Finite Group Databasebeta

← #161821

Subgroups — #161821

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#7728C212yesmaximal, complemented
5#7727C212yesmaximal, complemented
6#7701C214yesmaximal, complemented
7#7589C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432—21no—
11C22×C4⋊C8—21no—
12C22×C22⋊C8—21no—
13#2857C411yescharacteristic
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2855—21no—
18#2855C2212yes—
19#2851—21no—
20#2851—21no—
21#2848—21no—
22#2848C412yes—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C412yes—
26C4×C22⋊Q8C2212yes—
27#2296—21no—
28#2285C411yescharacteristic
29#2280—21no—
30#1817C2212yes—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1779—21no—
38#1749—21no—
39#1749C2212yes—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1271C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C23×C8—22no—
47#533—44no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×#117—21no—
60C2×#117—21no—
61C22×C42—44no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#432—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—44no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—44no—
85C2×#97—22no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2312yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389C2×C412yes—
97#389C2312yes—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#381—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2×C412yes—
122#379—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#336—22no—
130#336—22no—
131#336C2314yes—
132#336C2314yes—
133C25C22⋊C411yescharacteristic
134C22×Q8—21no—
135C22×Q8C22×C412yes—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C22×C8—24no—
165C22×C8—24no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—44no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171#125—44no—
172C22⋊Q8—28no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C4×Q8—44no—
180C4×Q8—44no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—44no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243#97—44no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264C2×Q8—24no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—416no—
329C22×C4—416no—
330C22×C4—48no—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4#13114yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13712yes—
343C22×C4#13714yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—88no—
354C4⋊C4—24no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—44no—
393C42—44no—
394C42#9914yes—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—28no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23#40812yes—
419C23#41712yes—
420C23#41912yes—
421C23#42112yes—
422C23#42112yes—
423C23C22×C22⋊C411yescharacteristic
424Q8—84no—
425Q8—84no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4#40812yes—
486C2×C4C2×#9912yes—
487C2×C4C2×#9912yes—
488C2×C4C2×#9912yes—
489C2×C4#41312yes—
490C2×C4#41712yes—
491C2×C4#41912yes—
492C2×C4C2×#13112yes—
493C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
494C2×C4C2×#13712yes—
495C2×C4C2×#13711yescharacteristic
496C8—84no—
497C8—84no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—416no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22#149714yes—
520C22C2×#40811yescharacteristic
521C22C22×#9911yescharacteristic
522C22C2×#41311yescharacteristic
523C22C2×#41911yescharacteristic
524C22C2×#41711yescharacteristic
525C22C2×#42111yescharacteristic
526C22C2×#42111yescharacteristic
527C4—22no—
528C4—22no—
529C4—22no—
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C2—24no—
545C2—42no—
546C2—42no—
547C2#1738111yescharacteristic
548C2#1746611yescharacteristic
549C2C2×#149711yescharacteristic
550C2#1757811yescharacteristic
551C2#1759211yescharacteristic
552C2#1767811yescharacteristic
553C2#1774911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7728C2semidirect
5. #7727C2semidirect
6. #7701C2semidirect
7. #7589C2semidirect
41. #1272C22semidirect
42. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#207152
C2×#17972
#77282
#77272
#77012
#75892