Finite Group Databasebeta

← #190077

Subgroups — #190077

Order 512.

969 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#17366C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8745C212yesmaximal, complemented
5#8744C212yesmaximal, complemented
6#8743C212yesmaximal, complemented
7#8742C212yesmaximal, complemented
8#7949C212yesmaximal, complemented
9#7948C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12C2×#432C2212yes—
13#2298C411yescharacteristic
14#2298C411yescharacteristic
15#2298C2211yescharacteristic
16#2298C2211yescharacteristic
17#2297C411yescharacteristic
18#2297C411yescharacteristic
19#2296C2211yescharacteristic
20#2296C2211yescharacteristic
21#2295—21no—
22#2295—21no—
23#2293—21no—
24#2293—21no—
25#2290—21no—
26#2290—21no—
27#2290—21no—
28#2290—21no—
29C4×#117C411yescharacteristic
30C4×#117C411yescharacteristic
31C4×#117C411yescharacteristic
32C4×#117C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C43C411yescharacteristic
35#1467—21no—
36#1466—21no—
37#1465—21no—
38#1465—21no—
39#1464C2212yes—
40#1464C2212yes—
41#1464C2212yes—
42#1464C2212yes—
43#1461—21no—
44#1460—21no—
45#1459—21no—
46#1459—21no—
47#1285C2214yes—
48#1285C2214yes—
49#1285C2214yes—
50#1285C2214yes—
51C22×#41C2×C411yescharacteristic
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#432—22no—
83#432—22no—
84#432C2314yes—
85#432C2314yes—
86#431—22no—
87#431—22no—
88#423—22no—
89C2×C4⋊C8—22no—
90#407—22no—
91C2×C22⋊C8—22no—
92C4×M16—22no—
93C2×#97—22no—
94#391—21no—
95#391—21no—
96#390—21no—
97#390—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#386—21no—
119#386—21no—
120#386—21no—
121#386—21no—
122#385—21no—
123#385—21no—
124#385—21no—
125#385—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4C2×C411yescharacteristic
147C4×C22⋊C4C2×C411yescharacteristic
148C4×C22⋊C4C2×C411yescharacteristic
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4C2×C411yescharacteristic
153C4×C22⋊C4C2×C411yescharacteristic
154#376—21no—
155#376—21no—
156#376—21no—
157#376—21no—
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160C43—21no—
161C43—21no—
162C43C2×C411yescharacteristic
163C43C2×C411yescharacteristic
164C43C2×C411yescharacteristic
165C43C2×C411yescharacteristic
166C43C2×C411yescharacteristic
167C43C2×C411yescharacteristic
168#324—22no—
169#324—22no—
170#324—22no—
171#324—22no—
172#323—22no—
173#323—22no—
174#323—22no—
175#323—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C22×C8—42no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—44no—
220#117—44no—
221#117—44no—
222#117—44no—
223#117—42no—
224#117—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C2×D411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4112yes—
301C2×C42#4112yes—
302C2×C42#4112yes—
303C2×C42#4112yes—
304C2×C42C2411yescharacteristic
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313#97—24no—
314#97—24no—
315#97—42no—
316C4×C8—42no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340#41—48no—
341#41—48no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—41no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4#11712yes—
402C22×C4#11712yes—
403C22×C4#12312yes—
404C22×C4#12312yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410M16—84no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—84no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C42—24no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42#11711yescharacteristic
481C42#11711yescharacteristic
482C42#11711yescharacteristic
483C42#11711yescharacteristic
484C42#12312yes—
485C42#12312yes—
486C42#12312yes—
487C42#12312yes—
488C42C2×#4112yes—
489C42C2×#4112yes—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#38112yes—
504C23#41312yes—
505C23C22×C22⋊C411yescharacteristic
506Q8—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4#38111yescharacteristic
576C2×C4#38111yescharacteristic
577C2×C4#38112yes—
578C2×C4#38111yescharacteristic
579C2×C4#38111yescharacteristic
580C2×C4#41312yes—
581C2×C4C2×#11711yescharacteristic
582C2×C4C2×#11711yescharacteristic
583C2×C4C2×#12312yes—
584C2×C4C2×#12312yes—
585C8—84no—
586C8—84no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22C2×#38111yescharacteristic
605C22C2×#41311yescharacteristic
606C22C2×#41311yescharacteristic
607C4—21no—
608C4—21no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4#182712yes—
629C4#182712yes—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—42no—
634C2—81no—
635C2#2069411yescharacteristic
636C2C2×#182711yescharacteristic
637C2C2×#182711yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8745C2semidirect
5. #8744C2semidirect
6. #8743C2semidirect
7. #8742C2semidirect
8. #7949C2semidirect
9. #7948C2semidirect

Maximal subgroups

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

subgroupindex
#340262
#173662
#173632
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