Finite Group Databasebeta

← #190364

Subgroups — #190364

Order 512.

1173 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#8777C214yesmaximal, complemented
5#8776C214yesmaximal, complemented
6#8051C212yesmaximal, complemented
7#7942C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C2×#407C2212yes—
11#2353C412yescomplemented
12#2353C412yescomplemented
13#2353C2212yes—
14#2353C2212yes—
15#2352—21no—
16#2352—21no—
17#2352—21no—
18#2352—21no—
19#2313—21no—
20#2312—21no—
21#2308—21no—
22#2283C411yescharacteristic, complemented
23#2283C411yescharacteristic, complemented
24#2280C2211yescharacteristic
25#2280C2211yescharacteristic
26C2×#380C411yescharacteristic, complemented
27C2×#380C411yescharacteristic, complemented
28#2273—21no—
29C2×C4×C22⋊C4C2212yes—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342—21no—
35#1465—21no—
36#1465—21no—
37#1465C2212yes—
38#1465C2212yes—
39C2×#324—21no—
40C2×#323—21no—
41#1297C2218yescomplemented
42#1284C2218yescomplemented
43C23×D4D411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—22no—
70C22×C42C2×C411yescharacteristic
71C2×#99—28no—
72C2×#99—22no—
73C2×#99—22no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407C2314yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#383C2×C412yescomplemented
93#383C2×C412yescomplemented
94#382—21no—
95#382—21no—
96#381—21no—
97#381—21no—
98#381C2×C412yes—
99#381C2312yes—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C411yescharacteristic
107#380C2×C411yescharacteristic
108#380C2×C411yescharacteristic
109#380C2×C411yescharacteristic
110#380C2×C412yescomplemented
111#380C2×C412yescomplemented
112#379—21no—
113#379—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4C2314yes—
129C4×C22⋊C4C2312yes—
130#376C2×C412yescomplemented
131#376C2×C412yescomplemented
132C2×#95—22no—
133C2×#95C2×C412yescomplemented
134C2×#95C2×C412yescomplemented
135#342—22no—
136#342—22no—
137#324—22no—
138#324—22no—
139#323—22no—
140#323—22no—
141C25C22⋊C411yescharacteristic
142C25C2×D411yescharacteristic
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—42no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22⋊C411yescharacteristic
149C22×D4C2×D411yescharacteristic
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4112yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C2×M16—24no—
178C2×M16—42no—
179C2×M16—84no—
180C22×C8—42no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197#117—82no—
198#117—82no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C2×D411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42—84no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256#99—28no—
257#99—28no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—24no—
273#95—42no—
274#95C22×C412yes—
275#95C22×C412yes—
276#95C22×C412yes—
277#95C22×C412yes—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—81no—
284C24—81no—
285C24—81no—
286C24—81no—
287C24#10414yes—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24#12312yes—
293C24#12312yes—
294C24C22×D411yescharacteristic
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4#11711yescharacteristic
368C22×C4#11711yescharacteristic
369C22×C4#11711yescharacteristic
370C22×C4#11711yescharacteristic
371C22×C4#11712yes—
372C22×C4#11712yes—
373C22×C4#12312yes—
374C22×C4#12312yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4#12312yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383M16—84no—
384M16—84no—
385C2×C8—44no—
386C2×C8—84no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—28no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—84no—
427C42—82no—
428C42—82no—
429C42—82no—
430C42—82no—
431C42—84no—
432C23—24no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23#38112yes—
467C23#38111yescharacteristic
468C23#38111yescharacteristic
469C23C2×#10412yes—
470C23C2×#10412yes—
471C23C2×#10412yes—
472C23#42112yes—
473C23C22×C22⋊C411yescharacteristic
474C23C2×#12312yes—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—44no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—816no—
541C2×C4—164no—
542C2×C4—164no—
543C2×C4#38111yescharacteristic
544C2×C4#38112yes—
545C2×C4#38111yescharacteristic
546C2×C4C2×#11711yescharacteristic
547C2×C4C2×#11711yescharacteristic
548C2×C4C2×#12312yes—
549C8—84no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—48no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22#182612yes—
583C22#182612yes—
584C22C2×#38111yescharacteristic
585C22C22×#10411yescharacteristic
586C22C2×#42111yescharacteristic
587C4—22no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—88no—
603C4#182812yes—
604C2—21no—
605C2—21no—
606C2—22no—
607C2—22no—
608C2—42no—
609C2—42no—
610C2—42no—
611C2—42no—
612C2C2×#182611yescharacteristic
613C2#2070811yescharacteristic
614C2C2×#182811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8777C2semidirect
5. #8776C2semidirect
6. #8051C2semidirect
7. #7942C2semidirect
11. #2353C4semidirect
12. #2353C4semidirect
22. #2283C4semidirect
23. #2283C4semidirect
26. C2×#380C4semidirect
27. C2×#380C4semidirect
41. #1297C22semidirect
42. #1284C22semidirect
92. #383C2×C4semidirect
93. #383C2×C4semidirect
110. #380C2×C4semidirect
111. #380C2×C4semidirect
130. #376C2×C4semidirect
131. #376C2×C4semidirect
133. C2×#95C2×C4semidirect
134. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341852
C2×C4×#992
C2×#14652
#87772
#87762
#80512
#79422