Finite Group Databasebeta

← #224381

Subgroups — #224381

Order 512.

1069 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#13935C211yesmaximal, characteristic, complemented
4#9913C212yesmaximal, complemented
5#9908C211yesmaximal, characteristic, complemented
6#9887C212yesmaximal, complemented
7#9884C211yesmaximal, characteristic, complemented
8#8571C211yesmaximal, characteristic
9#8571C212yesmaximal
10#8216C212yesmaximal
11C4×#327C211yesmaximal, characteristic
12C2×#554C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15#2857C2211yescharacteristic
16#2851C2212yes—
17#2584C412yescomplemented
18#2580C412yescomplemented
19#2567C411yescharacteristic, complemented
20#2538C411yescharacteristic, complemented
21#2465—22no—
22#2449—22no—
23#2423—21no—
24#2409—21no—
25#2360—21no—
26#2355C411yescharacteristic, complemented
27D4×C4⋊C4—21no—
28C4×#121C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31C2×#389—21no—
32#1985—22no—
33#1984—21no—
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1749C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
40C2×#327C2211yescharacteristic
41#1441—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#554—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×#121C2311yescharacteristic
62C2×C4×Q8—44no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D412yes—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#446—44no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8C2311yescharacteristic
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2311yescharacteristic
87C2×#97—22no—
88C2×C4×C8—21no—
89#399—22no—
90#398—22no—
91#396—22no—
92#395—22no—
93#395—22no—
94#393—22no—
95#391—21no—
96#391—21no—
97#391—21no—
98#391—44no—
99#390C2×C411yescharacteristic
100#390C2×C412yes—
101#389—22no—
102#389—22no—
103#389—21no—
104#389—44no—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C411yescharacteristic
109#389C2×C412yes—
110#388—22no—
111#388—21no—
112#387—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—21no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384C2×C411yescharacteristic
123#382C2×C412yes—
124#380—21no—
125#380—21no—
126#380C2311yescharacteristic
127#380C2312yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141#336—22no—
142#336—21no—
143#336—22no—
144#336—21no—
145#327—22no—
146#327—24no—
147#327Q814yescomplemented
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22⋊C412yes—
156C22×D4—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C4⋊Q8—42no—
172#125—44no—
173C22⋊Q8—48no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C22⋊Q8—81no—
177C22⋊Q8—81no—
178#121—21no—
179#121—21no—
180#121C2×Q812yes—
181#121#4112yes—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—44no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—82no—
193C4×D4—82no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C412yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C4⋊C8—42no—
250C4⋊C8—44no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—22no—
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8C2×Q812yes—
257#97—44no—
258C4×C8—42no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—21no—
274#95—22no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—81no—
280#95C2×Q812yes—
281C24—21no—
282C24C22×D411yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—81no—
296C2×Q8—81no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4C4×Q812yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C42−1+411yescharacteristic
370C22×C42−1+411yescharacteristic
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—42no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—81no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C4⋊C4#9914yes—
404C4⋊C4C4×Q812yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—81no—
416C22⋊C4—82no—
417C42—22no—
418C42—24no—
419C42—24no—
420C42—22no—
421C42—41no—
422C42—42no—
423C42—41no—
424C42—42no—
425C42—42no—
426C42—82no—
427C42—84no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C22×C22⋊C411yescharacteristic
440C23#54811yescharacteristic
441C23D4×Q811yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13711yescharacteristic
444C23#57611yescharacteristic
445C23#57711yescharacteristic
446Q8—48no—
447Q8—44no—
448Q8—44no—
449Q8—162no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—164no—
529C2×C4C2×#9914yes—
530C2×C4C2×#9912yes—
531C2×C4#41712yes—
532C2×C4#41712yes—
533C2×C4#42112yes—
534C2×C4C2×C4×Q811yescharacteristic
535C2×C4#51911yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C8—42no—
539C8—81no—
540C8—82no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22Q8×C22⋊C411yescharacteristic
561C22C22×#9911yescharacteristic
562C22#288411yescharacteristic
563C22C2×#41711yescharacteristic
564C22#289711yescharacteristic
565C22C2×#42111yescharacteristic
566C22#290111yescharacteristic
567C22#336611yescharacteristic
568C22#337211yescharacteristic
569C22#338411yescharacteristic
570C22#338511yescharacteristic
571C4—22no—
572C4—24no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—81no—
586C4—82no—
587C4—82no—
588C4—81no—
589C4—162no—
590C4#151314yes—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—82no—
596C2C2×#151311yescharacteristic
597C2#1760211yescharacteristic
598C2#1760311yescharacteristic
599C2#1762811yescharacteristic
600C2Q8×#9911yescharacteristic
601C2#2252211yescharacteristic
602C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. #13935C2semidirect
4. #9913C2semidirect
5. #9908C2semidirect
6. #9887C2semidirect
7. #9884C2semidirect
12. C2×#554C4semidirect
17. #2584C4semidirect
18. #2580C4semidirect
19. #2567C4semidirect
20. #2538C4semidirect
26. #2355C4semidirect
147. #327Q8semidirect

Maximal subgroups

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

subgroupindex
#365442
#225102
#139352
#99132
#99082
#98872
#98842
#85712
#82162
C4×#3272