Finite Group Databasebeta

← #6484801

Subgroups — #6484801

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#33146C214yesmaximal
4#29966C214yesmaximal
5#29286C214yesmaximal, complemented
6#29272C214yesmaximal, complemented
7#29219C212yesmaximal, complemented
8#29208C212yesmaximal, complemented
9#29178C214yesmaximal, complemented
10#21982C212yesmaximal
11#21833C212yesmaximal
12#20808C211yesmaximal, characteristic
13C2×#533—21no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8—21no—
16C2×#431C2212yes—
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2862—21no—
20#2859—21no—
21#2859C2212yes—
22#2857—21no—
23#2857—21no—
24#2857C2212yes—
25#2857C2212yes—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8—21no—
28C4×C4⋊Q8—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yes—
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C2×#391—21no—
39C2×#389—21no—
40C2×#389C2212yes—
41#2285—21no—
42#2285C2212yes—
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4C2212yes—
46C2×C43C2211yescharacteristic
47#2099C2218yes—
48#2086C22116yes—
49#2057C2218yes—
50#1990C2214yes—
51#1990C2214yes—
52#1988C2218yes—
53#1988C2218yes—
54#1947C2218yes—
55#1947C2218yes—
56#1918C22116yes—
57#1852C2218yes—
58#1845C2214yes—
59#1842C2212yes—
60#1834C2214yes—
61#1831C2212yes—
62#1823C2218yes—
63#1797C2212yes—
64#1797C2212yes—
65C4×C4⋊C8C2212yes—
66C4×C4⋊C8C2212yes—
67C23×Q8—21no—
68#533—48no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8Q818yescomplemented
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×#125—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—48no—
91C2×#117—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C42—22no—
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#431—28no—
107C2×#107C23116yes—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C4×C8C2314yes—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389C2314yes—
134#389C2314yes—
135#389C2314yes—
136#389C2314yes—
137#384—21no—
138#384—21no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2312yes—
144#384C2312yes—
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#382—22no—
150#382—22no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C2×#95—22no—
168C2×#95—22no—
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176C43C2311yescharacteristic
177#340C23116yes—
178#340C23116yes—
179#340C23116yes—
180#336C2314yes—
181#336C2314yes—
182C22×Q8—28no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—44no—
188C22×Q8—44no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C22×C8C2418yes—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203C4⋊Q8—28no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8C2×Q818yes—
210C4⋊Q8C2×Q818yes—
211#125—48no—
212C22⋊Q8—48no—
213C22⋊Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—48no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220#117—48no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
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—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×Q818yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C22⋊C4—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—48no—
264C2×C42D818yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×Q814yes—
288C2×C42#4114yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295#107—416no—
296C4⋊C8—28no—
297C4⋊C8—28no—
298C4⋊C8—44no—
299C22⋊C8—28no—
300C4×C8—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309C24C22×D411yescharacteristic
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—432no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4C2×D814yes—
350C22×C4C2×D814yes—
351C22×C4C2×D814yes—
352C22×C4#13714yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q812yes—
367C22×C4C2×#4112yes—
368C22×C4C2511yescharacteristic
369C2×C8—28no—
370C2×C8—28no—
371C2×C8—28no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4C22×Q818yes—
399C22⋊C4—48no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—48no—
419C42—48no—
420C42—48no—
421C42C22⋊Q814yes—
422C42C22⋊Q814yes—
423C42C2×D814yes—
424C42C2×D814yes—
425C42C2×D814yes—
426C42C2×D814yes—
427C42#13714yes—
428C42C22×Q812yes—
429C42C22×Q812yes—
430C42C2×#4112yes—
431C42C2×#4112yes—
432C422−1+414yes—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23C2×C22⋊Q812yes—
438C23C22×D812yes—
439C23C2×#13712yes—
440C23C23×D411yescharacteristic
441Q8—416no—
442Q8—416no—
443Q8—88no—
444Q8—88no—
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—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—416no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—88no—
486C2×C4#474116yes—
487C2×C4C2×C22⋊Q812yes—
488C2×C4C2×C22⋊Q812yes—
489C2×C4C2×C22⋊Q812yes—
490C2×C4C2×C22⋊Q812yes—
491C2×C4C2×C22⋊Q812yes—
492C2×C4#53612yes—
493C2×C4#53612yes—
494C2×C4C22×D812yes—
495C2×C4C22×D812yes—
496C2×C4C22×D812yes—
497C2×C4C22×D812yes—
498C2×C4C22×D812yes—
499C2×C4C22×D812yes—
500C2×C4C2×#13712yes—
501C2×C4C2×#13712yes—
502C2×C4C2×#13712yes—
503C2×C4C2×#13712yes—
504C2×C4C2×#13712yes—
505C2×C4#57512yes—
506C2×C4C23×Q811yescharacteristic
507C2×C4C22×#4111yescharacteristic
508C2×C4C2×2−1+412yes—
509C2×C4#58512yes—
510C8—48no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22C2×#47414yes—
518C22C22×C22⋊Q811yescharacteristic
519C22C2×#53611yescharacteristic
520C22#344911yescharacteristic
521C22C23×D811yescharacteristic
522C22C22×#13711yescharacteristic
523C22C22×#13711yescharacteristic
524C22C2×#57511yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—41no—
533C4—48no—
534C4—41no—
535C4—84no—
536C4—84no—
537C4—84no—
538C4C2×#47414yes—
539C4C2×#47414yes—
540C4#307814yes—
541C4#322614yes—
542C4#350212yes—
543C4#350712yes—
544C2—24no—
545C2#3517911yescharacteristic
546C2C22×#47411yescharacteristic
547C2C2×#307811yescharacteristic
548C2C2×#322611yescharacteristic
549C2#6086511yescharacteristic
550C2#6094011yescharacteristic
551C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29286C2semidirect
6. #29272C2semidirect
7. #29219C2semidirect
8. #29208C2semidirect
9. #29178C2semidirect
76. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C2×C4×C4⋊Q82
#331462
#299662
#292862
#292722
#292192
#292082
#291782
#219822
#218332
#208082