Finite Group Databasebeta

← #206946

Subgroups — #206946

Order 512.

1002 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
1C2×#2550C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3#20535C211yesmaximal, characteristic
4#9883C212yesmaximal
5#9882C212yesmaximal
6#9879C212yesmaximal
7#9878C212yesmaximal
8#7914C211yesmaximal, characteristic
9#7914C211yesmaximal, characteristic
10#7912C211yesmaximal, characteristic
11#7908C211yesmaximal, characteristic
12C22×C4⋊Q8C411yescharacteristic
13C22×#125C411yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2550—21no—
19#2550—21no—
20#2550—22no—
21#2550—21no—
22#2550—21no—
23#2550—22no—
24C2×#398C412yes—
25C2×#395C412yes—
26C2×#391C2212yes—
27C2×#389C411yescharacteristic
28C2×#389C411yescharacteristic
29C2×#389C2212yes—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—22no—
34#1939—21no—
35#1939—21no—
36#1797C2212yes—
37#1797C2212yes—
38#1771—22no—
39#1770—21no—
40C8×C4⋊C4—21no—
41#1749C2212yes—
42C4×C22⋊C8C2212yes—
43#1280C2211yescharacteristic
44#1280C2212yes—
45#1280C2212yes—
46#1280C2212yes—
47#1280C2211yescharacteristic
48#1280C2212yes—
49#1280C2212yes—
50#1280C2211yescharacteristic
51#1280C2212yes—
52#1280C2211yescharacteristic
53C23×Q8C2×C412yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—44no—
58C2×#125—22no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—44no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#432—22no—
80#432Q814yescomplemented
81#431—22no—
82#431—22no—
83#431—22no—
84#431Q814yescomplemented
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2312yes—
97C2×#97—21no—
98C2×#97—21no—
99C2×#97—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—22no—
103#398—22no—
104#398—22no—
105#398—22no—
106#398—22no—
107#395—22no—
108#395—22no—
109#395—22no—
110#395—22no—
111#391—22no—
112#391—22no—
113#391—22no—
114#391—22no—
115#389—21no—
116#389—22no—
117#389—21no—
118#389—21no—
119#389—22no—
120#389—21no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#382—21no—
126#382—21no—
127#382—21no—
128#382—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D412yes—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—44no—
155C22×Q8—82no—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C4⋊Q8—48no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#125—48no—
177#125—44no—
178#125—44no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—82no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×Q812yes—
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C4⋊C8—24no—
237C4⋊C8—24no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—82no—
241C4⋊C8—82no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—22no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8C2×Q812yes—
252C22⋊C8C2×Q812yes—
253C22⋊C8C2×Q814yes—
254#97—24no—
255#97—42no—
256#97—82no—
257C4×C8—24no—
258C4×C8—42no—
259C4×C8—82no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276C24C23×C411yescharacteristic
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—88no—
284C2×Q8—88no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C4×Q812yes—
338C22×C4C4×Q812yes—
339C22×C4C4×Q812yes—
340C22×C4C4×Q812yes—
341C22×C4#13112yes—
342C22×C4#13112yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4#10014yes—
390C4⋊C4#10114yes—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42C22⋊Q812yes—
400C42C22⋊Q812yes—
401C42C22⋊Q812yes—
402C42C22⋊Q812yes—
403C42#12412yes—
404C42#12412yes—
405C42C4⋊Q812yes—
406C42C4⋊Q812yes—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23C22×C22⋊C411yescharacteristic
412C23C2×C4×Q811yescharacteristic
413C23C2×C4×Q811yescharacteristic
414C23C2×#13111yescharacteristic
415C23C2×#13111yescharacteristic
416C23#56811yescharacteristic
417C23#56811yescharacteristic
418Q8—88no—
419Q8—88no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#39112yes—
485C2×C4#39112yes—
486C2×C4#39112yes—
487C2×C4#39112yes—
488C2×C4C2×#10014yes—
489C2×C4C2×#10012yes—
490C2×C4C2×#10114yes—
491C2×C4C2×#10112yes—
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×#12411yescharacteristic
495C2×C4C2×C4⋊Q811yescharacteristic
496C8—42no—
497C8—42no—
498C8—44no—
499C8—81no—
500C8—81no—
501C8—82no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22C2×#39111yescharacteristic
515C22#288011yescharacteristic
516C22#288011yescharacteristic
517C22#288311yescharacteristic
518C22#288411yescharacteristic
519C22C22×#10011yescharacteristic
520C22C22×#10111yescharacteristic
521C22#296011yescharacteristic
522C22#296111yescharacteristic
523C22#296211yescharacteristic
524C22#296211yescharacteristic
525C4—24no—
526C4—24no—
527C4—22no—
528C4—22no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—81no—
539C4—82no—
540C4—81no—
541C4—82no—
542C4#199714yes—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2#1742111yescharacteristic
548C2#1742511yescharacteristic
549C2#1742611yescharacteristic
550C2#1743011yescharacteristic
551C2#2130411yescharacteristic
552C2#2133411yescharacteristic
553C2C2×#199711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
80. #432Q8semidirect
84. #431Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
#219532
#205352
#98832
#98822
#98792
#98782
#79142
#79122
#79082