Finite Group Databasebeta

← #25159673

Subgroups — #25159673

Order 11008.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583190C211yesmaximal, characteristic
2C2×C4×#10621873C211yesmaximal, characteristic
3#19138434C211yesmaximal, characteristic, complemented
4C4⋊C4×D172C211yesmaximal, characteristic
5#19136137C211yesmaximal, characteristic, complemented
6#19136073C211yesmaximal, characteristic, complemented
7#19135975C211yesmaximal, characteristic, complemented
8#19135929C211yesmaximal, characteristic, complemented
9#19135323C211yesmaximal, characteristic, complemented
10#19134928C211yesmaximal, characteristic
11#19134928C211yesmaximal, characteristic
12#19134866C211yesmaximal, characteristic, complemented
13#19134859C211yesmaximal, characteristic
14#19134859C211yesmaximal, characteristic
15C43×#1916C211yesmaximal, characteristic, complemented
16C22×#10621873C2211yescharacteristic
17C2×#13063325C2211yescharacteristic
18C2×C4×Dic86C2211yescharacteristic
19C2×C4×D172C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21C2×C4×D172C2211yescharacteristic
22C2×C4⋊C4×D43—21no—
23#14583190—24no—
24C4×#10621873—24no—
25C2×#13063257C2211yescharacteristic
26C2×#13063257C2211yescharacteristic
27C2×#13063257C2211yescharacteristic
28C2×#13063257C2211yescharacteristic
29C2×#13063257C2211yescharacteristic
30C2×#13063250C2211yescharacteristic
31C2×#13063250C2211yescharacteristic
32C2×#13063250C2211yescharacteristic
33C2×#13063247C2211yescharacteristic
34C2×#13063247C2211yescharacteristic
35C2×#13063247C2211yescharacteristic
36#14582833—21no—
37#14582819—21no—
38#14582805C2211yescharacteristic
39#14582805C2211yescharacteristic
40#14582803—21no—
41#14582797C2211yescharacteristic
42#14582797C2211yescharacteristic
43#14582797C2211yescharacteristic
44#14582797C2211yescharacteristic
45#14582785—21no—
46#14582773—21no—
47#14582773—21no—
48#14582769C2211yescharacteristic
49#14582756—21no—
50#14582662C2211yescharacteristic
51#14582662C2211yescharacteristic
52#14582662C2211yescharacteristic
53#14582662C2211yescharacteristic
54C2×C4⋊C8×C43C2211yescharacteristic
55C2×C4×C8×C43C2211yescharacteristic
56C4×C4⋊C4×C43C2211yescharacteristic
57C43×#336C2211yescharacteristic
58C43×#336C2211yescharacteristic
59C43×#336C2211yescharacteristic
60C43×#336C2211yescharacteristic
61C22×Dic86—21no—
62C22×Dic86C2311yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65C22×C4×D43—21no—
66C22×C4×D43—21no—
67C2×#10621873—22no—
68C2×#10621873—22no—
69C2×#10621873Q814yescomplemented
70#13063325—22no—
71#13063325—41no—
72C4×Dic86—22no—
73C4×Dic86—41no—
74C4×D172—22no—
75C4×D172—22no—
76C4×D172—22no—
77C4×D172—22no—
78C4×D172—41no—
79C4×D172—41no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43—21no—
82C2×C4⋊Dic43—21no—
83C2×C4⋊Dic43C2311yescharacteristic
84C2×C4⋊Dic43C2311yescharacteristic
