Finite Group Databasebeta

← #117596

Subgroups — #117596

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21940C211yesmaximal, characteristic
2C2×#1787C211yesmaximal, characteristic
3C2×#1787C211yesmaximal, characteristic
4#13922C211yesmaximal, characteristic
5#10854C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C24×Q8C2211yescharacteristic
9C22×#103—22no—
10C22×#103—22no—
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#391—22no—
14C2×#380C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×#380C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1852—22no—
19#1852—22no—
20#1808—22no—
21#1787—22no—
22#1787—22no—
23#1787—22no—
24#1787C414yescomplemented
25C2×#342—22no—
26#1446—22no—
27C22⋊#95—22no—
28#1294—22no—
29#1282—22no—
30C23×Q8—21no—
31C23×Q8—21no—
32C23×Q8—21no—
33C23×Q8—21no—
34C23×Q8—21no—
35C23×Q8—21no—
36C23×Q8—21no—
37C23×Q8—21no—
38C23×Q8—21no—
39C23×Q8—21no—
40C23×Q8—21no—
41C23×Q8—48no—
42C23×Q8C2×C412yes—
43C23×Q8D412yes—
44C23×Q8Q812yes—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—21no—
48C23×C8D412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D412yes—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42—21no—
57C2×#107—24no—
58C2×#103—28no—
59C2×#103—28no—
60C2×#103—48no—
61C2×#103—48no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8C2×C412yes—
72C2×C22⋊C8D412yes—
73C2×C22⋊C8D412yes—
74#391—48no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380C2×C412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C2×#95—21no—
92C2×#95—28no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96#342—44no—
97#340—24no—
98C25C2×D411yescharacteristic
99C22×Q8—28no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—28no—
104C22×Q8—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—48no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—48no—
122C22×Q8—48no—
123C22×Q8—48no—
124C22×Q8—48no—
125C22×Q8—48no—
126C22×Q8—48no—
127C22×Q8—48no—
128C22×Q8—48no—
129C22×Q8—48no—
130C22×Q8—48no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×Q8C4⋊C414yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—48no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×Q811yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169C23×C4#4111yescharacteristic
170C23×C4#4111yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8—41no—
177C22×C8—41no—
178C22×C8C22⋊C414yes—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—41no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4Q1614yes—
206C2×C42—44no—
207C2×C42—41no—
208C2×C42—41no—
209#107—84no—
210#103—416no—
211#103—416no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220#95—22no—
221#95—416no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229C24—42no—
230C24—42no—
231C24C4×D411yescharacteristic
232C24#12011yescharacteristic
233C24#12111yescharacteristic
234C24#12111yescharacteristic
235C24C22⋊Q811yescharacteristic
236C24#12311yescharacteristic
237C24#12711yescharacteristic
238C24C2×Q1612yes—
239C24#13512yes—
240C24AGL(1,ℤ/8)12yes—
241C24#13712yes—
242C2×Q8—28no—
243C2×Q8—28no—
244C2×Q8—28no—
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—28no—
248C2×Q8—28no—
249C2×Q8—28no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—48no—
253C2×Q8—44no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—48no—
257C2×Q8—44no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—48no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—44no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C4⋊C411yescharacteristic
335C22×C4#11711yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4C22⋊Q811yescharacteristic
350C22×C4C22⋊Q811yescharacteristic
351C22×C4C22⋊Q811yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12411yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12611yescharacteristic
360C22×C4#12611yescharacteristic
361C2×C8—24no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—82no—
367C2×C8—82no—
368C2×C8—82no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—81no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C22⋊C4—81no—
387C42—82no—
388C42—82no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23#38511yescharacteristic
408C23#38811yescharacteristic
409C23#39011yescharacteristic
410C23#39211yescharacteristic
411C23#39411yescharacteristic
412C23#39611yescharacteristic
413C23#39911yescharacteristic
414C23C4×Q1612yes—
415C23#44012yes—
416C23#44912yes—
417C23#45112yes—
418C23#46212yes—
419C23#46312yes—
420C23#46712yes—
421C23#46812yes—
422C23#47712yes—
423C23#47912yes—
424C23#48412yes—
425C23#48512yes—
426C23#49411yescharacteristic
427C23#49511yescharacteristic
428C23#49611yescharacteristic
429C23#49711yescharacteristic
430Q8—28no—
431Q8—28no—
432Q8—48no—
433Q8—44no—
434Q8—44no—
435Q8—48no—
436Q8—48no—
437Q8—48no—
438Q8—48no—
439Q8—48no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—162no—
492C2×C4—162no—
493C2×C4—164no—
494C2×C4#38011yescharacteristic
495C2×C4#38511yescharacteristic
496C2×C4#38611yescharacteristic
497C2×C4#38811yescharacteristic
498C2×C4#39311yescharacteristic
499C2×C4#39411yescharacteristic
500C2×C4#39611yescharacteristic
501C2×C4#39711yescharacteristic
502C2×C4B(4,𝔽2)14yes—
503C8—44no—
504C8—82no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—24no—
518C22—24no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22#143911yescharacteristic
534C22#186312yes—
535C22#192812yes—
536C22#196911yescharacteristic
537C22#197311yescharacteristic
538C22#200912yes—
539C22#201211yescharacteristic
540C22#201611yescharacteristic
541C22#202112yes—
542C22#203612yes—
543C22#206812yes—
544C22#209511yescharacteristic
545C22#209711yescharacteristic
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—48no—
553C4—84no—
554C4—162no—
555C4—162no—
556C4—162no—
557C4#188912yes—
558C4#202912yes—
559C2—22no—
560C2—21no—
561C2—22no—
562C2—21no—
563C2—21no—
564C2—22no—
565C2—22no—
566C2—21no—
567C2#994812yes—
568C2#995712yes—
569C2#1106911yescharacteristic
570C2#1112411yescharacteristic
571C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1787C4semidirect

Maximal subgroups

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

subgroupindex
#219402
C2×#17872
#139222
#108542
#80232
C2×#12822