Finite Group Databasebeta

← #214126

Subgroups — #214126

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#9074C214yesmaximal, complemented
5#8939C214yesmaximal, complemented
6#8710C212yesmaximal, complemented
7#8313C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12C22×C4⋊C8C2212yes—
13C22×C22⋊C8—21no—
14#2857C411yescharacteristic
15#2857C411yescharacteristic
16#2856C411yescharacteristic
17#2856C411yescharacteristic
18#2854—21no—
19#2854C412yes—
20#2850—21no—
21#2850—21no—
22#2847—21no—
23#2847C412yes—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C2212yes—
27C4×#121C2212yes—
28#2297—21no—
29#2285C2211yescharacteristic
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1779—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#336—21no—
41C2×D4⋊C8C2218yescomplemented
42#1279C2218yescomplemented
43C22×#41—21no—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C23×C8—22no—
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#120—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2312yes—
69C2×#117—21no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#431—44no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—44no—
87C2×C4⋊C8C2314yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—44no—
94C2×C4×C8C2314yes—
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C412yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2312yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134D4⋊C8—48no—
135C25C22⋊C411yescharacteristic
136C2×#41—44no—
137C2×#41—44no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8—21no—
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—22no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
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×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8—44no—
181C4⋊Q8—42no—
182#127—42no—
183#124—44no—
184#123—48no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187#121—28no—
188#121—28no—
189#121—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#120—48no—
196C4×D4—28no—
197C4×D4—44no—
198C4×D4—48no—
199C4×D4—48no—
200#117—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
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×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2411yescharacteristic
251C4⋊C8—28no—
252C4⋊C8—44no—
253C4⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C4×C8—28no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277#41—88no—
278#41—88no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
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—21no—
308C22×C4—21no—
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—21no—
323C22×C4—22no—
324C22×C4—21no—
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—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
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—48no—
356C22×C4—48no—
357C22×C4—416no—
358C22×C4—416no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
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×C4#13114yes—
373C22×C4#13114yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4AGL(1,ℤ/8)14yes—
376C22×C4#13712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—88no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—42no—
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—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—44no—
424C42H(5,𝔽2)14yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—28no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23#40812yes—
456C23#40812yes—
457C23#41712yes—
458C23#41812yes—
459C23#42112yes—
460C23C22×C22⋊C411yescharacteristic
461Q8—84no—
462D4—48no—
463D4—84no—
464D4—84no—
465D4—88no—
466D4—88no—
467D4—88no—
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—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
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—42no—
511C2×C4—42no—
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—84no—
523C2×C4—84no—
524C2×C4#40812yes—
525C2×C4#40812yes—
526C2×C4#41712yes—
527C2×C4#41812yes—
528C2×C4#42112yes—
529C2×C4#51812yes—
530C2×C4#53412yes—
531C2×C4#53412yes—
532C2×C4C2×#13112yes—
533C2×C4C2×#13112yes—
534C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
535C2×C4C2×AGL(1,ℤ/8)12yes—
536C2×C4C2×#13711yescharacteristic
537C8—84no—
538C8—84no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—416no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22#149614yes—
568C22#230811yescharacteristic
569C22C2×#40811yescharacteristic
570C22C2×#40811yescharacteristic
571C22C2×#41811yescharacteristic
572C22C2×#41711yescharacteristic
573C22C2×#42111yescharacteristic
574C22C2×#42111yescharacteristic
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#297714yes—
589C4#319212yes—
590C4#319312yes—
591C2—24no—
592C2—42no—
593C2—42no—
594C2—162no—
595C2#1746311yescharacteristic
596C2#1746611yescharacteristic
597C2C2×#149611yescharacteristic
598C2#1748111yescharacteristic
599C2#2173911yescharacteristic
600C2#2176411yescharacteristic
601C2#2177411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×#1797C2semidirect
4. #9074C2semidirect
5. #8939C2semidirect
6. #8710C2semidirect
7. #8313C2semidirect
41. C2×D4⋊C8C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
C2×#17972
#90742
#89392
#87102
#83132