Finite Group Databasebeta

← #365180

Subgroups — #365180

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#11315C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#381—21no—
19C2×#381—21no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1852—24no—
26#1823C418yescomplemented
27C2×#340—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1445—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2×C414yes—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2312yes—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—24no—
82C2×#107—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8C2×C414yes—
86#381—44no—
87#381—44no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380C2×C414yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—42no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—44no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340C2×C418yescomplemented
114#340C2×C418yescomplemented
115#336—24no—
116#336—42no—
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—28no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—48no—
166C22×C8—48no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C22×C8#4114yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4D814yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—82no—
233#107—44no—
234#107—44no—
235C22⋊C8—28no—
236#95—24no—
237#95—24no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—42no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—82no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24C4×D412yes—
251C24#12112yes—
252C24#12312yes—
253C24C2×D812yes—
254C24#13512yes—
255C24AGL(1,ℤ/8)12yes—
256C24#13712yes—
257C24C22×D411yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—48no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—84no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4#11714yes—
334C22×C4C4×D414yes—
335C22×C4C4×D412yes—
336C22×C4#12112yes—
337C22×C4#12312yes—
338C22×C4#12414yes—
339C22×C4#12614yes—
340C22×C4C2×D814yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—416no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—48no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4#12314yes—
391C22⋊C4—28no—
392C22⋊C4—48no—
393C22⋊C4—44no—
394C22⋊C4—84no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—88no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23#38814yes—
417C23C4×D814yes—
418C23#44014yes—
419C23#45914yes—
420C23#46414yes—
421C23#48014yes—
422C23#48214yes—
423C23C2×C4×D411yescharacteristic
424C23C2×#12111yescharacteristic
425C23C2×#12311yescharacteristic
426C23#54612yes—
427C23#54712yes—
428C23C22×D811yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×#13711yescharacteristic
432C23#57712yes—
433C23#57812yes—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—816no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—816no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4#38814yes—
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C4×#4112yes—
498C2×C4#52012yes—
499C2×C4C2×#12311yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4C2×#12411yescharacteristic
502C2×C4C2×#12611yescharacteristic
503C2×C4#52912yes—
504C2×C4#53411yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53912yes—
507C2×C4#55112yes—
508C2×C4#55112yes—
509C2×C4#55112yes—
510C2×C4#55212yes—
511C2×C4#55212yes—
512C2×C4#55212yes—
513C8—44no—
514C8—42no—
515C8—42no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—28no—
520C22—28no—
521C22—28no—
522C22—24no—
523C22—28no—
524C22—24no—
525C22—24no—
526C22#192518yes—
527C22C2×#38811yescharacteristic
528C22#235912yes—
529C22#241211yescharacteristic
530C22#241311yescharacteristic
531C22#241912yes—
532C22C2×C4×D811yescharacteristic
533C22C2×#44011yescharacteristic
534C22#294712yes—
535C22C2×#45911yescharacteristic
536C22C2×#46411yescharacteristic
537C22#304211yescharacteristic
538C22#304411yescharacteristic
539C22C2×#48011yescharacteristic
540C22C2×#48211yescharacteristic
541C22#309712yes—
542C22#329412yes—
543C22#330012yes—
544C22#330612yes—
545C22#331212yes—
546C22#333412yes—
547C22#333712yes—
548C22#334312yes—
549C22#334412yes—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—84no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—161no—
558C4—161no—
559C4C4×#12311yescharacteristic
560C4#232211yescharacteristic
561C4#234312yes—
562C4#249312yes—
563C4#250311yescharacteristic
564C4#256111yescharacteristic
565C2—28no—
566C2#1397211yescharacteristic
567C2C2×#192511yescharacteristic
568C2#2106411yescharacteristic
569C2#2291612yes—
570C2#2293412yes—
571C2#2452611yescharacteristic
572C2#2452911yescharacteristic
573C2#2457611yescharacteristic
574C2#2457911yescharacteristic
575C2#2493312yes—
576C2#2494512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11315C2semidirect
7. #11256C2semidirect
8. #10907C2semidirect
9. #10806C2semidirect
26. #1823C4semidirect
113. #340C2×C4semidirect
114. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
C2×#18232
C22×#3402
#139042
C22⋊C4×C4⋊C42
#113152
#112562
#109072
#108062
#107212
#107142