Finite Group Databasebeta

← #375788

Subgroups — #375788

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C212yesmaximal
2C2×#2001C211yesmaximal, characteristic
3C2×#1878C212yesmaximal, complemented
4C2×#1865C212yesmaximal, complemented
5#11095C211yesmaximal, characteristic
6#10970C212yesmaximal
7#10932C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10713C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×C4×Q8C2212yes—
13C22×#117C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2323—28no—
21#2323—28no—
22C2×#391C2212yes—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#388—22no—
26C2×#388—22no—
27C2×#387—22no—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30#2001C418yescomplemented
31#1878—28no—
32#1865—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40C23×Q8C2312yes—
41C24×C4C2311yescharacteristic
42C22×Q16—24no—
43C22×Q16—24no—
44C22×Q16—41no—
45C22×Q16—41no—
46C22×Q16C2×C414yes—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8C2×C418yes—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—41no—
65C2×#117—41no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42—22no—
79C22×C42C2312yes—
80C2×#106—24no—
81C2×#106—41no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103C2×C418yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—41no—
95C2×C22⋊C8—41no—
96C2×C22⋊C8C2×C414yes—
97#391—24no—
98#391D418yes—
99#389—216no—
100#389—24no—
101#389D418yes—
102#388—216no—
103#388—216no—
104#387—216no—
105#380—24no—
106#380—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—41no—
118#340C2×C414yes—
119#336—24no—
120#336—24no—
121C25C2×D411yescharacteristic
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22×C414yes—
134C22×Q8C2×D414yes—
135C22×Q8#4114yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C2×Q16—216no—
155C2×Q16—44no—
156C2×Q16—44no—
157C2×Q16—48no—
158C2×Q16—44no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22⋊Q8—216no—
177C22⋊Q8—216no—
178C22⋊Q8—216no—
179C22⋊Q8—216no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184#117—44no—
185#117—48no—
186#117—44no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C2×D414yes—
238C2×C42C2×D414yes—
239#106—44no—
240#103—216no—
241#103—48no—
242#103—48no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24C4×D412yes—
259C24C22×D411yescharacteristic
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8C4×D418yes—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
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—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C4×D412yes—
351C22×C4#12014yes—
352C22×C4#12014yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363Q16—416no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—232no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—48no—
390C4⋊C4—416no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C4⋊C4C4×D418yes—
397C22⋊C4—232no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—88no—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—416no—
406C42—88no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23C2×C4×D411yescharacteristic
429C23C2×#12012yes—
430C23#52512yes—
431C23#53411yescharacteristic
432C23#53711yescharacteristic
433C23#57712yes—
434C23#57712yes—
435C23#57812yes—
436C23#57812yes—
437Q8—28no—
438Q8—28no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4C2×C4×D412yes—
499C2×C4C2×C4×D412yes—
500C2×C4#51811yescharacteristic
501C2×C4#52011yescharacteristic
502C2×C4#52512yes—
503C2×C4#52512yes—
504C2×C4#53811yescharacteristic
505C2×C4#53911yescharacteristic
506C2×C4D4211yescharacteristic
507C2×C4D4212yes—
508C2×C4D4211yescharacteristic
509C2×C4#54612yes—
510C2×C4#54712yes—
511C2×C4#54812yes—
512C2×C4#55012yes—
513C2×C4#55111yescharacteristic
514C2×C4#55312yes—
515C2×C4#55911yescharacteristic
516C8—42no—
517C8—42no—
518C8—82no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C4×#12012yes—
529C22#232511yescharacteristic
530C22#240311yescharacteristic
531C22#240712yes—
532C22#260011yescharacteristic
533C22#294712yes—
534C22#295011yescharacteristic
535C22#295211yescharacteristic
536C22#300912yes—
537C22#301112yes—
538C22#301312yes—
539C22#301312yes—
540C22#312012yes—
541C22#312112yes—
542C22#312512yes—
543C22#312712yes—
544C22#320112yes—
545C22#320112yes—
546C22#325812yes—
547C22#326111yescharacteristic
548C22#326311yescharacteristic
549C4—42no—
550C4—48no—
551C4—44no—
552C4—44no—
553C4—48no—
554C4—84no—
555C4—84no—
556C4#235811yescharacteristic
557C4#236112yes—
558C4#236611yescharacteristic
559C4#245511yescharacteristic
560C4#245612yes—
561C4#246311yescharacteristic
562C2—82no—
563C2#1417111yescharacteristic
564C2#2146112yes—
565C2#2146312yes—
566C2#2219211yescharacteristic
567C2#2219811yescharacteristic
568C2#2392712yes—
569C2#2416512yes—
570C2#2416912yes—
571C2#2942512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1878C2semidirect
4. C2×#1865C2semidirect
7. #10932C2semidirect
8. #10885C2semidirect
30. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
C2×#20012
C2×#18782
C2×#18652
#110952
#109702
#109322
#108852
#107132