Finite Group Databasebeta

← #215112

Subgroups — #215112

Order 512.

1416 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
1C2×#2854C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#20714C211yesmaximal, characteristic, complemented
4#9651C214yesmaximal, complemented
5#9650C212yesmaximal, complemented
6#9649C212yesmaximal, complemented
7#8304C214yesmaximal, complemented
8C22×#123C2212yes—
9C22×#121C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#431—21no—
13#2854—21no—
14#2854—21no—
15#2854—21no—
16#2854—21no—
17#2854—21no—
18#2854—21no—
19#2854—21no—
20#2854—21no—
21C2×#390C411yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C412yes—
26C2×#388C2212yes—
27C2×#386C412yes—
28C2×#380C411yescharacteristic
29#1817—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1779C2212yes—
36#1779C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41C2×#324C2218yes—
42C2×#323C2218yes—
43C23×D4C2×C411yescharacteristic
44C23×D4C2×C412yes—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#123—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—48no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—44no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2312yes—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#431—44no—
83#431—44no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2314yes—
91C2×C22⋊C8C2314yes—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#324—48no—
134#323—48no—
135C25C22×C411yescharacteristic
136C25C22×C412yes—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—24no—
142C22×D4—44no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—24no—
178C22×C8—24no—
179#123—48no—
180#123—48no—
181#123—48no—
182#123—48no—
183#121—44no—
184#121—44no—
185#121—44no—
186#121—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C4⋊C8—44no—
249C4⋊C8—44no—
250C22⋊C8—24no—
251C22⋊C8—28no—
252C22⋊C8—24no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C4×C8—44no—
258C4×C8—44no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282C24C23×C411yescharacteristic
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—88no—
296C2×D4—816no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
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—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4#9914yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4#13114yes—
374C22×C4#13114yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4H(5,𝔽2)14yes—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—28no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—48no—
453C23—84no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23C2×#9912yes—
460C23#41012yes—
461C23C22×C22⋊C411yescharacteristic
462C23#51812yes—
463C23C2×#13112yes—
464C23C2×#13112yes—
465D4—84no—
466D4—88no—
467D4—88no—
468D4—84no—
469C2×C4—28no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4#40812yes—
526C2×C4#40812yes—
527C2×C4#40812yes—
528C2×C4#40812yes—
529C2×C4C2×#9912yes—
530C2×C4C2×#9912yes—
531C2×C4#41312yes—
532C2×C4#41312yes—
533C2×C4#53512yes—
534C2×C4#53512yes—
535C8—84no—
536C8—84no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—416no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—168no—
566C22#230911yescharacteristic
567C22C2×#40811yescharacteristic
568C22C2×#40811yescharacteristic
569C22C22×#9911yescharacteristic
570C22C2×#41011yescharacteristic
571C22C2×#41311yescharacteristic
572C22C2×#41311yescharacteristic
573C22#297914yes—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4#146814yes—
589C2—24no—
590C2—42no—
591C2—42no—
592C2—162no—
593C2C2×#146811yescharacteristic
594C2#1737611yescharacteristic
595C2#1738111yescharacteristic
596C2#1738311yescharacteristic
597C2#2178711yescharacteristic
598C2#2179311yescharacteristic
599C2#2180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ 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
2. #20714C2semidirect
3. #20714C2semidirect
4. #9651C2semidirect
5. #9650C2semidirect
6. #9649C2semidirect
7. #8304C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#207142
#96512
#96502
#96492
#83042