85C2×C4⋊Dic43C2311yescharacteristic
86C2×C4⋊Dic43C2311yescharacteristic
87C2×C4⋊Dic43C2311yescharacteristic
88C2×C4⋊Dic43C2311yescharacteristic
89C2×Dic43⋊C4—21no—
90C2×Dic43⋊C4—21no—
91C2×Dic43⋊C4—21no—
92C2×Dic43⋊C4—21no—
93C2×Dic43⋊C4—21no—
94C4⋊C4×D43—44no—
95C2×(C2×D43)⋊C4—21no—
96C2×(C2×D43)⋊C4—21no—
97C2×(C2×D43)⋊C4—21no—
98C2×C4×Dic43—21no—
99#13063257—22no—
100#13063257—22no—
101#13063257—22no—
102#13063257—22no—
103#13063257—41no—
104#13063250—22no—
105#13063250—22no—
106#13063250—41no—
107#13063247—22no—
108#13063247—22no—
109#13063247D414yes—
110C43⋊#95—21no—
111C43⋊#95—21no—
112C43⋊#95—21no—
113C43⋊#95—21no—
114C43⋊#95—21no—
115C43⋊#95—21no—
116C43⋊#95—21no—
117C43⋊#95—21no—
118C22×C8×C43C2311yescharacteristic
119C22×C8×C43C2311yescharacteristic
120C22×C8×C43C2311yescharacteristic
121C22×C8×C43C2311yescharacteristic
122C2×C4⋊C4×C43D411yescharacteristic
123C2×C4⋊C4×C43D411yescharacteristic
124C2×C42×C43C2311yescharacteristic
125C2×C42×C43C2311yescharacteristic
126C2×C42×C43C2311yescharacteristic
127C4⋊C8×C43—22no—
128C4×C8×C43—22no—
129#95×C43D411yescharacteristic
130#95×C43D411yescharacteristic
131C23×D43—21no—
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2411yescharacteristic
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C2×D172—21no—
147C2×D172—21no—
148C2×D172—21no—
149C2×D172C2×D412yes—
150C2×D172C2×Q812yes—
151C2×D172#4112yes—
152C2×C4×D43—24no—
153C2×C4×D43—42no—
154C2×C4×D43—42no—
155C2×C4×D43—42no—
156C2×C4×D43—42no—
157C2×C4×D43—42no—
158C2×C4×D43—42no—
159C2×C4×D43—81no—
160C2×C4×D43—81no—
161C2×Dic86—21no—
162C2×Dic86—21no—
163C2×Dic86—21no—
164C2×Dic86—21no—
165C2×Dic86—21no—
166C2×Dic86—42no—
167C2×Dic86—42no—
168C2×Dic86—81no—
169C2×Dic86C2×Q812yes—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43—21no—
177C2×C8×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43C2×D412yes—
181C2×C8×C43C2×Q812yes—
182C4⋊C4×C43—41no—
183C4⋊C4×C43SD1614yescomplemented
184C42×C43—21no—
185C42×C43—21no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—21no—
189C42×C43—21no—
190(C2×D43)⋊C4—24no—
191(C2×D43)⋊C4—42no—
192(C2×D43)⋊C4—42no—
193C4⋊Dic43—21no—
194C4⋊Dic43—21no—
195C4⋊Dic43—21no—
196C4⋊Dic43—21no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—42no—
204C4⋊Dic43—42no—
205C4⋊Dic43—81no—
206C4⋊Dic43C2×D412yes—
207C4⋊Dic43#4112yes—
208Dic43⋊C4—42no—
209Dic43⋊C4—42no—
210Dic43⋊C4—42no—
211Dic43⋊C4—81no—
212Dic43⋊C4—81no—
213C4×Dic43—42no—
214#10621873—28no—
215C23×C43C22×D411yescharacteristic
216C22×D43—22no—
217C22×D43—22no—
218C22×D43—22no—
219C22×D43—41no—
220C22×D43—41no—
221C22×D43—41no—
222C22×D43—41no—
223C2×C4×C43—21no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—21no—
238C2×C4×C43C2×SD1612yes—
239C2×C4×C43C2×SD1612yes—
240C2×C4×C43C2×SD1612yes—
