Finite Group Databasebeta

← #238271

Subgroups — #238271

Order 512.

1111 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#10375C218yesmaximal, complemented
6#8023C211yesmaximal, characteristic
7C2×#1289C211yesmaximal, characteristic
8C2×#1282C211yesmaximal, characteristic
9C2×#543—21no—
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2857—22no—
14#2659—22no—
15#2549—22no—
16#2549C2214yes—
17#2440—22no—
18#2440—22no—
19#2315—21no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1996—22no—
25#1996C2214yes—
26C2×#342—21no—
27C2×#342—21no—
28#1520—22no—
29#1520—22no—
30#1520—22no—
31#1520C2214yes—
32C2×#328—22no—
33C2×#328C2214yes—
34#1294C2214yes—
35#1289C2214yes—
36#1282C2214yes—
37C23×Q8Q811yescharacteristic, complemented
38C24×C4C2311yescharacteristic
39C23×C8—21no—
40C23×C8—21no—
41#543—44no—
42#543—44no—
43C2×C4⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8Q812yescomplemented
58C2×C22⋊Q8C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4Q811yescharacteristic, complemented
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83#398—24no—
84#398—24no—
85#395—24no—
86#395—24no—
87#391—24no—
88#389—22no—
89#389—22no—
90#389—24no—
91#385—24no—
92#385—24no—
93#384—22no—
94#380—24no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380D412yes—
103#380D412yes—
104#380D412yes—
105#380C2312yes—
106#380C2312yes—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4C2312yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95D412yes—
114C2×#95C2312yes—
115#342—24no—
116#342—24no—
117#328—28no—
118#328—28no—
119C25C2×D411yescharacteristic
120C22×Q8—24no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—82no—
127C22×Q8C2×Q812yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—41no—
135C23×C4—82no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×Q811yescharacteristic
142C23×C4C2×Q811yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4#4111yescharacteristic
146C23×C4#4111yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—82no—
157C22×C8—82no—
158C4⋊Q8—84no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—48no—
168C22⋊Q8—416no—
169C22⋊Q8—416no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—48no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×Q812yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4SD1614yes—
228C2×C22⋊C4Q1614yes—
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—82no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95C2×D414yes—
251#95C2×D414yes—
252C24—81no—
253C24—81no—
254C24#12011yescharacteristic
255C24#12011yescharacteristic
256C24C22⋊Q811yescharacteristic
257C24C22⋊Q811yescharacteristic
258C24#12411yescharacteristic
259C24#12411yescharacteristic
260C24C2×SD1612yes—
261C24C2×Q1612yes—
262C24AGL(1,ℤ/8)12yes—
263C24#13712yes—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—48no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—82no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4C22⋊Q811yescharacteristic
325C22×C4C22⋊Q811yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C22⋊Q811yescharacteristic
328C22×C4C22⋊Q811yescharacteristic
329C22×C4C22⋊Q811yescharacteristic
330C22×C4C22⋊Q811yescharacteristic
331C22×C4C22⋊Q811yescharacteristic
332C22×C4C22⋊Q811yescharacteristic
333C22×C4#12411yescharacteristic
334C22×C4#12411yescharacteristic
335C22×C4#12411yescharacteristic
336C22×C4#12411yescharacteristic
337C22×C4#12411yescharacteristic
338C22×C4#12411yescharacteristic
339C22×C4#12411yescharacteristic
340C22×C4#12411yescharacteristic
341C22×C4#12411yescharacteristic
342C22×C4#12411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—82no—
352C2×C8—82no—
353C2×C8—82no—
354C2×C8—82no—
355C2×C8—82no—
356C2×C8—82no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C4⋊C4—82no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4C2×SD1614yes—
393C22⋊C4C2×Q1614yes—
394C42—44no—
395C42—82no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#39311yescharacteristic
415C23#39311yescharacteristic
416C23#39311yescharacteristic
417C23#39311yescharacteristic
418C23#45012yes—
419C23#45012yes—
420C23#45112yes—
421C23#45112yes—
422C23#47512yes—
423C23#47512yes—
424C23#47712yes—
425C23#47712yes—
426C23#48712yes—
427C23#48712yes—
428C23#48912yes—
429C23#48912yes—
430C23C2×#12011yescharacteristic
431C23C2×C22⋊Q811yescharacteristic
432C23C2×#12411yescharacteristic
433C23C22×SD1611yescharacteristic
434C23C22×Q1611yescharacteristic
435C23C2×AGL(1,ℤ/8)11yescharacteristic
436C23C2×#13711yescharacteristic
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—168no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—162no—
496C2×C4—162no—
497C2×C4—164no—
498C2×C4—164no—
499C2×C4#39311yescharacteristic
500C2×C4#39311yescharacteristic
501C2×C4#39311yescharacteristic
502C2×C4#39311yescharacteristic
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×#12411yescharacteristic
506C2×C4C2×#12411yescharacteristic
507C8—82no—
508C8—82no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22#202412yes—
540C22#202412yes—
541C22#202412yes—
542C22#202412yes—
543C22C2×#39311yescharacteristic
544C22C2×#45011yescharacteristic
545C22C2×#45111yescharacteristic
546C22C2×#47511yescharacteristic
547C22C2×#47711yescharacteristic
548C22C2×#48711yescharacteristic
549C22C2×#48911yescharacteristic
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—44no—
558C4—44no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4—162no—
564C4#203212yes—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—22no—
572C2—22no—
573C2—22no—
574C2—22no—
575C2C2×#202411yescharacteristic
576C2#2347211yescharacteristic
577C2C2×#203211yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10375C2semidirect
37. C23×Q8Q8semidirect
57. C2×C22⋊Q8Q8semidirect
63. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#19962
C2×#15202
#103752
#80232
C2×#12892
C2×#12822