Finite Group Databasebeta

← #192386

Subgroups — #192386

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#14275C211yesmaximal, characteristic
6#8872C214yesmaximal, complemented
7#8871C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9C2×#1280C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2329—22no—
14#2329—22no—
15#2329—22no—
16#2329C2214yes—
17C2×#391—21no—
18C2×#389—21no—
19C2×#387—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27C22×#95C2211yescharacteristic
28#1892—22no—
29#1872C2212yes—
30#1872C2212yes—
31#1853—21no—
32#1853—21no—
33#1853C2212yes—
34#1853C2212yes—
35#1824—21no—
36#1807—22no—
37#1787—21no—
38#1520C418yescomplemented
39C2×#328C2214yes—
40#1444C2212yes—
41#1442C2212yes—
42#1441—22no—
43C22⋊#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46C23×Q8D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—21no—
51C23×C8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2×C414yes—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8Q814yes—
59C2×C22⋊Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—21no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2×C414yes—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88#391—24no—
89#391—24no—
90#389—24no—
91#389—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#385—24no—
97#385—24no—
98#385—44no—
99#380—21no—
100#380—21no—
101#380—22no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C414yes—
107#380D412yes—
108#380C2312yes—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D412yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#328C2×C4116yescomplemented
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—81no—
136C22×Q8—81no—
137C22×Q8C4⋊C414yescomplemented
138C22×Q8C2×D412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4—81no—
159C23×C4SD1614yes—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q812yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—81no—
177C22×C8—81no—
178C22×C8—81no—
179C22×C8—81no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8C4⋊C418yescomplemented
188C22⋊Q8C22×C414yes—
189C22⋊Q8C2×Q814yes—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C4⋊C414yescomplemented
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—82no—
238C2×C42—82no—
239#106—82no—
240#106—82no—
241#103—82no—
242#103—82no—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—42no—
248C22⋊C8C22×C414yes—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—41no—
261#95—41no—
262#95—44no—
263#95—82no—
264#95—82no—
265#95C22×C414yes—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24C4×D412yes—
271C24#12011yescharacteristic
272C24#12011yescharacteristic
273C24C22⋊Q812yes—
274C24C2×SD1612yes—
275C24#13512yes—
276C24#13712yes—
277C24#13712yes—
278C24C22×D411yescharacteristic
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8C2×C4⋊C414yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—84no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4C2×C4⋊C412yes—
363C22×C4C2×C4⋊C412yes—
364C22×C4C4×D412yes—
365C22×C4C4×D412yes—
366C22×C4C4×D412yes—
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4C22⋊Q812yes—
370C22×C4C22⋊Q812yes—
371C22×C4C22⋊Q812yes—
372C22×C4#12312yes—
373C22×C4#12312yes—
374C22×C4C2×SD1614yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—84no—
388C2×C8—84no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4—88no—
406C4⋊C4C2×C4⋊C414yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—28no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—48no—
413C22⋊C4—44no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C42—84no—
417C42—84no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23#38512yes—
443C23#38512yes—
444C23C4×SD1614yes—
445C23#44114yes—
446C23#44812yes—
447C23#44812yes—
448C23#45212yes—
449C23#45212yes—
450C23#47514yes—
451C23#47914yes—
452C23C2×C4×D411yescharacteristic
453C23C2×#12011yescharacteristic
454C23C2×C22⋊Q811yescharacteristic
455C23C22×SD1611yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×#13711yescharacteristic
458C23C2×#13711yescharacteristic
459Q8—44no—
460Q8—82no—
461Q8—84no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4#38512yes—
532C2×C4#38512yes—
533C2×C4C22×C4⋊C411yescharacteristic
534C2×C4C2×C4×D411yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4C2×#12311yescharacteristic
537C8—44no—
538C8—82no—
539C22—21no—
540C22—22no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—22no—
545C22—21no—
546C22—22no—
547C22—21no—
548C22—21no—
549C22—22no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—21no—
556C22—22no—
557C22—21no—
558C22—22no—
559C22—21no—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22#186514yes—
568C22#186514yes—
569C22C2×#38511yescharacteristic
570C22C2×C4×SD1611yescharacteristic
571C22C2×#44111yescharacteristic
572C22C2×#44811yescharacteristic
573C22C2×#45211yescharacteristic
574C22#301911yescharacteristic
575C22#302011yescharacteristic
576C22#302611yescharacteristic
577C22#302611yescharacteristic
578C22C2×#47511yescharacteristic
579C22C2×#47911yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—22no—
585C4—22no—
586C4—44no—
587C4—44no—
588C4—44no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—161no—
594C4—161no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—24no—
599C2—24no—
600C2—21no—
601C2#1828611yescharacteristic
602C2#1828911yescharacteristic
603C2#1829411yescharacteristic
604C2#1829711yescharacteristic
605C2#2086411yescharacteristic
606C2C2×#186511yescharacteristic
607C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8872C2semidirect
7. #8871C2semidirect
38. #1520C4semidirect
131. #328C2×C4semidirect
137. C22×Q8C4⋊C4semidirect
187. C22⋊Q8C4⋊C4semidirect
216. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#219382
#217592
C2×#15202
#142752
#88722
#88712
#80232
C2×#12802