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43C22×Q811yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C43C2×#4111yescharacteristic
245C2×C4×C43H(5,𝔽2)11yescharacteristic
246C2×C4×C432−1+411yescharacteristic
247C2×Dic43—22no—
248C2×Dic43—22no—
249C2×Dic43—22no—
250C2×Dic43—41no—
251C2×Dic43—41no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256C2×Dic43—41no—
257C2×Dic43—41no—
258C2×Dic43—41no—
259C2×Dic43—41no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—41no—
265C2×Dic43—41no—
266C2×Dic43—41no—
267C2×Dic43—41no—
268D172—22no—
269D172—22no—
270D172—22no—
271D172C22⋊Q814yes—
272C4×D43—48no—
273C4×D43—84no—
274C4×D43—84no—
275Dic86—22no—
276Dic86—22no—
277Dic86—41no—
278Dic86—41no—
279Dic86—84no—
280C8×C43—22no—
281C8×C43—22no—
282C8×C43—41no—
283C8×C43—41no—
284#26376—431nomaximal, Sylow
285C22×C43#52511yescharacteristic
286C22×C43D4211yescharacteristic
287C22×C43D4×Q811yescharacteristic
288C22×C43C22×SD1611yescharacteristic
289C22×C43#57511yescharacteristic
290C22×C43#57711yescharacteristic
291C22×C43#57711yescharacteristic
292C2×D43—24no—
293C2×D43—42no—
294C2×D43—42no—
295C2×D43—42no—
296C2×D43—42no—
297C2×D43—42no—
298C2×D43—42no—
299C4×C43—22no—
300C4×C43—22no—
301C4×C43—22no—
302C4×C43—41no—
303C4×C43—41no—
304C4×C43—41no—
305C4×C43C2×C22⋊Q811yescharacteristic
306C4×C43#53911yescharacteristic
307C4×C43#54711yescharacteristic
308C4×C43#55411yescharacteristic
309Dic43—42no—
310Dic43—42no—
311Dic43—42no—
312Dic43—81no—
313Dic43—81no—
314Dic43—81no—
315Dic43—81no—
316C2×#476—431no—
317C2×C4×SD16—431no—
318#2451—431no—
319D4×C4⋊C4—431no—
320#2096—431no—
321#2084—431no—
322#2066—431no—
323#2058—431no—
324#1941—431no—
325#1916D4311yescharacteristic, complemented
326#1877—431no—
327#1877—431no—
328#1865—431no—
329#1864—431no—
330#1864—431no—
331C2×C43#244911yescharacteristic
332C2×C43#310111yescharacteristic
333C2×C43#312511yescharacteristic
334C2×C43#326711yescharacteristic
335C2×C43D4×SD1611yescharacteristic
336C2×C43Q8×SD1611yescharacteristic
337C2×C43#338811yescharacteristic
338D43—44no—
339C22×SD16—431no—
340C2×C4⋊Q8—431no—
341C2×C4×Q8—431no—
342C2×C4×D4—431no—
343C2×C4×D4—431no—
344C2×C4×D4—431no—
345C22×C4⋊C4—861no—
346#476—864no—
347C4×SD16—864no—
348C2×#106—431no—
349C2×#106—431no—
350C2×#106—431no—
351C2×#106—431no—
352C2×#106—431no—
353C2×C4⋊C8C2×D4311yescharacteristic
354C2×#103—431no—
355C2×#103—431no—
356C2×#103—431no—
357C2×#102—431no—
358C2×#102—431no—
359C2×#102—431no—
360C2×C4×C8C2×D4311yescharacteristic
361#398—861no—
362#395—861no—
363#391—431no—
364#391—431no—
365#391—861no—
366#389—431no—
367#389—431no—
368#389—431no—
369#389—431no—
370#387—861no—
371#385—861no—
372#385—861no—
373#384—431no—
374#380—861no—
375C4×C4⋊C4C2×D4311yescharacteristic
376#340—431no—
377#340—431no—
378#340—431no—
379#340—431no—
380#336C2×D4311yescharacteristic
