Finite Group Databasebeta

← #106964

Subgroups — #106964

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4#10854C211yesmaximal, characteristic
5#8023C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11#2345—22no—
12#2345—22no—
13#2345—22no—
14#2345C414yescomplemented
15C2×#389—21no—
16C2×#387—22no—
17C2×#380—21no—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×C4×C4⋊C4—21no—
22C2×C4×C22⋊C4—21no—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1824—22no—
25#1824—22no—
26#1824—21no—
27#1824—21no—
28#1824C414yescomplemented
29C2×#342—21no—
30C2×#342—21no—
31C2×#340—22no—
32C2×#340C414yescomplemented
33#1294—22no—
34#1289—22no—
35#1282—22no—
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C23×C8—21no—
39C23×C8C2×C412yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4Q812yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8C2×C412yes—
78#389—24no—
79#389—24no—
80#387—28no—
81#387—28no—
82#380—24no—
83#380—21no—
84#380—21no—
85#380—24no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380C2×C412yes—
94#380C2×C412yes—
95#380D412yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4D412yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D412yes—
110#342—42no—
111#342—42no—
112#340—24no—
113#340—24no—
114#340—28no—
115C25C2×D411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—44no—
129C23×C4—41no—
130C23×C4—84no—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×Q811yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—44no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4C4214yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C414yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C4⋊C414yes—
195C2×C4⋊C4C4⋊C412yes—
196C2×C4⋊C4C4⋊C412yes—
197C2×C4⋊C4C4⋊C414yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4SD1612yes—
227C2×C22⋊C4Q1612yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—28no—
231C2×C42—44no—
232C2×C42—84no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241#95—28no—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95C22⋊C414yes—
251C24—42no—
252C24—42no—
253C24C2×C22⋊C411yescharacteristic
254C24C4×D411yescharacteristic
255C24C4×D411yescharacteristic
256C24#12011yescharacteristic
257C24#12011yescharacteristic
258C24#12111yescharacteristic
259C24C22⋊Q811yescharacteristic
260C24C2×SD1611yescharacteristic
261C24C2×Q1611yescharacteristic
262C24#13511yescharacteristic
263C24#13511yescharacteristic
264C24AGL(1,ℤ/8)11yescharacteristic
265C24#13711yescharacteristic
266C24#13711yescharacteristic
267C24#13711yescharacteristic
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C2×C4211yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C4⋊C411yescharacteristic
331C22×C4C2×C4⋊C411yescharacteristic
332C22×C4C2×C4⋊C411yescharacteristic
333C22×C4#11711yescharacteristic
334C22×C4#11711yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4C22⋊Q811yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C2×C8—44no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—82no—
358C2×C8—82no—
359C2×C8—82no—
360C2×C8—82no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—88no—
378C4⋊C4#9514yes—
379C4⋊C4#9514yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—81no—
394C22⋊C4—81no—
395C22⋊C4—81no—
396C22⋊C4—81no—
397C22⋊C4—81no—
398C22⋊C4—81no—
399C22⋊C4#10314yes—
400C42—44no—
401C42—44no—
402C42—88no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23C4×C22⋊C411yescharacteristic
426C23#37911yescharacteristic
427C23#38011yescharacteristic
428C23#38511yescharacteristic
429C23#38511yescharacteristic
430C23#38611yescharacteristic
431C23#38611yescharacteristic
432C23C2×#10311yescharacteristic
433C23#41611yescharacteristic
434C23#41711yescharacteristic
435C23#41911yescharacteristic
436C23C4×SD1611yescharacteristic
437C23C4×SD1611yescharacteristic
438C23C4×Q1611yescharacteristic
439C23C4×Q1611yescharacteristic
440C23#44011yescharacteristic
441C23#44011yescharacteristic
442C23#44111yescharacteristic
443C23#44111yescharacteristic
444C23#44811yescharacteristic
445C23#44811yescharacteristic
446C23#44911yescharacteristic
447C23#44911yescharacteristic
448C23#45111yescharacteristic
449C23#45111yescharacteristic
450C23#45211yescharacteristic
451C23#45211yescharacteristic
452C23#46011yescharacteristic
453C23#46211yescharacteristic
454C23#46411yescharacteristic
455C23#46411yescharacteristic
456C23#47511yescharacteristic
457C23#47711yescharacteristic
458C23#47911yescharacteristic
459C23#47911yescharacteristic
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—162no—
521C2×C4—162no—
522C2×C4—162no—
523C2×C4—162no—
524C2×C4C2×#9511yescharacteristic
525C2×C4C4×C22⋊C411yescharacteristic
526C2×C4#38011yescharacteristic
527C2×C4#38111yescharacteristic
528C2×C4#38511yescharacteristic
529C2×C4#38511yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4#38611yescharacteristic
532C2×C4#41014yes—
533C8—82no—
534C8—82no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C22⋊#9511yescharacteristic
576C22C4×#10311yescharacteristic
577C22#176311yescharacteristic
578C22#178711yescharacteristic
579C22#178811yescharacteristic
580C22#180311yescharacteristic
581C22#180511yescharacteristic
582C22#186311yescharacteristic
583C22#186311yescharacteristic
584C22#186511yescharacteristic
585C22#186511yescharacteristic
586C22#187811yescharacteristic
587C22#187811yescharacteristic
588C22#188011yescharacteristic
589C22#188011yescharacteristic
590C22#189912yes—
591C22#190412yes—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—162no—
604C4—162no—
605C4#174012yes—
606C4#189912yes—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2#925712yes—
620C2#926112yes—
621C2#1073611yescharacteristic
622C2#1078511yescharacteristic
623C2#1078911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
14. #2345C4semidirect
28. #1824C4semidirect
32. C2×#340C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#18242
#108542
#80232
C2×#12892
C2×#12822