Finite Group Databasebeta

← #188503

Subgroups — #188503

Order 512.

1198 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
1#34233C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3C2×#1516C211yesmaximal, characteristic
4C2×#1513C211yesmaximal, characteristic
5#14355C211yesmaximal, characteristic, complemented
6#8577C212yesmaximal, complemented
7#8575C212yesmaximal, complemented
8#8574C212yesmaximal, complemented
9#8572C212yesmaximal, complemented
10#8023C211yesmaximal, characteristic, complemented
11#8015C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2361C412yescomplemented
16#2360C412yescomplemented
17#2358—22no—
18#2353—21no—
19#2349—21no—
20#2347C411yescharacteristic, complemented
21#2346C411yescharacteristic, complemented
22#2341C411yescharacteristic, complemented
23#2339C411yescharacteristic, complemented
24#2335—22no—
25#2318—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28C2×#381C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2275—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1851—21no—
34#1851—21no—
35#1850—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1516—24no—
39#1513—24no—
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42C2×#328C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1444—21no—
47#1442—21no—
48#1439—22no—
49C4⋊#95C2212yes—
50#1434C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41C2×C411yescharacteristic
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59#525—22no—
60#525—22no—
61#525D414yes—
62#525Q814yes—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121C2311yescharacteristic
68C2×#120—21no—
69C2×C4×Q8C2×C412yes—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yes—
75C2×#117C2×C411yescharacteristic
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C412yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42C2312yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#407—22no—
87#407—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2312yes—
95#391C2×C412yes—
96#390—22no—
97#390C2×C412yes—
98#389—22no—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#383—21no—
114#382—21no—
115#381—21no—
116#381—22no—
117#381—22no—
118#380—21no—
119#380—21no—
120#380—22no—
121#380—22no—
122#380—21no—
123#380C2×C411yescharacteristic
124#380C2×C411yescharacteristic
125#380C2312yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4C2×C411yescharacteristic
135C4×C22⋊C4C2×C411yescharacteristic
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D411yescharacteristic
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D411yescharacteristic
140#376—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152#336—22no—
153#336—22no—
154#328—24no—
155#328—22no—
156#328—22no—
157#327—24no—
158#327—22no—
159#327—22no—
160C25C2×D411yescharacteristic
161C2×#41—22no—
162C2×#41—82no—
163C2×#41C4⋊C414yescomplemented
164C22×Q8C22×C411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—82no—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×Q812yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C2×M16—42no—
189C2×M16—82no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—42no—
194C22×C8—82no—
195#123—24no—
196#123—24no—
197C22⋊Q8—22no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8C4⋊C414yescomplemented
201C22⋊Q8C2×D412yes—
202C22⋊Q8C2×Q812yes—
203#121—22no—
204#121—21no—
205#121—21no—
206#121C4⋊C414yescomplemented
207#121C2×D412yes—
208#121C2×Q812yes—
209#120—24no—
210#120—24no—
211C4×Q8—82no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—48no—
217C4×D4—82no—
218C4×D4—82no—
219C4×D4—82no—
220#117—22no—
221#117C4⋊C414yescomplemented
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—81no—
253C2×C22⋊C4—81no—
254C2×C22⋊C4—81no—
255C2×C22⋊C4—81no—
256C2×C22⋊C4—82no—
257C2×C22⋊C4C22⋊C412yes—
258C2×C22⋊C4C22⋊C412yes—
259C2×C22⋊C4C22⋊C412yes—
260C2×C22⋊C4C22⋊C412yes—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—82no—
266C2×C42C22×C412yes—
267C2×C42C22×C412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C4⋊C8—82no—
271C4⋊C8—82no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—22no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—44no—
296#95—42no—
297#95—82no—
298#95—82no—
299#95C22⋊C411yescharacteristic
300#95C22⋊C411yescharacteristic
301#95C22⋊C411yescharacteristic
302#95C22⋊C411yescharacteristic
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24#12112yes—
315C24C22⋊Q812yes—
316C24C22×D411yescharacteristic
317#41—84no—
318#41—84no—
319C2×Q8—21no—
320C2×Q8—42no—
321C2×Q8C2×C4⋊C412yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—21no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—82no—
334C2×D4C2×C4⋊C412yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4#11712yes—
392C22×C4#11712yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C22⋊Q812yes—
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q811yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405M16—84no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—84no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C4⋊C4C2×C4⋊C412yes—
428C4⋊C4C2×C4⋊C412yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C42—24no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—44no—
455C42—82no—
456C42—82no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#38012yes—
485C23#38012yes—
486C23C22×C22⋊C411yescharacteristic
487C23C2×#12111yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23#57611yescharacteristic
490C23#57711yescharacteristic
491C23#57711yescharacteristic
492C23#57811yescharacteristic
493Q8—42no—
494Q8—42no—
495D4—42no—
496D4—42no—
497D4—48no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—84no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—164no—
568C2×C4—164no—
569C2×C4#38012yes—
570C2×C4#38012yes—
571C2×C4#42114yes—
572C2×C4C22×C4⋊C411yescharacteristic
573C2×C4C2×#11711yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C8—81no—
577C8—81no—
578C8—82no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C2×#38011yescharacteristic
607C22#288312yes—
608C22#289711yescharacteristic
609C22#289811yescharacteristic
610C22C2×#42112yes—
611C22#304511yescharacteristic
612C22#304611yescharacteristic
613C22#308311yescharacteristic
614C22#308411yescharacteristic
615C4—21no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—161no—
631C4—162no—
632C4—161no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—42no—
638C2—42no—
639C2—82no—
640C2#1759812yes—
641C2#1759912yes—
642C2#2063311yescharacteristic
643C2#2065211yescharacteristic
644C2#2065411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
5. #14355C2semidirect
6. #8577C2semidirect
7. #8575C2semidirect
8. #8574C2semidirect
9. #8572C2semidirect
10. #8023C2semidirect
11. #8015C2semidirect
15. #2361C4semidirect
16. #2360C4semidirect
20. #2347C4semidirect
21. #2346C4semidirect
22. #2341C4semidirect
23. #2339C4semidirect
163. C2×#41C4⋊C4semidirect
200. C22⋊Q8C4⋊C4semidirect
206. #121C4⋊C4semidirect
221. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342332
#219322
C2×#15162
C2×#15132
#143552
#85772
#85752
#85742
#85722
#80232
#80152