381#336C2×D4311yescharacteristic
382#336C2×D4311yescharacteristic
383#336C2×D4311yescharacteristic
384C43#2637611yescharacteristic, complemented, Sylow
385C22×Q8—431no—
386C22×Q8—861no—
387C22×D4—431no—
388C23×C4—861no—
389C23×C4—861no—
390C23×C4—861no—
391C2×SD16—434no—
392C2×SD16—862no—
393C2×SD16—862no—
394C22×C8C22×D4311yescharacteristic
395C22×C8C22×D4311yescharacteristic
396C22×C8C22×D4311yescharacteristic
397C22×C8C22×D4311yescharacteristic
398C4⋊Q8—862no—
399C4⋊Q8—1721no—
400C4×Q8—862no—
401C4×Q8—1721no—
402C4×D4—862no—
403C4×D4—862no—
404C4×D4—862no—
405C4×D4—862no—
406C4×D4—1721no—
407C4×D4—1721no—
408C2×C4⋊C4—431no—
409C2×C4⋊C4—431no—
410C2×C4⋊C4—431no—
411C2×C4⋊C4—431no—
412C2×C4⋊C4—431no—
413C2×C4⋊C4—431no—
414C2×C4⋊C4—861no—
415C2×C4⋊C4—861no—
416C2×C4⋊C4—861no—
417C2×C4⋊C4—861no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4—861no—
421C2×C4⋊C4—861no—
422C2×C4⋊C4—1724no—
423C2×C4⋊C4D17211yescharacteristic
424C2×C4⋊C4D17211yescharacteristic
425C2×C22⋊C4—861no—
426C2×C22⋊C4—861no—
427C2×C22⋊C4—861no—
428C2×C42—861no—
429C2×C42C22×D4311yescharacteristic
430C2×C42C22×D4311yescharacteristic
431C2×C42C22×D4311yescharacteristic
432#106—862no—
433#106—862no—
434#106—862no—
435#106—862no—
436#106—1721no—
437C4⋊C8—22no—
438#103—862no—
439#103—862no—
440#103—1721no—
441#102—434no—
442#102—862no—
443#102—862no—
444C4×C8—22no—
445#95—861no—
446#95—861no—
447#95—861no—
448#95—861no—
449#95—861no—
450#95—861no—
451#95—861no—
452#95—861no—
453#95D17211yescharacteristic
454#95D17211yescharacteristic
455C24—861no—
456C2×Q8—432no—
457C2×Q8—861no—
458C2×Q8—861no—
459C2×Q8—861no—
460C2×Q8—861no—
461C2×Q8—861no—
462C2×Q8—3441no—
463C2×Q8—1722no—
464C2×Q8—1722no—
465C2×D4—432no—
466C2×D4—432no—
467C2×D4—432no—
468C2×D4—861no—
469C2×D4—861no—
470C2×D4—861no—
471C22×C4—864no—
472C22×C4—861no—
473C22×C4—861no—
474C22×C4—861no—
475C22×C4—861no—
476C22×C4—861no—
477C22×C4—861no—
478C22×C4—861no—
479C22×C4—3441no—
480C22×C4—3441no—
481C22×C4—1722no—
482C22×C4—1722no—
483C22×C4—1722no—
484C22×C4—1722no—
485C22×C4—1722no—
486C22×C4—1722no—
487C22×C4C2×D17211yescharacteristic
488C22×C4C2×D17211yescharacteristic
489C22×C4C2×D17211yescharacteristic
490C22×C4C2×D17211yescharacteristic
491C22×C4C2×D17211yescharacteristic
492C22×C4C2×D17211yescharacteristic
493C22×C4C23×D4311yescharacteristic
494SD16—868no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8D4×D4312yes—
506C2×C8Q8×D4312yes—
507C4⋊C4—41no—
508C4⋊C4—432no—
509C4⋊C4—432no—
510C4⋊C4—861no—
511C4⋊C4—861no—
512C4⋊C4—861no—
513C4⋊C4—861no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—3441no—
521C4⋊C4—3441no—
522C4⋊C4—3441no—
523C4⋊C4—1722no—
524C4⋊C4—1722no—
525C4⋊C4—1722no—
526C4⋊C4—1722no—
527C4⋊C4—1722no—
528C4⋊C4#1062187314yescomplemented
529C22⋊C4—864no—
530C22⋊C4—1722no—
