Finite Group Databasebeta

← #495396

Subgroups — #495396

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13294C214yesmaximal, complemented
5#13293C212yesmaximal, complemented
6#13292C212yesmaximal, complemented
7#12573C214yesmaximal
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C2×#413C2212yes—
11C2×#410C2212yes—
12C2×C4×M16—21no—
13#2582—21no—
14#2581—21no—
15#2580—21no—
16#2579C412yescomplemented
17#2579C2212yes—
18#2538—21no—
19#2487C411yescharacteristic, complemented
20#2487C2211yescharacteristic
21#2409C411yescharacteristic, complemented
22#2409C2211yescharacteristic
23#2396—21no—
24#2396—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2297C2212yes—
28#2296C412yescomplemented
29#2285—21no—
30#2285C411yescharacteristic, complemented
31#1799—21no—
32#1799—21no—
33#1799C2212yes—
34#1799C2212yes—
35C4×#101—21no—
36C4×#101—21no—
37C4×#100C2212yes—
38C4×#100C2212yes—
39#1742—21no—
40#1740—21no—
41#1390C2218yes—
42#1390C2218yes—
43C22×#41C2311yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C22×M16—22no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C4×#41—41no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—22no—
69C2×#117C2×C412yes—
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C412yes—
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#413—24no—
80#413Q818yescomplemented
81C2×#101—22no—
82C2×#100C2314yes—
83#410—28no—
84C2×#99—22no—
85C2×#99C2314yes—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—41no—
89C4×M16—41no—
90C2×#97—41no—
91C2×C4×C8—41no—
92#400—21no—
93#400—21no—
94#399—21no—
95#399—21no—
96#398—22no—
97#398—22no—
98#396—22no—
99#396—22no—
100#395—21no—
101#395—21no—
102#393—21no—
103#393—21no—
104#391—22no—
105#391—21no—
106#391—21no—
107#390—22no—
108#389—22no—
109#389—22no—
110#389—21no—
111#389—21no—
112#387—22no—
113#387—22no—
114#386C2×C412yes—
115#386C2×C412yes—
116#386C2312yes—
117#386C2312yes—
118#385—22no—
119#385—22no—
120#385C2×C412yes—
121#385C2×C412yes—
122#385C2×C412yes—
123#385C2×C412yes—
124#384—21no—
125#384—41no—
126#382—21no—
127#382—41no—
128#380—21no—
129#380—21no—
130#380—22no—
131#380—22no—
132#380C2×C412yes—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—41no—
140C4×C4⋊C4—41no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143#343—22no—
144#343—22no—
145#342—22no—
146#342—22no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41C2×Q814yes—
151C2×#41#4114yes—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—44no—
157C22×Q8C2×D411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C22×D4C2411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×M16—28no—
173C2×M16—24no—
174C2×M16—44no—
175C2×M16—42no—
176C22×C8—42no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—44no—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22×C412yes—
247C2×C22⋊C4C22×C412yes—
248C2×C22⋊C4C22×C411yescharacteristic
249C2×C22⋊C4C22×C411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280#101—28no—
281#100—24no—
282#100C2×Q818yes—
283#99—28no—
284#99—44no—
285#97—42no—
286#97—42no—
287C4×C8—42no—
288C4×C8—42no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—21no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—42no—
301#95—42no—
302C24C2×C22⋊C411yescharacteristic
303C24C23×C411yescharacteristic
304C24C22×D411yescharacteristic
305#41—44no—
306#41—44no—
307#41—44no—
308#41—44no—
309C2×Q8—24no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8C22⋊Q814yes—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4C4×Q814yes—
328C2×D4#12412yes—
329C2×D4C4⋊Q812yes—
330C2×D4C22×Q812yes—
331C2×D4C2×#4112yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—82no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C4×Q814yes—
380C22×C4C22⋊Q814yes—
381C22×C4#12412yes—
382C22×C4C4⋊Q812yes—
383C22×C4C22×D411yescharacteristic
384M16—28no—
385M16—44no—
386M16—48no—
387C2×C8—44no—
388C2×C8—48no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—84no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—44no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42—84no—
449C42#9911yescharacteristic
450C42#9911yescharacteristic
451C42#9911yescharacteristic
452C42#9911yescharacteristic
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23#39112yes—
466C23C22×C22⋊C411yescharacteristic
467C23C2×C4×Q812yes—
468C23C2×#12411yescharacteristic
469C23C2×C4⋊Q811yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—48no—
473Q8—44no—
474D4—44no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4#39112yes—
538C2×C4#39112yes—
539C2×C4#39112yes—
540C2×C4C2×#9911yescharacteristic
541C2×C4C2×#9911yescharacteristic
542C2×C4C2×#9911yescharacteristic
543C2×C4C2×#9911yescharacteristic
544C2×C4C2×#9911yescharacteristic
545C2×C4C2×#9911yescharacteristic
546C2×C4#41012yes—
547C2×C4C2×C22⋊Q812yes—
548C8—48no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—22no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—81no—
567C22—81no—
568C22C2×#39111yescharacteristic
569C22C22×#9911yescharacteristic
570C22C2×#41011yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—44no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—164no—
588C4#199612yes—
589C4#199612yes—
590C4#212012yes—
591C2—21no—
592C2—21no—
593C2—41no—
594C2—42no—
595C2—41no—
596C2—42no—
597C2C2×#199611yescharacteristic
598C2C2×#212011yescharacteristic
599C2#3334711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
socleC22C22×#99
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13294C2semidirect
5. #13293C2semidirect
6. #13292C2semidirect
9. C22×C4×Q8C4semidirect
16. #2579C4semidirect
19. #2487C4semidirect
21. #2409C4semidirect
28. #2296C4semidirect
30. #2285C4semidirect
80. #413Q8semidirect

Maximal subgroups

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

subgroupindex
#349522
#206322
C4×#4132
#132942
#132932
#132922
#125732