Finite Group Databasebeta

← #185034

Subgroups — #185034

Order 512.

4136 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
1C23×C22⋊Q8C211yesmaximal, characteristic
2#20579C211yesmaximal, characteristic
3C2×#1511C214yesmaximal, complemented
4#14681C211yesmaximal, characteristic
5#8329C218yesmaximal, complemented
6C24×Q8C411yescharacteristic, complemented
7C25×C4C2211yescharacteristic
8C22×C22⋊Q8—21no—
9C22×C22⋊Q8—22no—
10C22×C22⋊Q8—21no—
11C22×C22⋊Q8—22no—
12C22×C22⋊Q8—216no—
13C22×C22⋊Q8C414yescomplemented
14C22×C22⋊Q8C412yescomplemented
15C22×C22⋊Q8C2214yes—
16C22×C22⋊Q8C2212yes—
17C23×C4⋊C4—21no—
18C23×C4⋊C4C411yescharacteristic, complemented
19C23×C22⋊C4—21no—
20C2×#407—21no—
21C2×#407—21no—
22C2×#407C2212yes—
23C2×#381—21no—
24C2×#381C2212yes—
25C2×#379—21no—
26#1779—22no—
27#1779C2214yes—
28#1511—216no—
29C2×#328C22116yescomplemented
30C22⋊#95—22no—
31C22⋊#95C2214yes—
32C26D411yescharacteristic
33C23×Q8—28no—
34C23×Q8—22no—
35C23×Q8—21no—
36C23×Q8—44no—
37C23×Q8—44no—
38C23×Q8C2×C414yes—
39C23×Q8C2×C412yes—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4—28no—
43C24×C4—44no—
44C24×C4—44no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D412yes—
52C24×C4D412yes—
53C24×C4C2312yes—
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C2×C22⋊Q8—232no—
57C2×C22⋊Q8—232no—
58C2×C22⋊Q8—216no—
59C2×C22⋊Q8—216no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—232no—
70C2×C22⋊Q8—232no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8C2×C418yes—
73C2×C22⋊Q8C2×C414yes—
74C2×C22⋊Q8D4116yes—
75C2×C22⋊Q8C2314yes—
76C22×C4⋊C4—216no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4C2×C414yes—
85C22×C4⋊C4C2×C412yes—
86C22×C22⋊C4—216no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—22no—
93#407—28no—
94#407—44no—
95#407—44no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100#381—28no—
101#381—44no—
102#379—44no—
103C2×#95—22no—
104C2×#95—24no—
105C2×#95C2314yes—
106#328—232no—
107C25—24no—
108C25—28no—
109C25—24no—
110C25—42no—
111C25—44no—
112C25C22⋊C411yescharacteristic
113C25C22⋊C411yescharacteristic
114C25C22⋊C412yes—
115C25C2×D411yescharacteristic
116C25C2×D412yes—
117C22×Q8—28no—
118C22×Q8—216no—
119C22×Q8—216no—
120C22×Q8—22no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—48no—
124C22×Q8—464no—
125C22×Q8—48no—
126C22×Q8—44no—
127C22×Q8—48no—
128C22×Q8—48no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8C22⋊C418yescomplemented
133C22×Q8C22×C414yes—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—28no—
138C23×C4—28no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—28no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—24no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—464no—
171C23×C4—48no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C412yes—
176C23×C4C22⋊C412yes—
177C23×C4C22⋊C412yes—
178C23×C4C22⋊C412yes—
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D418yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8—24no—
197C22⋊Q8—232no—
198C22⋊Q8—232no—
199C22⋊Q8—232no—
200C22⋊Q8—232no—
201C22⋊Q8—232no—
202C22⋊Q8—216no—
203C22⋊Q8—28no—
204C22⋊Q8—216no—
205C22⋊Q8—216no—
206C22⋊Q8—28no—
207C22⋊Q8—216no—
208C22⋊Q8C22⋊C4116yescomplemented
209C22⋊Q8C2×D4116yes—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—232no—
213C2×C4⋊C4—232no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—216no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4C22⋊C418yescomplemented
224C2×C4⋊C4C22×C414yes—
225C2×C22⋊C4—216no—
226C2×C22⋊C4—232no—
227C2×C22⋊C4—232no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—48no—
238C22⋊C8—28no—
239C22⋊C8—48no—
240C22⋊C8—48no—
241C22⋊C8—44no—
242#95—28no—
243#95—48no—
244#95—44no—
245C24—22no—
246C24—22no—
247C24—24no—
248C24—24no—
249C24—232no—
250C24—24no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—22no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—24no—
