Finite Group Databasebeta

← #114385

Subgroups — #114385

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#14159C211yesmaximal, characteristic
6C2×#1294C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊Q8C2211yescharacteristic
10#3189—22no—
11#3189—22no—
12#3189—22no—
13#3189—22no—
14C2×#484—21no—
15C2×#484C412yescomplemented
16C2×#467—22no—
17C2×#451—21no—
18C2×#451C412yescomplemented
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C22×#95C2211yescharacteristic
24#1996—22no—
25#1996—22no—
26C2×#342—22no—
27#1520—22no—
28#1520—22no—
29C2×#328—22no—
30C2×#328—22no—
31C2×#328—22no—
32C2×#328—22no—
33#1446—21no—
34#1445—21no—
35#1436—22no—
36C22⋊#95—22no—
37#1294—22no—
38#1282—22no—
39C23×Q8C2×C411yescharacteristic
40C23×Q8D411yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×Q16—22no—
44C23×C8—21no—
45C23×C8—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⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8D412yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80#484—24no—
81#484—24no—
82#484—24no—
83#484—24no—
84#467—28no—
85#467—28no—
86#451—24no—
87#451—24no—
88#451—24no—
89#451—24no—
90C2×#107—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8C2×C412yes—
102C2×C22⋊C8D412yes—
103C2×C22⋊C8D412yes—
104C2×C22⋊C8D412yes—
105#380—21no—
106#380—21no—
107#380—21no—
108#380D412yes—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—24no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95D412yes—
123#342—28no—
124#328—28no—
125#328—28no—
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—82no—
138C22×Q8—82no—
139C22×Q8C22⋊C412yes—
140C22×Q8C22⋊C412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—44no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4#4111yescharacteristic
169C2×Q16—28no—
170C2×Q16—84no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—41no—
176C22×C8—41no—
177C22×C8—82no—
178C22×C8—82no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22⋊C412yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4Q1614yescomplemented
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—82no—
263#107—28no—
264#103—28no—
265#103—28no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8C22⋊C414yes—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—48no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—84no—
284C24—42no—
285C24—42no—
286C24C4×D411yescharacteristic
287C24#12011yescharacteristic
288C24#12011yescharacteristic
289C24#12111yescharacteristic
290C24#12111yescharacteristic
291C24#12111yescharacteristic
292C24#12711yescharacteristic
293C24C2×Q1612yes—
294C24#13512yes—
295C24#13712yes—
296C24#13712yes—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×Q8—82no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4#12011yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12011yescharacteristic
381C22×C4#12011yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12311yescharacteristic
392C22×C4#12311yescharacteristic
393C22×C4#12311yescharacteristic
394C22×C4#12311yescharacteristic
395C22×C4#12411yescharacteristic
396C22×C4#12411yescharacteristic
397C22×C4#12711yescharacteristic
398Q16—88no—
399C2×C8—24no—
400C2×C8—44no—
401C2×C8—42no—
402C2×C8—82no—
403C2×C8—82no—
404C2×C8—82no—
405C2×C8—82no—
406C2×C8—82no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C42—82no—
441C42—82no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—44no—
455C23—44no—
456C23—41no—
457C23—41no—
458C23—44no—
459C23—44no—
460C23#38611yescharacteristic
461C23#38611yescharacteristic
462C23#39011yescharacteristic
463C23#39211yescharacteristic
464C23#39211yescharacteristic
465C23#39211yescharacteristic
466C23#39411yescharacteristic
467C23C4×Q1612yes—
468C23#44112yes—
469C23#45112yes—
470C23#45112yes—
471C23#45212yes—
472C23#45212yes—
473C23#46212yes—
474C23#46212yes—
475C23#46412yes—
476C23#46412yes—
477C23#46712yes—
478C23#47012yes—
479C23#49411yescharacteristic
480C23#49511yescharacteristic
481C23#49711yescharacteristic
482C23#49711yescharacteristic
483Q8—44no—
484Q8—44no—
485Q8—82no—
486Q8—82no—
487Q8—82no—
488Q8—82no—
489Q8—82no—
490Q8—82no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—88no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—164no—
552C2×C4#37911yescharacteristic
553C2×C4#38611yescharacteristic
554C2×C4#38611yescharacteristic
555C2×C4#39011yescharacteristic
556C2×C4#39211yescharacteristic
557C2×C4#39411yescharacteristic
558C2×C4#39411yescharacteristic
559C2×C4#39411yescharacteristic
560C8—44no—
561C8—82no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22#143811yescharacteristic
591C22#187812yes—
592C22#187812yes—
593C22#190612yes—
594C22#191312yes—
595C22#196911yescharacteristic
596C22#197111yescharacteristic
597C22#200112yes—
598C22#201511yescharacteristic
599C22#201511yescharacteristic
600C22#201611yescharacteristic
601C22#201611yescharacteristic
602C22#205612yes—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—84no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—162no—
618C2—21no—
619C2—22no—
620C2—21no—
621C2—22no—
622C2—22no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—21no—
627C2#927614yes—
628C2#1097211yescharacteristic
629C2#1101911yescharacteristic
630C2#1102411yescharacteristic
subgroupG/H
center, socleC23#386
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×#484C4semidirect
18. C2×#451C4semidirect
259. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#19962
C2×#15202
C22×#3282
#141592
C2×#12942
C2×#12822