Finite Group Databasebeta

← #358041

Subgroups — #358041

Order 512.

2242 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C22×#423C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#13756C211yesmaximal, characteristic
7C2×C4⋊#95C211yesmaximal, characteristic
8#11232C214yesmaximal, complemented
9#11231C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#428—24no—
18C2×#428—24no—
19C2×#428—24no—
20C22×#107C2212yes—
21C22×#106C2212yes—
22C2×#423—24no—
23C2×#423—24no—
24C2×#423C418yescomplemented
25C22×C4⋊C8C2214yes—
26C22×#95—22no—
27#1736—24no—
28#1736—24no—
29#1736—24no—
30C2×#340C2212yes—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C2×#340C2212yes—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2214yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—22no—
40C4×#95—21no—
41C4×#95—21no—
42C24×C4—21no—
43C24×C4—21no—
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16C2×C414yes—
57C22×M16C2×C414yes—
58C22×M16D414yes—
59C22×M16D414yes—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117D414yes—
73C2×#117Q814yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D414yes—
87C22×C4⋊C4Q814yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—21no—
100C22×C42—21no—
101C22×C42C2×C414yes—
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42Q814yes—
105C22×C42C2312yes—
106#428—232no—
107C2×#107—24no—
108C2×#107—24no—
109C2×#107—24no—
110C2×#106—24no—
111C2×#106—24no—
112C2×#106—24no—
113#423—232no—
114C2×C4⋊C8—28no—
115C2×C4⋊C8—28no—
116C2×C4⋊C8C2×C4116yescomplemented
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—48no—
124#340—28no—
125#340—24no—
126#340—24no—
127#340—28no—
128C25C2×D411yescharacteristic
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C4⋊C418yes—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×Q812yes—
160C23×C4C2×Q812yes—
161C23×C4C2×Q812yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—28no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—28no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—44no—
179C2×M16—44no—
180C2×M16—44no—
181C2×M16—44no—
182C2×M16C22⋊C4116yes—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C414yes—
194C22×C8C22×C414yes—
195C22×C8C2×D414yes—
196C22×C8C2×D414yes—
197#117—28no—
198#117—28no—
199#117—28no—
200#117—28no—
201#117—44no—
202#117—44no—
203#117—44no—
204#117—44no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C4⋊C4C2×Q814yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C42—28no—
256C2×C42—28no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C4⋊C414yes—
274C2×C42C4⋊C414yes—
275C2×C42C22×C414yes—
276C2×C42C2×Q814yes—
277#107—216no—
278#106—216no—
279C4⋊C8—232no—
280#95—44no—
281#95—44no—
282#95—48no—
283#95—44no—
284#95—48no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24C2×C4⋊C412yes—
291C24#12112yes—
292C24#12112yes—
293C24#12112yes—
294C24#12112yes—
295C24C22⋊Q812yes—
296C24C22⋊Q812yes—
297C24C22×D411yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—216no—
309C22×C4—216no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C4⋊C414yes—
354C22×C4C2×C4⋊C414yes—
355C22×C4C2×C4⋊C412yes—
356C22×C4#11714yes—
357C22×C4C4×D414yes—
358C22×C4C4×D414yes—
359C22×C4C4×Q814yes—
360C22×C4C4×Q814yes—
361C22×C4#12014yes—
362C22×C4#12114yes—
363C22×C4#12114yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C22⋊Q814yes—
369C22×C4C22⋊Q814yes—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q814yes—
373C22×C4C22⋊Q814yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4#12314yes—
377C22×C4#12314yes—
378C22×C4#12314yes—
379C22×C4#12414yes—
380C22×C4#12514yes—
381C22×C4#12514yes—
