Finite Group Databasebeta

← #213610

Subgroups — #213610

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4#9120C214yesmaximal, complemented
5#9119C214yesmaximal, complemented
6#8721C212yesmaximal, complemented
7#8337C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525C412yes—
10C23×C42C2211yescharacteristic
11C2×#407—21no—
12C22×C22⋊C8—21no—
13C2×C4×M16C2212yes—
14C22×#97C2212yes—
15#2857C411yescharacteristic
16#2857C411yescharacteristic
17#2856C411yescharacteristic
18#2856C411yescharacteristic
19#2854—21no—
20#2854—21no—
21#2850C2212yes—
22#2850C2212yes—
23#2847—21no—
24#2847—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28C4×#121—21no—
29#2297C412yes—
30#2285C2211yescharacteristic
31#1749—21no—
32#1749—21no—
33#1749C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39#1728—21no—
40C2×#336—21no—
41C2×#329C2218yescomplemented
42#1295C2218yescomplemented
43C22×#41C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49#532—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124C2312yes—
53#525—48no—
54C2×#123—22no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×#121C2×C412yes—
65C2×#120—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×#117C2×C412yes—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#407—28no—
85C2×C22⋊C8—28no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C4×M16—44no—
91C2×#97—22no—
92C2×#97—44no—
93C2×#97C2314yes—
94C2×#97C2314yes—
95C2×C4×C8—22no—
96#391C2×C412yes—
97#391C2×C412yes—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2×C412yes—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382C2312yes—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#329—48no—
136C25C22⋊C411yescharacteristic
137C2×#41—44no—
138C2×#41—44no—
139C2×#41—44no—
140C22×Q8C22×C412yes—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C2×M16—28no—
176C2×M16—44no—
177C2×M16—44no—
178C22×C8—28no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C4⋊Q8—42no—
184#127—42no—
185#124—28no—
186#123—48no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#121—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#121—44no—
196#121—44no—
197#120—48no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—48no—
201C4×D4—48no—
202#117—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2411yescharacteristic
253C22⋊C8—28no—
254C22⋊C8—28no—
255C22⋊C8—28no—
256C22⋊C8—28no—
257#97—28no—
258#97—28no—
259#97—44no—
260C4×C8—44no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278#41—816no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—42no—
299C2×D4—42no—
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—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
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—416no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4#10014yes—
361C22×C4#10414yes—
362C22×C4#10414yes—
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×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375M16—416no—
376C2×C8—28no—
377C2×C8—28no—
378C2×C8—416no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C22⋊C4—44no—
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—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—44no—
420C42H(5,𝔽2)14yes—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—28no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23C2×#10012yes—
452C23#41312yes—
453C23C2×#10412yes—
454C23C2×#10412yes—
455C23#42112yes—
456C23C22×C22⋊C411yescharacteristic
457Q8—84no—
458D4—84no—
459D4—84no—
460D4—84no—
461D4—88no—
462D4—88no—
463D4—88no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
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—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
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—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—88no—
518C2×C4C2×#10012yes—
519C2×C4C2×#10012yes—
520C2×C4C2×#10412yes—
521C2×C4C2×#10412yes—
522C2×C4C2×#10412yes—
523C2×C4C2×#10412yes—
524C2×C4#42112yes—
525C2×C4#51812yes—
526C2×C4#53412yes—
527C2×C4#53412yes—
528C8—48no—
529C8—48no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—416no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22#152514yes—
559C22#230811yescharacteristic
560C22C22×#10011yescharacteristic
561C22C2×#41311yescharacteristic
562C22C22×#10411yescharacteristic
563C22C22×#10411yescharacteristic
564C22C2×#42111yescharacteristic
565C22C2×#42111yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—82no—
579C4—82no—
580C4—82no—
581C2—24no—
582C2—42no—
583C2—42no—
584C2—162no—
585C2C2×#152511yescharacteristic
586C2#1764611yescharacteristic
587C2#1765111yescharacteristic
588C2#1767111yescharacteristic
589C2#2175211yescharacteristic
590C2#2177311yescharacteristic
591C2#2177311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1749C2semidirect
4. #9120C2semidirect
5. #9119C2semidirect
6. #8721C2semidirect
7. #8337C2semidirect
41. C2×#329C22semidirect
42. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C4×#4072
C2×#17492
#91202
#91192
#87212
#83372