Finite Group Databasebeta

← #188486

Subgroups — #188486

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4#14682C211yesmaximal, characteristic
5#8576C214yesmaximal, complemented
6#8574C214yesmaximal, complemented
7#8023C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C22×C4⋊C8—21no—
12C22×C22⋊C8C2212yes—
13C4×C22⋊Q8—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C412yescomplemented
18C4×C22⋊Q8C412yescomplemented
19C4×C22⋊Q8C2212yes—
20C4×C22⋊Q8C2212yes—
21C2×#391C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#387—21no—
24C2×#385—21no—
25C2×#382—21no—
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35C2×#336—21no—
36#1520—28no—
37C2×#328C2218yes—
38C4⋊#95—21no—
39#1434C2212yes—
40#1434C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43#1294C2218yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8D414yes—
55C2×C22⋊Q8Q814yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—44no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—44no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94#391—21no—
95#391—21no—
96#391C2×C412yes—
97#391C2×C412yes—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#382—44no—
111#380—21no—
112#380—21no—
113#380—24no—
114#380—24no—
115#380C2×C412yes—
116#380C2312yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95C2314yes—
139#336—22no—
140#336—22no—
141#328—216no—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8C4⋊C414yes—
148C22×Q8C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×Q812yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184C22×C8—24no—
185C22×C8—44no—
186C22×C8—44no—
187C22×C8—44no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8C4⋊C418yes—
196C22⋊Q8C2×D414yes—
197C22⋊Q8C2×Q814yes—
198C4×Q8—44no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8—44no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C4⋊C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C4⋊C8—44no—
270C4⋊C8—44no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—44no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24#12112yes—
297C24C22⋊Q812yes—
298C24#13514yes—
299C24#13714yes—
300C24C22×D411yescharacteristic
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8C2×C4⋊C414yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—416no—
369C22×C4—416no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C4⋊C412yes—
398C22×C4C2×C4⋊C412yes—
399C22×C4#11712yes—
400C22×C4#11712yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4C22⋊Q812yes—
405C22×C4C22⋊Q812yes—
406C22×C4C22⋊Q812yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—88no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—88no—
431C4⋊C4C2×C4⋊C414yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—88no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—44no—
454C42—44no—
455C42—44no—
456C42—44no—
457C42#9914yes—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23#38012yes—
479C23#38012yes—
480C23#41612yes—
481C23#41612yes—
482C23#41912yes—
483C23#41912yes—
484C23#46418yes—
485C23#47918yes—
486C23C22×C22⋊C411yescharacteristic
487C23C2×#12111yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23C2×#13512yes—
490C23C2×#13712yes—
491Q8—44no—
492Q8—44no—
493Q8—48no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4#38012yes—
570C2×C4#38012yes—
571C2×C4C2×#9912yes—
572C2×C4C2×#9912yes—
573C2×C4C2×#9912yes—
574C2×C4#41012yes—
575C2×C4#42112yes—
576C2×C4#42112yes—
577C2×C4C22×C4⋊C411yescharacteristic
578C2×C4C2×#11711yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C8—84no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—48no—
603C22—48no—
604C22#180514yes—
605C22#180514yes—
606C22C2×#38011yescharacteristic
607C22C22×#9911yescharacteristic
608C22C2×#41011yescharacteristic
609C22C2×#41611yescharacteristic
610C22C2×#41911yescharacteristic
611C22C2×#42111yescharacteristic
612C22C2×#42111yescharacteristic
613C22C2×#46412yes—
614C22C2×#47912yes—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—44no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—164no—
633C4#179914yes—
634C2—22no—
635C2—22no—
636C2—24no—
637C2—24no—
638C2#1756711yescharacteristic
639C2#1757211yescharacteristic
640C2#1757811yescharacteristic
641C2#1759611yescharacteristic
642C2C2×#179911yescharacteristic
643C2C2×#180511yescharacteristic
644C2#2065411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8576C2semidirect
6. #8574C2semidirect
9. C22×C4×Q8C4semidirect
17. C4×C22⋊Q8C4semidirect
18. C4×C22⋊Q8C4semidirect
21. C2×#391C4semidirect
22. C2×#389C4semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#17972
C2×#15202
#146822
#85762
#85742
#80232