Finite Group Databasebeta

← #6490673

Subgroups — #6490673

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 541 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#30091C214yesmaximal, complemented
5#29272C214yesmaximal, complemented
6#29219C212yesmaximal, complemented
7#28689C212yesmaximal, complemented
8#27960C212yesmaximal
9#27610C214yesmaximal
10#25813C212yesmaximal
11#22247C214yesmaximal, complemented
12#20777C211yesmaximal, characteristic
13C2×#533—21no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8—21no—
16C2×#500C2214yes—
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#2099C2218yes—
49#2083C2218yes—
50#2057C2218yes—
51#1990C2214yes—
52#1990C2214yes—
53#1988C2218yes—
54#1947C2218yes—
55#1918C22116yes—
56#1918C22116yes—
57#1852C2218yes—
58#1839C2212yes—
59#1834C2214yes—
60#1831C2212yes—
61#1823C2218yes—
62#1817C2212yes—
63#1797C2212yes—
64C4×C4⋊C8C2212yes—
65C4×C22⋊C8C2212yes—
66#1744C2212yes—
67C23×Q8—21no—
68#533—48no—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8C2312yes—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×#125—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C4×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—48no—
90C2×#117—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C42—22no—
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42D411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#500—216no—
106C2×#107C23116yes—
107C2×C4⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C4×C8C2314yes—
110#391—22no—
111#391—22no—
112#391—22no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—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#389C2314yes—
131#389C2314yes—
132#389C2314yes—
133#389C2314yes—
134#384—21no—
135#384—21no—
136#384—21no—
137#384—21no—
138#384—21no—
139#384—21no—
140#384C2312yes—
141#384C2312yes—
142#384C2312yes—
143#384C2312yes—
144#384C2312yes—
145#384C2312yes—
146#382—22no—
147#382—22no—
148#380—22no—
149#380—22no—
150#380—22no—
151#380C2314yes—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
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⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C2×#95—22no—
165C2×#95—22no—
166C43D411yescharacteristic
167C43D411yescharacteristic
168C43D411yescharacteristic
169C43D411yescharacteristic
170C43C2311yescharacteristic
171C43C2311yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174#340C23116yes—
175#340C23116yes—
176#340C23116yes—
177#336C2314yes—
178#336C2314yes—
179#336C2314yes—
180#336C2314yes—
181C22×Q8—28no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—44no—
187C22×Q8—44no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C22×C8C2418yes—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—24no—
202C4⋊Q8—28no—
203C4⋊Q8—28no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8#4118yes—
209#125—48no—
210C22⋊Q8—48no—
211C22⋊Q8—48no—
212C4×Q8—48no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—48no—
217C4×Q8—48no—
218#117—48no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
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—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42—48no—
262C2×C42D818yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—232no—
292C4⋊C8—28no—
293C22⋊C8—28no—
294C4×C8—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303C24C22×D411yescharacteristic
304C2×Q8—28no—
305C2×Q8—28no—
306C2×Q8—44no—
307C2×Q8—432no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4C2×D814yes—
344C22×C4C2×D814yes—
345C22×C4C2×D814yes—
346C22×C4#13714yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C42−1+414yes—
361C22×C4C2511yescharacteristic
362C2×C8—28no—
363C2×C8—28no—
364C2×C8—28no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4C2×#4118yes—
392C22⋊C4—48no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
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—41no—
413C42—41no—
414C42—48no—
415C42—48no—
416C42—48no—
417C42C2×D814yes—
418C42C2×D814yes—
419C42C2×D814yes—
420C42C2×D814yes—
421C42#13714yes—
422C42C2×#4114yes—
423C422−1+414yes—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23#53612yes—
429C23C22×D812yes—
430C23C2×#13712yes—
431C23C23×D411yescharacteristic
432Q8—416no—
433Q8—416no—
434Q8—88no—
435Q8—88no—
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—28no—
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—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—416no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—88no—
479C2×C4#52512yes—
480C2×C4#52512yes—
481C2×C4#53612yes—
482C2×C4#53612yes—
483C2×C4#53612yes—
484C2×C4C22×D812yes—
485C2×C4C22×D812yes—
486C2×C4C22×D812yes—
487C2×C4C22×D812yes—
488C2×C4C22×D812yes—
489C2×C4C22×D812yes—
490C2×C4C2×#13712yes—
491C2×C4C2×#13712yes—
492C2×C4C2×#13712yes—
493C2×C4C2×#13712yes—
494C2×C4C2×#13712yes—
495C2×C4#57512yes—
496C2×C4C22×#4112yes—
497C2×C4C2×2−1+412yes—
498C2×C4C2×2−1+412yes—
499C8—48no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22#322614yes—
507C22C2×#52511yescharacteristic
508C22C2×#53611yescharacteristic
509C22C2×#53611yescharacteristic
510C22C23×D811yescharacteristic
511C22C22×#13711yescharacteristic
512C22C22×#13711yescharacteristic
513C22C2×#57511yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—48no—
523C4—41no—
524C4—41no—
525C4—84no—
526C4—84no—
527C4—84no—
528C4#310014yes—
529C4#311314yes—
530C4#322614yes—
531C4#352314yes—
532C2—24no—
533C2#3524811yescharacteristic
534C2C2×#310011yescharacteristic
535C2C2×#311311yescharacteristic
536C2C2×#322611yescharacteristic
537C2C2×#322611yescharacteristic
538C2#6086511yescharacteristic
539C2#6086511yescharacteristic
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
4. #30091C2semidirect
5. #29272C2semidirect
6. #29219C2semidirect
7. #28689C2semidirect
11. #22247C2semidirect

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
#300912
#292722
#292192
#286892
#279602
#276102
#258132
#222472
#207772