Finite Group Databasebeta

← #115339

Subgroups — #115339

Order 512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2#21759C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#483—22no—
11C2×#483—22no—
12C2×#483—22no—
13C2×#483—22no—
14C2×#483—22no—
15C2×#483C414yescomplemented
16C22×#106—21no—
17C22×#103—21no—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#381—22no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1996—22no—
25#1996—22no—
26#1892—22no—
27#1872—22no—
28#1824—21no—
29#1823—22no—
30#1823—22no—
31#1787—21no—
32C2×#342—22no—
33C2×#340—22no—
34#1445—22no—
35C4⋊#95—22no—
36#1294—22no—
37#1282—22no—
38C23×Q8D411yescharacteristic
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8—21no—
42C23×C8D412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8D412yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78#483—28no—
79#483—28no—
80#483—28no—
81#483—28no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C412yes—
99C2×C22⋊C8D412yes—
100C2×C22⋊C8D412yes—
101#381—44no—
102#381—44no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380D412yes—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C2×#95—21no—
116C2×#95—28no—
117C2×#95—21no—
118C2×#95—21no—
119#342—28no—
120#340—28no—
121#340—24no—
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—44no—
129C22×Q8C22⋊C412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—41no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C23×C4#4111yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—44no—
168C22×C8—44no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C22⋊C414yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4SD1614yes—
251C2×C42—44no—
252C2×C42—41no—
253C2×C42—41no—
254#106—216no—
255#103—216no—
256#103—84no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264C22⋊C8C22⋊C414yes—
265#95—22no—
266#95—416no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272C24—42no—
273C24—42no—
274C24C4×D411yescharacteristic
275C24#12011yescharacteristic
276C24#12011yescharacteristic
277C24#12111yescharacteristic
278C24#12111yescharacteristic
279C24#12111yescharacteristic
280C24#12711yescharacteristic
281C24C2×SD1612yes—
282C24#13512yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—44no—
293C2×Q8—82no—
294C2×Q8—82no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4#12311yescharacteristic
378C22×C4#12311yescharacteristic
379C22×C4#12411yescharacteristic
380C22×C4#12411yescharacteristic
381C22×C4#12711yescharacteristic
382C2×C8—24no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—42no—
388C2×C8—44no—
389C2×C8—82no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—81no—
422C22⋊C4—81no—
423C42—82no—
424C42—82no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23#38611yescharacteristic
444C23#38611yescharacteristic
445C23#39011yescharacteristic
446C23#39211yescharacteristic
447C23#39211yescharacteristic
448C23#39211yescharacteristic
449C23#39411yescharacteristic
450C23C4×SD1612yes—
451C23#44012yes—
452C23#44812yes—
453C23#44912yes—
454C23#45012yes—
455C23#45212yes—
456C23#46012yes—
457C23#46112yes—
458C23#46312yes—
459C23#46412yes—
460C23#46512yes—
461C23#46812yes—
462C23#49211yescharacteristic
463C23#49511yescharacteristic
464C23#49611yescharacteristic
465C23#49711yescharacteristic
466Q8—44no—
467Q8—44no—
468Q8—82no—
469Q8—82no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—88no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—162no—
529C2×C4—162no—
530C2×C4#37911yescharacteristic
531C2×C4#38611yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#39011yescharacteristic
534C2×C4#39211yescharacteristic
535C2×C4#39411yescharacteristic
536C2×C4#39411yescharacteristic
537C2×C4#39411yescharacteristic
538C2×C4#45814yes—
539C8—44no—
540C8—82no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—22no—
551C22—24no—
552C22—22no—
553C22—24no—
554C22—22no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22#143811yescharacteristic
570C22#187712yes—
571C22#188012yes—
572C22#196811yescharacteristic
573C22#197311yescharacteristic
574C22#200012yes—
575C22#200912yes—
576C22#201211yescharacteristic
577C22#201311yescharacteristic
578C22#201411yescharacteristic
579C22#201511yescharacteristic
580C22#202212yes—
581C22#205512yes—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—84no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—162no—
593C4—162no—
594C4#189012yes—
595C4#202212yes—
596C2—22no—
597C2—21no—
598C2—22no—
599C2—22no—
600C2—22no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2#996012yes—
605C2#996112yes—
606C2#1099211yescharacteristic
607C2#1102311yescharacteristic
608C2#1103111yescharacteristic
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
15. C2×#483C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
#217592
C2×#19962
C2×#18232
#139042
#80232
C2×#12822