382C22×C4#12514yes—
383C22×C4#12514yes—
384C22×C4#12614yes—
385C22×C4#12614yes—
386C22×C4C4⋊Q814yes—
387C22×C4C4⋊Q814yes—
388C22×C4C4⋊Q814yes—
389C22×C4AGL(1,ℤ/8)14yes—
390C22×C4#13714yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405M16—232no—
406M16—416no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8C2×C22⋊C418yes—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—28no—
450C42—28no—
451C42—48no—
452C42—48no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23#38014yes—
470C23#38914yes—
471C23#38914yes—
472C23#39414yes—
473C23#39614yes—
474C23#39614yes—
475C23#39814yes—
476C23C22×C4⋊C411yescharacteristic
477C23C2×#12111yescharacteristic
478C23C2×#12111yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×#12111yescharacteristic
481C23C2×C22⋊Q811yescharacteristic
482C23C2×C22⋊Q811yescharacteristic
483C23C2×AGL(1,ℤ/8)12yes—
484C23C2×#13712yes—
485C23#57512yes—
486C23#57512yes—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—432no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—816no—
515C2×C4—816no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#38014yes—
531C2×C4#38618yes—
532C2×C4#38718yes—
533C2×C4#38718yes—
534C2×C4#38914yes—
535C2×C4#38914yes—
536C2×C4#39118yes—
537C2×C4#39414yes—
538C2×C4#39518yes—
539C2×C4#39614yes—
540C2×C4#39614yes—
541C2×C4#39818yes—
542C2×C4#39818yes—
543C2×C4#39814yes—
544C2×C4#40018yes—
545C2×C4#42818yescomplemented
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4C2×C4×Q811yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4C2×#12311yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4C2×#12511yescharacteristic
566C2×C4C2×#12511yescharacteristic
567C2×C4C2×#12511yescharacteristic
568C2×C4C2×#12511yescharacteristic
569C2×C4C2×#12611yescharacteristic
570C2×C4C2×#12611yescharacteristic
571C2×C4C2×C4⋊Q811yescharacteristic
572C2×C4C2×C4⋊Q811yescharacteristic
573C2×C4C2×C4⋊Q811yescharacteristic
574C8—28no—
575C8—44no—
576C22—28no—
577C22—24no—
578C22—24no—
579C22—28no—
580C22—28no—
581C22—24no—
582C22—24no—
583C22—28no—
584C22—24no—
585C22—24no—
586C22#144118yes—
587C22C2×#38011yescharacteristic
588C22C2×#38911yescharacteristic
589C22C2×#38911yescharacteristic
590C22C2×#39411yescharacteristic
591C22C2×#39611yescharacteristic
592C22C2×#39611yescharacteristic
593C22C2×#39811yescharacteristic
594C22C2×#42812yes—
595C22#291112yes—
596C22#303612yes—
597C22#303712yes—
598C22#303811yescharacteristic
599C22#303911yescharacteristic
600C22#304011yescharacteristic
601C22#304011yescharacteristic
602C22#305512yes—
603C22#305512yes—
604C22#305612yes—
605C22#305612yes—
606C22#307712yes—
607C22#307812yes—
608C22#307912yes—
609C22#308012yes—
610C4—28no—
611C4—44no—
612C4—84no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—82no—
617C4#144118yes—
618C4C2×#38611yescharacteristic
619C4C2×#38711yescharacteristic
620C4C2×#38711yescharacteristic
621C4C2×#39111yescharacteristic
622C4C2×#39511yescharacteristic
623C4C2×#39811yescharacteristic
624C4C2×#39811yescharacteristic
625C4C2×#40011yescharacteristic
626C2—28no—
627C2C2×#144111yescharacteristic
628C2#2066112yes—
629C2#2117012yes—
630C2#2117111yescharacteristic
631C2#2117211yescharacteristic
632C2#2354611yescharacteristic
633C2#2354711yescharacteristic
634C2#2361712yes—
635C2#2361811yescharacteristic
636C2#2361911yescharacteristic
637C2#2367012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11232C2semidirect
9. #11231C2semidirect
24. C2×#423C4semidirect
116. C2×C4⋊C8C2×C4semidirect
545. C2×C4#428semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4232
C2×#17362
#137562
C2×C4⋊#952
#112322
#112312