Finite Group Databasebeta

← #204246

Subgroups — #204246

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3#20525C211yesmaximal, characteristic, complemented
4C2×#1472C211yesmaximal, characteristic, complemented
5C2×#1472C211yesmaximal, characteristic, complemented
6C2×#1462C211yesmaximal, characteristic
7C22×#323C211yesmaximal, characteristic
8#9652C214yesmaximal, complemented
9#9651C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C23×C22⋊C4C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2561—22no—
17#2561—22no—
18#2561—21no—
19#2561—21no—
20#2561—22no—
21C2×#397C412yes—
22C2×#393C412yes—
23C2×#388C2214yes—
24C2×#381C412yes—
25C2×#379C412yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1851—22no—
28#1850—22no—
29#1797C2214yescomplemented
30#1753—22no—
31#1752—21no—
32C8×C22⋊C4—21no—
33#1749C2212yescomplemented
34C4×C22⋊C8C2212yescomplemented
35#1472—24no—
36#1472—24no—
37#1462—24no—
38C2×#324C2212yescomplemented
39C2×#324C2214yescomplemented
40C2×#324C2212yescomplemented
41C2×#323—24no—
42C2×#323C2212yes—
43C2×#323C2214yes—
44C2×#323C2212yes—
45C26C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C22×M16—21no—
51C23×C8—21no—
52C23×C8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C414yes—
74C22×C22⋊C4C2×C414yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2312yes—
77C22×C42C2312yes—
78C2×C4⋊C8—24no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407D414yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8D414yes—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2312yes—
96C2×#97—22no—
97C2×C4×C8—22no—
98#397—24no—
99#397—22no—
100#397—22no—
101#393—24no—
102#393—22no—
103#393—22no—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#381—22no—
109#381—22no—
110#381—24no—
111#379—22no—
112#379—22no—
113#379—24no—
114C4×C22⋊C4D412yes—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D411yescharacteristic
117C4×C22⋊C4D411yescharacteristic
118C4×C22⋊C4D412yes—
119C2×#95C2×C414yes—
120C2×#95C2×C414yes—
121C2×#95C2×C412yes—
122#336—24no—
123#336—22no—
124#336—22no—
125#324—28no—
126#324—24no—
127#324—24no—
128#323—24no—
129#323—24no—
130#323—28no—
131C25—22no—
132C25—24no—
133C25—21no—
134C25—21no—
135C25—22no—
136C25—42no—
137C25—44no—
138C25—82no—
139C25C22×C411yescharacteristic
140C25C22×C412yes—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—82no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—82no—
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C414yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—82no—
169C2×M16—82no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—82no—
178C22×C8—82no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—84no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C414yes—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D414yes—
234C4⋊C8—84no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8C2×D412yes—
247C22⋊C8C2×D412yes—
248C22⋊C8C2×D414yes—
249#97—82no—
250C4×C8—82no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95C22⋊C414yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—22no—
259C24—22no—
260C24—24no—
261C24—28no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—42no—
269C24—44no—
270C24—44no—
271C24—48no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—42no—
277C24—42no—
278C24—44no—
279C24—82no—
280C24—82no—
281C24—84no—
282C24—84no—
283C24—816no—
284C24—816no—
285C24—84no—
286C24—84no—
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289C24C4×D412yes—
290C24C4×D412yes—
291C24#13114yes—
292C24C23×C411yescharacteristic
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—84no—
298C2×Q8—84no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C22⋊C414yes—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12412yes—
343C22×C4#12412yes—
344C22×C4#12712yes—
345C22×C4#12712yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352M16—84no—
353M16—84no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—84no—
366C2×C8—84no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—42no—
389C42—42no—
390C42—44no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—48no—
409C23—48no—
410C23—48no—
411C23—48no—
412C23—48no—
413C23—48no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—816no—
421C23—84no—
422C23—84no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—84no—
428C23—84no—
429C23—84no—
430C23—816no—
431C23—816no—
432C23—832no—
433C23—816no—
434C23—84no—
435C23#39012yes—
436C23#39012yes—
437C23#41314yes—
438C23C22×C22⋊C411yescharacteristic
439C23C2×C4×D411yescharacteristic
440C23C2×C4×D411yescharacteristic
441C23C2×#13112yes—
442C23#56812yes—
443Q8—84no—
444Q8—84no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#39012yes—
487C2×C4#39012yes—
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12411yescharacteristic
491C2×C4C2×#12711yescharacteristic
492C8—42no—
493C8—42no—
494C8—44no—
495C8—82no—
496C8—82no—
497C22—24no—
498C22—24no—
499C22—28no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—44no—
510C22—44no—
511C22—48no—
512C22—44no—
513C22—48no—
514C22—44no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—816no—
520C22—816no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22#196714yes—
525C22C2×#39011yescharacteristic
526C22#288012yes—
527C22#288312yes—
528C22C2×#41312yes—
529C22#292911yescharacteristic
530C22#293011yescharacteristic
531C22#293312yes—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C2—21no—
544C2—21no—
545C2—22no—
546C2—24no—
547C2—42no—
548C2—84no—
549C2#1742214yes—
550C2#2119111yescharacteristic
551C2#2121311yescharacteristic
552C2C2×#196711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
3. #20525C2semidirect
4. C2×#1472C2semidirect
5. C2×#1472C2semidirect
8. #9652C2semidirect
9. #9651C2semidirect
29. #1797C22semidirect
33. #1749C22semidirect
34. C4×C22⋊C8C22semidirect
38. C2×#324C22semidirect
39. C2×#324C22semidirect
40. C2×#324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
#219322
#205252
C2×#14722
C2×#14622
C22×#3232
#96522
#96512