Finite Group Databasebeta

← #190326

Subgroups — #190326

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8775C212yesmaximal, complemented
5#8774C212yesmaximal, complemented
6#8773C212yesmaximal, complemented
7#8772C212yesmaximal, complemented
8#8072C212yesmaximal, complemented
9#7976C212yesmaximal, 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#2346—21no—
22#2343—21no—
23#2343—21no—
24#2339—21no—
25#2295—21no—
26#2290—21no—
27#2290—21no—
28#2283C411yescharacteristic
29#2283C411yescharacteristic
30#2283C2211yescharacteristic
31C42×D4—21no—
32C4×#117C411yescharacteristic
33C2×#376C411yescharacteristic
34C2×#376C2211yescharacteristic
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×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#432—22no—
78#432C2314yes—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#423—22no—
84#423—22no—
85C2×#101—22no—
86C2×#100—22no—
87#407—22no—
88#407—22no—
89C4×M16—22no—
90C4×M16C2314yes—
91#391—21no—
92#391—21no—
93#389—21no—
94#389—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#387—21no—
100#387—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#386—21no—
108#386—21no—
109#386—21no—
110#386—21no—
111#385—21no—
112#385—21no—
113#385—21no—
114#385—21no—
115#383—21no—
116#383C2×C411yescharacteristic
117#383C2×C411yescharacteristic
118#383C2311yescharacteristic
119#383C2311yescharacteristic
120#382—21no—
121#382—21no—
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#381—21no—
125#381—21no—
126#381C2×C411yescharacteristic
127#381C2×C411yescharacteristic
128#380—21no—
129#380C2×C411yescharacteristic
130#380C2×C411yescharacteristic
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic
151#376—21no—
152#376—21no—
153#376—21no—
154#376—21no—
155#376C2×C411yescharacteristic
156#376C2×C411yescharacteristic
157#376C2×C411yescharacteristic
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#376C2×C411yescharacteristic
161#376C2311yescharacteristic
162#376C2311yescharacteristic
163C2×#95—21no—
164C2×#95—21no—
165C2×#95C2×C411yescharacteristic
166C2×#95C2×C411yescharacteristic
167C2×#95C2×C411yescharacteristic
168C2×#95C2×C411yescharacteristic
169C43—21no—
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—
223#117—44no—
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⋊C4—21no—
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—21no—
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
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×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4112yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286C2×C42#4112yes—
287C2×C42C2411yescharacteristic
288C4⋊C8—24no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292#101—44no—
293#100—44no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#97—24no—
299#97—24no—
300C4×C8—24no—
301C4×C8—42no—
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#95—21no—
324#95—21no—
325#95—21no—
326#95C22×C411yescharacteristic
327#95C22×C411yescharacteristic
328#95C22×C411yescharacteristic
329#95C22×C411yescharacteristic
330#95C22×C411yescharacteristic
331#95C22×C411yescharacteristic
332#95C22×C411yescharacteristic
333#95C22×C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337#41—48no—
338#41—48no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
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—22no—
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—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4#11712yes—
405C22×C4#11712yes—
406C22×C4#12312yes—
407C22×C4#12312yes—
408C22×C4#12312yes—
409C22×C4#12312yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413M16—44no—
414M16—44no—
415M16—84no—
416C2×C8—44no—
417C2×C8—44no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C42—24no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—81no—
469C42—81no—
470C42#11711yescharacteristic
471C42#11711yescharacteristic
472C42#11711yescharacteristic
473C42#11711yescharacteristic
474C42#12312yes—
475C42#12312yes—
476C42#12312yes—
477C42#12312yes—
478C42C2×#4112yes—
479C42C2×#4112yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23#38112yes—
494C23#42112yes—
495C23C22×C22⋊C411yescharacteristic
496Q8—44no—
497D4—28no—
498D4—44no—
499D4—44no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
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—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4#38111yescharacteristic
568C2×C4#38111yescharacteristic
569C2×C4#38111yescharacteristic
570C2×C4#38112yes—
571C2×C4#38111yescharacteristic
572C2×C4#42112yes—
573C2×C4C2×#11711yescharacteristic
574C2×C4C2×#11711yescharacteristic
575C2×C4C2×#12312yes—
576C2×C4C2×#12312yes—
577C8—84no—
578C8—84no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—81no—
595C22—81no—
596C22C2×#38111yescharacteristic
597C22C2×#42111yescharacteristic
598C22C2×#42111yescharacteristic
599C4—22no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—22no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4#182712yes—
622C2—21no—
623C2—21no—
624C2—42no—
625C2—42no—
626C2—81no—
627C2#2070511yescharacteristic
628C2C2×#182711yescharacteristic
629C2#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. #8072C2semidirect
9. #7976C2semidirect

Maximal subgroups

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

subgroupindex
#341232
#207282
#173612
#87752
#87742
#87732
#87722
#80722
#79762