect
556. C23C2×#10621885semidirect
607. C2×C4C2×#10621885semidirect
623. C22#14583054semidirect

Maximal subgroups

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

subgroupindex
C23×#106218852
C2×D4×Dic862
C2×#145830552
C2×#145830542
#191356942
#191356032
#191350162
#191349392
#191348882
#191345292
C43×#24092
#2385643