264C24—22no—
265C24—24no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—416no—
273C24—432no—
274C24—416no—
275C24—432no—
276C24#9914yes—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C412yes—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C412yes—
281C24#12014yes—
282C24#13718yes—
283C24C22×D411yescharacteristic
284C2×Q8—216no—
285C2×Q8—24no—
286C2×Q8—28no—
287C2×Q8—464no—
288C2×Q8—464no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—432no—
293C2×Q8—416no—
294C2×Q8—416no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—48no—
298C2×Q8C2×C22⋊C418yes—
299C22×C4—232no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—216no—
313C22×C4—216no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—416no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—432no—
358C22×C4—432no—
359C22×C4—432no—
360C22×C4—432no—
361C22×C4—416no—
362C22×C4—432no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—432no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4#9914yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C414yes—
383C22×C4#12014yes—
384C22×C4#12014yes—
385C22×C4#12014yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D414yes—
389M16—88no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C4⋊C4—216no—
394C4⋊C4—232no—
395C4⋊C4—24no—
396C4⋊C4—216no—
397C4⋊C4—216no—
398C4⋊C4—28no—
399C4⋊C4C2×C22⋊C418yes—
400C22⋊C4—232no—
401C22⋊C4—216no—
402C22⋊C4—216no—
403C22⋊C4—816no—
404C22⋊C4—88no—
405C23—28no—
406C23—28no—
407C23—216no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—216no—
418C23—28no—
419C23—28no—
420C23—216no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—28no—
431C23—216no—
432C23—22no—
433C23—216no—
434C23—28no—
435C23—44no—
436C23—48no—
437C23—48no—
438C23—48no—
439C23—48no—
440C23—48no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23—48no—
445C23—48no—
446C23—48no—
447C23—48no—
448C23—48no—
449C23—4128no—
450C23—44no—
451C23—48no—
452C23—48no—
453C23—44no—
454C23#37912yes—
455C23#37912yes—
456C23C2×#9914yes—
457C23C2×#9912yes—
458C23C2×#9912yes—
459C23#41914yes—
460C23#41914yes—
461C23#42114yes—
462C23C22×C22⋊C411yescharacteristic
463C23C2×#12012yes—
464C23C2×#13714yes—
465Q8—48no—
466Q8—48no—
467Q8—416no—
468Q8—432no—
469C2×C4—216no—
470C2×C4—216no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—416no—
502C2×C4—432no—
503C2×C4—432no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—44no—
508C2×C4—416no—
509C2×C4—84no—
510C2×C4—88no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—84no—
514C2×C4#37912yes—
515C2×C4#37912yes—
516C2×C4C2×#9914yes—
517C2×C4C22×C22⋊C412yes—
518C2×C4C2×#12012yes—
519C8—84no—
520C22—232no—
521C22—24no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—22no—
526C22—24no—
527C22—22no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—432no—
542C22—416no—
543C22—432no—
544C22—416no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22#151118yescomplemented
552C22#178114yes—
553C22C2×#37911yescharacteristic
554C22C22×#9911yescharacteristic
555C22C22×#9911yescharacteristic
556C22C2×#41912yes—
557C22C2×#42112yes—
558C22#300414yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—24no—
563C4—22no—
564C4—42no—
565C4—48no—
566C4—48no—
567C4—164no—
568C2—24no—
569C2—28no—
570C2—24no—
571C2—42no—
572C2—44no—
573C2C2×#151112yes—
574C2#1757812yes—
575C2C2×#178111yescharacteristic
576C2#2059911yescharacteristic
577C2#2060511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1511C2semidirect
5. #8329C2semidirect
6. C24×Q8C4semidirect
13. C22×C22⋊Q8C4semidirect
14. C22×C22⋊Q8C4semidirect
18. C23×C4⋊C4C4semidirect
29. C2×#328C22semidirect
132. C22×Q8C22⋊C4semidirect
208. C22⋊Q8C22⋊C4semidirect
223. C2×C4⋊C4C22⋊C4semidirect
551. C22#1511semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
#205792
C2×#15112
#146812
#83292