Finite Group Databasebeta

← #364115

Subgroups — #364115

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#1897C211yesmaximal, characteristic
3C2×#1896C211yesmaximal, characteristic
4#20627C211yesmaximal, characteristic, complemented
5#13756C211yesmaximal, characteristic, complemented
6#11209C212yesmaximal, complemented
7#11053C212yesmaximal, complemented
8#10808C212yesmaximal, complemented
9#10790C212yesmaximal, complemented
10#10720C211yesmaximal, characteristic, complemented
11#10719C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#423C2212yes—
18C22×#104C2212yes—
19C22×#104C2212yes—
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2324C414yescomplemented
23#2323C414yescomplemented
24#2322—22no—
25#2321—22no—
26#2297C2211yescharacteristic
27#2296C2211yescharacteristic
28#2293—22no—
29#2290—22no—
30#2280C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1897—24no—
33#1896—24no—
34#1798—22no—
35#1798—22no—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42#1728C2212yes—
43C4⋊#95—22no—
44C22⋊#95—22no—
45C4×#95C2212yes—
46C4×#95C2212yes—
47C22×#41C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52#525—22no—
53#525—22no—
54#525D414yes—
55#525Q814yes—
56C2×#123—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×#121C2×C412yes—
59C2×#120—21no—
60C4×#41—42no—
61C4×#41—42no—
62C2×C4×Q8C2×C412yescomplemented
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yescomplemented
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—21no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117—42no—
73C2×#117C2×C414yes—
74C2×#117D412yes—
75C2×#117D412yes—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—24no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—42no—
89C22×C42C2312yes—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#423—42no—
94#423—42no—
95C2×C4⋊C8—24no—
96C2×#104—24no—
97C2×#104—24no—
98C2×#104—24no—
99C2×#104—24no—
100C2×#104—42no—
101C2×#104—42no—
102#407—22no—
103#407—22no—
104#407—22no—
105#407—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108#391C2×C412yescomplemented
109#390—24no—
110#390C2×C412yescomplemented
111#389—24no—
112#389C2×C414yescomplemented
113#388—24no—
114#388—24no—
115#388—24no—
116#387—24no—
117#386—22no—
118#386—24no—
119#385—22no—
120#385—24no—
121#381—21no—
122#380C2×C414yes—
123#379—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C414yescomplemented
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#376—21no—
130C2×#95—24no—
131C2×#95—24no—
132C43—21no—
133#339—24no—
134#339—24no—
135#339—24no—
136#339—24no—
137#339—42no—
138#339—42no—
139#336—22no—
140#336—22no—
141C25C2×D411yescharacteristic
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41C2×D412yes—
149C2×#41C2×Q812yes—
150C22×Q8C22×C411yescharacteristic
151C22×D4—21no—
152C22×D4—22no—
153C22×D4C22×C411yescharacteristic
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—48no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×Q812yes—
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C22×C8—22no—
189C22×C8—22no—
190#123—24no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8C4⋊C418yes—
194#121—24no—
195#121C4⋊C418yes—
196#120—24no—
197#120—24no—
198C4×Q8—44no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204#117—21no—
205#117—21no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—48no—
211#117—44no—
212#117—44no—
213#117—48no—
214#117—48no—
215#117—44no—
216#117C2×D412yes—
217#117C2×Q812yes—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4C22⋊C414yes—
245C2×C22⋊C4C22⋊C414yes—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—48no—
267C2×C42—48no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C22×C412yes—
272C2×C42C22×C412yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C4⋊C8—48no—
279#104—48no—
280#104—48no—
281C22⋊C8—44no—
282C22⋊C8—44no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—48no—
288#95—48no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—44no—
295C24C2×C22⋊C412yes—
296C24#12112yes—
297C24C22⋊Q812yes—
298C24C22×D411yescharacteristic
299#41—24no—
300#41—24no—
301#41—24no—
302#41—24no—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8C2×C4⋊C412yes—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—28no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—84no—
322C2×D4C2×C4⋊C412yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—416no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—416no—
359C22×C4—416no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C4⋊C414yes—
374C22×C4#11714yes—
375C22×C4#12114yes—
376C22×C4#12112yes—
377C22×C4C22⋊Q814yes—
378C22×C4C22⋊Q812yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389M16—42no—
390M16—42no—
391M16—42no—
392M16—42no—
393M16—42no—
394M16—42no—
395M16—42no—
396M16—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—22no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—84no—
412C4⋊C4C2×C4⋊C414yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—48no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C42—48no—
439C42—44no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—84no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—41no—
464C23—44no—
465C23—41no—
466C23#38014yes—
467C23C22×C22⋊C411yescharacteristic
468C23C2×#12111yescharacteristic
469C23C2×C22⋊Q811yescharacteristic
470C23#52511yescharacteristic
471C23#52511yescharacteristic
472C23#53411yescharacteristic
473C23#53411yescharacteristic
474Q8—24no—
475Q8—42no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—88no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—432no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—88no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4#38014yes—
528C2×C4C22×C4⋊C411yescharacteristic
529C2×C4C2×#11711yescharacteristic
530C2×C4C4×#4112yes—
531C2×C4#51812yes—
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4#53212yes—
535C2×C4#53412yes—
536C2×C4#54612yes—
537C2×C4#54612yes—
538C2×C4#55112yes—
539C2×C4#55112yes—
540C2×C4#55312yes—
541C2×C4#55312yes—
542C2×C4#55412yes—
543C2×C4#55412yes—
544C8—82no—
545C8—82no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—44no—
563C22—82no—
564C22—82no—
565C22C2×#38011yescharacteristic
566C22#229711yescharacteristic
567C22#230811yescharacteristic
568C22#253411yescharacteristic
569C22#253411yescharacteristic
570C22#257911yescharacteristic
571C22#257911yescharacteristic
572C22#290114yes—
573C4—24no—
574C4—48no—
575C4—42no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—44no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4#234912yes—
584C4#235312yes—
585C4#254312yes—
586C4#256412yes—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—44no—
592C2—82no—
593C2#1396011yescharacteristic
594C2#2065512yes—
595C2#2141912yes—
596C2#2177512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20627C2semidirect
5. #13756C2semidirect
6. #11209C2semidirect
7. #11053C2semidirect
8. #10808C2semidirect
9. #10790C2semidirect
10. #10720C2semidirect
11. #10719C2semidirect
22. #2324C4semidirect
23. #2323C4semidirect
62. C2×C4×Q8C2×C4semidirect
65. C2×C4×D4C2×C4semidirect
108. #391C2×C4semidirect
110. #390C2×C4semidirect
112. #389C2×C4semidirect
125. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#18972
C2×#18962
#206272
#137562
#112092
#110532
#108082
#107902
#107202
#107192