Finite Group Databasebeta

← #190329

Subgroups — #190329

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34195C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8775C212yesmaximal, complemented
5#8774C212yesmaximal, complemented
6#8773C212yesmaximal, complemented
7#8772C212yesmaximal, complemented
8#8070C212yesmaximal, complemented
9#7975C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12#2920C2212yes—
13#2356C2211yescharacteristic
14#2356C2211yescharacteristic
15#2353C411yescharacteristic
16#2353C411yescharacteristic
17#2353C2211yescharacteristic
18#2353C2211yescharacteristic
19#2349C411yescharacteristic
20#2349C411yescharacteristic
21#2347—21no—
22#2343—21no—
23#2343—21no—
24#2339—21no—
25#2320—21no—
26#2318—21no—
27#2318—21no—
28#2283C411yescharacteristic
29#2283C411yescharacteristic
30#2283C411yescharacteristic
31#2283C2211yescharacteristic
32C2×#383C411yescharacteristic
33C2×#383C2211yescharacteristic
34#2275—21no—
35#1822—21no—
36#1801—21no—
37#1800—21no—
38#1785—21no—
39C4×#101C2212yes—
40C4×#100C2212yes—
41#1467C2212yes—
42#1466C2212yes—
43#1465—21no—
44#1465—21no—
45#1458—21no—
46#1458—21no—
47#1300C2214yes—
48#1300C2214yes—
49#1288C2214yes—
50#1288C2214yes—
51C22×#41D411yescharacteristic
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#432—22no—
75#432C2314yes—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—22no—
81#423—22no—
82C2×#101—22no—
83C2×#100—22no—
84#407—22no—
85#407—22no—
86C4×M16—22no—
87C4×M16C2314yes—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
96#387—21no—
97#387—21no—
98#387—21no—
99#387—21no—
100#387—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#386—21no—
105#386—21no—
106#386—21no—
107#386—21no—
108#385—21no—
109#385—21no—
110#385—21no—
111#385—21no—
112#383—21no—
113#383—21no—
114#383—21no—
115#383—21no—
116#383C2×C411yescharacteristic
117#383C2×C411yescharacteristic
118#383C2×C411yescharacteristic
119#383C2×C411yescharacteristic
120#383C2×C411yescharacteristic
121#383C2×C411yescharacteristic
122#383C2311yescharacteristic
123#383C2311yescharacteristic
124#382—21no—
125#382—21no—
126#382—21no—
127#382C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#381—21no—
130#381—21no—
131#381—21no—
132#381C2×C411yescharacteristic
133#381C2×C411yescharacteristic
134#380—21no—
135#380—21no—
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4C2×C411yescharacteristic
155C4×C22⋊C4C2×C411yescharacteristic
156#376—21no—
157#376—21no—
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#376C2311yescharacteristic
161#376C2311yescharacteristic
162C2×#95—21no—
163C2×#95—21no—
164C2×#95—21no—
165C2×#95—21no—
166C2×#95C2×C411yescharacteristic
167C2×#95C2×C411yescharacteristic
168C2×#95C2×C411yescharacteristic
169C2×#95C2×C411yescharacteristic
170#343—22no—
171#343—22no—
172#324—22no—
173#324—22no—
174#323—22no—
175#323—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×D411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—24no—
198C2×M16—42no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—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—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—44no—
216C4×D4—44no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
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×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4112yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280C2×C42C2411yescharacteristic
281C4⋊C8—24no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285#101—44no—
286#100—44no—
287C22⋊C8—24no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291#97—24no—
292#97—24no—
293C4×C8—24no—
294C4×C8—42no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325#95C22×C411yescharacteristic
326#95C22×C411yescharacteristic
327#95C22×C411yescharacteristic
328#95C22×C411yescharacteristic
329#95C22×C411yescharacteristic
330#95C22×C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334#41—48no—
335#41—48no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
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—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
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—44no—
411M16—44no—
412M16—84no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42—42no—
457C42—81no—
458C42—81no—
459C42—81no—
460C42—81no—
461C42—81no—
462C42—81no—
463C42#11711yescharacteristic
464C42#11711yescharacteristic
465C42#11711yescharacteristic
466C42#11711yescharacteristic
467C42#12312yes—
468C42#12312yes—
469C42#12312yes—
470C42#12312yes—
471C42C2×#4112yes—
472C42C2×#4112yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23#38112yes—
487C23#42112yes—
488C23C22×C22⋊C411yescharacteristic
489Q8—44no—
490D4—28no—
491D4—44no—
492D4—44no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—162no—
557C2×C4—162no—
558C2×C4—162no—
559C2×C4—162no—
560C2×C4#38111yescharacteristic
561C2×C4#38111yescharacteristic
562C2×C4#38111yescharacteristic
563C2×C4#38111yescharacteristic
564C2×C4#38112yes—
565C2×C4#42112yes—
566C2×C4C2×#11711yescharacteristic
567C2×C4C2×#11711yescharacteristic
568C2×C4C2×#12312yes—
569C2×C4C2×#12312yes—
570C8—84no—
571C8—84no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—81no—
588C22—81no—
589C22C2×#38111yescharacteristic
590C22C2×#42111yescharacteristic
591C22C2×#42111yescharacteristic
592C4—22no—
593C4—21no—
594C4—21no—
595C4—22no—
596C4—22no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C2—21no—
616C2—21no—
617C2—42no—
618C2—42no—
619C2—81no—
620C2#2070511yescharacteristic
621C2#2070811yescharacteristic
622C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8775C2semidirect
5. #8774C2semidirect
6. #8773C2semidirect
7. #8772C2semidirect
8. #8070C2semidirect
9. #7975C2semidirect

Maximal subgroups

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

subgroupindex
#341952
#207282
#173612
#87752
#87742
#87732
#87722
#80702
#79752