Finite Group Databasebeta

← #358407

Subgroups — #358407

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1762C211yesmaximal, characteristic
2C2×C4×#102C211yesmaximal, characteristic
3C2×#1746C211yesmaximal, characteristic, complemented
4#13785C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#11189C212yesmaximal
9#11066C212yesmaximal
10#10703C211yesmaximal, characteristic, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#390C2212yes—
19C2×#389C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1762—24no—
25C4×#102C418yescomplemented
26#1746—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1437—22no—
37#1436—22no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102C2×C414yes—
75C2×#102C2×C414yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97D414yes—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8C2×C414yes—
82#390—24no—
83#390—24no—
84#389—24no—
85#389—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92#376—22no—
93#376—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340C2×C414yes—
111#340C2×C414yes—
112#336—24no—
113#336—24no—
114C25—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4C22×C412yes—
124C22×D4#4114yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156C22×C8C2×D414yes—
157C4×D4—28no—
158C4×D4—28no—
159C4×D4—44no—
160C4×D4—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4#4114yes—
188C2×C4⋊C4#4114yes—
189C2×C4⋊C4#4114yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—82no—
209C2×C42—81no—
210C2×C42—81no—
211C2×C42—81no—
212C2×C42—81no—
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219#102—216no—
220#102—28no—
221#102—28no—
222#97—28no—
223C4×C8—28no—
224#95—28no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—24no—
229#95—42no—
230#95—42no—
231#95—82no—
232#95—82no—
233#95—82no—
234#95—82no—
235C24—22no—
236C24—24no—
237C24—41no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24C22×D411yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4#11714yes—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—48no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—88no—
318C22×C4—88no—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12412yes—
324C22×C4#12412yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C4⋊C4#11714yes—
371C22⋊C4—48no—
372C22⋊C4—88no—
373C22⋊C4—88no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—84no—
379C42—84no—
380C42—84no—
381C42#11714yes—
382C42C4×D414yes—
383C42#12314yes—
384C42#12614yes—
385C23—24no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—81no—
405C23—82no—
406C23—81no—
407C23—81no—
408C23C2×C4×D411yescharacteristic
409C23C2×#12111yescharacteristic
410C23#52512yes—
411C23#52512yes—
412C23C2×#12411yescharacteristic
413C23#57612yes—
414C23#57611yescharacteristic
415C23#57711yescharacteristic
416C23#57711yescharacteristic
417C23#57712yes—
418C23#57811yescharacteristic
419D4—24no—
420D4—24no—
421D4—44no—
422D4—44no—
423D4—42no—
424D4—42no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
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—42no—
466C2×C4—816no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#38814yes—
480C2×C4#38814yes—
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×#11711yescharacteristic
483C2×C4C2×#11711yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C4×#4111yescharacteristic
486C2×C4C4×#4111yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52512yes—
489C2×C4C2×#12611yescharacteristic
490C2×C4#52912yes—
491C2×C4#52912yes—
492C2×C4#52912yes—
493C2×C4#52912yes—
494C2×C4#52912yes—
495C2×C4#52912yes—
496C2×C4#52912yes—
497C2×C4#52912yes—
498C2×C4#53212yes—
499C8—44no—
500C8—42no—
501C8—42no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—84no—
506C22—84no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—84no—
513C22C2×#38811yescharacteristic
514C22#229011yescharacteristic
515C22#229011yescharacteristic
516C22#239212yes—
517C22#239812yes—
518C22#294811yescharacteristic
519C22#294911yescharacteristic
520C22#295111yescharacteristic
521C22#295211yescharacteristic
522C22#306211yescharacteristic
523C22#306212yes—
524C22#306311yescharacteristic
525C22#311812yes—
526C22#311912yes—
527C22#311912yes—
528C22#312112yes—
529C22#312212yes—
530C22#312312yes—
531C22#312412yes—
532C22#312512yes—
533C22#313612yes—
534C22#313611yescharacteristic
535C22#313711yescharacteristic
536C4—24no—
537C4—24no—
538C4—44no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4—82no—
545C4#228311yescharacteristic
546C4#229011yescharacteristic
547C4#229311yescharacteristic
548C4#229811yescharacteristic
549C4#239512yes—
550C4#240112yes—
551C2—84no—
552C2#1384211yescharacteristic
553C2#2108711yescharacteristic
554C2#2108911yescharacteristic
555C2#2112911yescharacteristic
556C2#2113311yescharacteristic
557C2#2113411yescharacteristic
558C2#2113611yescharacteristic
559C2#2371212yes—
560C2#2371412yes—
561C2#2382712yes—
562C2#2383012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1746C2semidirect
5. #13783C2semidirect
6. #11315C2semidirect
7. #11293C2semidirect
10. #10703C2semidirect
11. C4×#340C2semidirect
25. C4×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17622
C2×C4×#1022
C2×#17462
#137852
#137832
#113152
#112932
#111892
#110662
#107032
C4×#3402