531C22⋊C4—1722no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—1722no—
539C23—862no—
540C23—862no—
541C23—862no—
542C23—1721no—
543C23—1721no—
544C23—1721no—
545C23—1721no—
546C23C22×D17211yescharacteristic
547Q8—862no—
548Q8—862no—
549Q8—3444no—
550Q8—1721no—
551Q8—1721no—
552D4—434no—
553D4—862no—
554D4—862no—
555D4—862no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—862no—
572C2×C4—862no—
573C2×C4—862no—
574C2×C4—3444no—
575C2×C4—3444no—
576C2×C4—1728no—
577C2×C4—1721no—
578C2×C4—1721no—
579C2×C4—1721no—
580C2×C4—1721no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4—1721no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4—1721no—
590C2×C4—1721no—
591C2×C4—1721no—
592C2×C4—1721no—
593C2×C4—1721no—
594C2×C4—1721no—
595C2×C4C2×#1062187312yes—
596C2×C4C2×#1062187312yes—
597C2×C4C2×#1062187312yes—
598C2×C4C2×D4×D4311yescharacteristic
599C2×C4C2×Q8×D4311yescharacteristic
600C2×C4#41×D4311yescharacteristic
601C2×C4#41×D4311yescharacteristic
602C2×C4#1306337111yescharacteristic
603C2×C4#1306337211yescharacteristic
604C8—22no—
605C8—22no—
606C8—41no—
607C8—41no—
608C22—864no—
609C22—1722no—
610C22—1722no—
611C22—1722no—
612C22—1722no—
613C22—1722no—
614C22—1722no—
615C22#1458341111yescharacteristic
616C22D4×D17211yescharacteristic
617C22Q8×D17211yescharacteristic
618C22C22×#1062187311yescharacteristic
619C22#1458369911yescharacteristic
620C22#1458371111yescharacteristic
621C22#1458371111yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—3441no—
629C4—3441no—
630C4—3441no—
631C4—3441no—
632C4—1722no—
633C4—1722no—
634C4—1722no—
635C4C22⋊Q8×D4311yescharacteristic
636C4#1458349011yescharacteristic
637C4#1458354711yescharacteristic
638C4#1458360511yescharacteristic
639C2—1724no—
640C2#1913840911yescharacteristic
641C2#1914460411yescharacteristic
642C2#1914487411yescharacteristic
643C2#1914612511yescharacteristic
644C2D4×#1062187311yescharacteristic
645C2Q8×#1062187311yescharacteristic
646C2#1914754211yescharacteristic
subgroupG/H
centerC23C22×D172
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1916C2
socleC23×C43C22×D4
prime radicalC22×D172C23
hypercenter#1916D43
hypocenterC43#26376
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
3. #19138434C2semidirect
5. #19136137C2semidirect
6. #19136073C2semidirect
7. #19135975C2semidirect
8. #19135929C2semidirect
9. #19135323C2semidirect
12. #19134866C2semidirect
15. C43×#1916C2semidirect
69. C2×#10621873Q8semidirect
183. C4⋊C4×C43SD16semidirect
325. #1916D43semidirect
384. C43#26376semidirect
528. C4⋊C4#10621873semidirect

Maximal subgroups

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

subgroupindex
C2×#145831902
C2×C4×#106218732
#191384342
C4⋊C4×D1722
#191361372
#191360732
#191359752
#191359292
#191353232
#191349282
#191348662
#191348592
C43×#19162
#2637643