Finite Group Databasebeta

← #214577

Subgroups — #214577

Order 512.

1101 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#21853C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#9139C212yesmaximal, complemented
5#9138C212yesmaximal, complemented
6#9137C212yesmaximal, complemented
7#9131C212yesmaximal, complemented
8#8727C212yesmaximal, complemented
9#8340C212yesmaximal, complemented
10C2×#536—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#408—21no—
18#2857C2212yes—
19#2857C2212yes—
20#2621—21no—
21#2621—21no—
22#2547—21no—
23#2547C412yes—
24#2546—21no—
25#2546C412yes—
26#2486—21no—
27#2486—21no—
28#2486C412yes—
29#2486C412yes—
30#2310—21no—
31#2285C2211yescharacteristic
32#1985—21no—
33#1955—21no—
34#1870—21no—
35#1870—21no—
36#1798—21no—
37#1798—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1797C2212yes—
42#1729—21no—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×#41D411yescharacteristic
52C23×Q8—21no—
53C22×M16—21no—
54C22×M16—21no—
55C23×C8—21no—
56#536—44no—
57#536—44no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—41no—
62C2×C4⋊Q8—41no—
63C2×C4⋊Q8—41no—
64C2×C4⋊Q8—41no—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8C2312yes—
67C2×#124—22no—
68C2×#123—22no—
69C2×#123—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C4×#41—41no—
73C4×#41—41no—
74C4×#41—41no—
75C4×#41—41no—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C2×#117D411yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#432—41no—
94#432—41no—
95#423—41no—
96#423—41no—
97#423—41no—
98#423—41no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108#408—44no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×#97—21no—
115#398—22no—
116#398—22no—
117#398—22no—
118#398—22no—
119#397—22no—
120#397—22no—
121#397—22no—
122#397—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#389—22no—
128#389—22no—
129#389C2×C414yes—
130#389C2×C414yes—
131#384C2312yes—
132#382—21no—
133#381—22no—
134#381—22no—
135#380—22no—
136#380—22no—
137#380—21no—
138#380C2×C414yes—
139#380C2×C414yes—
140#380C2×C412yes—
141C4×C4⋊C4—21no—
142C4×C4⋊C4C2×C412yes—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#332—42no—
148#332—42no—
149#332—42no—
150#332—42no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—41no—
156C2×#41—41no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8C22⋊C411yescharacteristic
166C22×D4C22⋊C411yescharacteristic
167C22×D4C22⋊C411yescharacteristic
168C22×D4C22⋊C411yescharacteristic
169C23×C4—22no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—82no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8—42no—
201C4⋊Q8—48no—
202#124—48no—
203#123—48no—
204#123—48no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230#117—41no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C22×C414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295C22⋊C8—44no—
296C22⋊C8—42no—
297C22⋊C8—44no—
298C22⋊C8—82no—
299C22⋊C8—82no—
300#97—42no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310#41—44no—
311#41—44no—
312#41—44no—
313#41—44no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—44no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
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—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4#13712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395M16—84no—
396M16—84no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—84no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—48no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—44no—
457C42H(5,𝔽2)11yescharacteristic
458C42H(5,𝔽2)11yescharacteristic
459C42H(5,𝔽2)11yescharacteristic
460C42H(5,𝔽2)11yescharacteristic
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23#41712yes—
475C23#41812yes—
476C23C22×C22⋊C411yescharacteristic
477Q8—44no—
478Q8—88no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482Q8—82no—
483D4—44no—
484D4—44no—
485D4—44no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—88no—
540C2×C4#41712yes—
541C2×C4#41812yes—
542C2×C4#51811yescharacteristic
543C2×C4#51811yescharacteristic
544C2×C4#53411yescharacteristic
545C2×C4#53411yescharacteristic
546C2×C4#53411yescharacteristic
547C2×C4#53411yescharacteristic
548C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
549C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
550C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
551C2×C4C2×#13711yescharacteristic
552C2×C4#57712yes—
553C2×C4#57812yes—
554C8—82no—
555C8—82no—
556C8—82no—
557C8—82no—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22#230811yescharacteristic
574C22#288912yes—
575C22C2×#41811yescharacteristic
576C22C2×#41711yescharacteristic
577C22#289812yes—
578C4—22no—
579C4—22no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—84no—
590C4—84no—
591C4#319211yescharacteristic
592C4#319211yescharacteristic
593C4#319311yescharacteristic
594C4#319311yescharacteristic
595C4#320011yescharacteristic
596C4#320112yes—
597C4#320211yescharacteristic
598C2—42no—
599C2—42no—
600C2—81no—
601C2#1775912yes—
602C2#1776312yes—
603C2#2175611yescharacteristic
604C2#2176411yescharacteristic
605C2#2177111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9139C2semidirect
5. #9138C2semidirect
6. #9137C2semidirect
7. #9131C2semidirect
8. #8727C2semidirect
9. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#218532
#206292
#91392
#91382
#91372
#91312
#87272
#83402