Finite Group Databasebeta

← #428421

Subgroups — #428421

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic, complemented
3C2×#2033C212yesmaximal
4C2×#1852C212yesmaximal, complemented
5C2×#1845C211yesmaximal, characteristic, complemented
6#15150C211yesmaximal, characteristic, complemented
7#11373C211yesmaximal, characteristic, complemented
8#11324C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#10925C211yesmaximal, characteristic, complemented
11#10921C211yesmaximal, characteristic, complemented
12C22×#123—22no—
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#107C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2501—24no—
23#2501—24no—
24C2×#399—22no—
25C2×#397—22no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2093—24no—
34#2033—28no—
35#1852—28no—
36#1845—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1445—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#123—216no—
52C2×#123—48no—
53C2×#123—48no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—24no—
57C2×#121—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×#107—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8D418yes—
96C2×C4×C8—24no—
97C2×C4×C8—22no—
98#399—216no—
99#397—48no—
100#390—22no—
101#390—22no—
102#390—22no—
103#390D414yes—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384—22no—
109#384D414yes—
110#380—24no—
111#380—24no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—24no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—24no—
122#336—22no—
123C25—21no—
124C25C2×D411yescharacteristic
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—42no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—44no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D414yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C2×D414yes—
171C22×C8C2×D414yes—
172C22×C8#4114yes—
173C22×C8#4114yes—
174#123—464no—
175#121—48no—
176#121—48no—
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—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4D818yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#107—48no—
238#107—48no—
239#107—48no—
240#102—48no—
241#102—48no—
242C22⋊C8—216no—
243C4×C8—28no—
244#95—24no—
245#95—44no—
246#95—44no—
247#95—48no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—24no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—44no—
262C24#12712yes—
263C24C2×D814yes—
264C24C22×D411yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—48no—
270C2×D4—416no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—84no—
284C2×D4—88no—
285C2×D4—84no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4—88no—
341C22×C4—84no—
342C22×C4#12014yes—
343C22×C4#12712yes—
344C22×C4C2×D814yes—
345C22×C4C2×D814yes—
346C22×C4AGL(1,ℤ/8)14yes—
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C22×C42−1+412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8#12318yescomplemented
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—416no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—416no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—88no—
387C22⋊C4—216no—
388C22⋊C4—48no—
389C22⋊C4—416no—
390C22⋊C4—416no—
391C22⋊C4—88no—
392C42—22no—
393C42—22no—
394C42—24no—
395C42—41no—
396C42—41no—
397C42—42no—
398C23—28no—
399C23—28no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—44no—
407C23—44no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—84no—
419C23—84no—
420C23—84no—
421C23—84no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23#49314yes—
430C23C2×#12011yescharacteristic
431C23C2×#12711yescharacteristic
432C23#53511yescharacteristic
433C23#54712yes—
434C23#54712yes—
435C23C22×D812yes—
436C23C2×AGL(1,ℤ/8)12yes—
437C23#57814yes—
438D4—44no—
439D4—44no—
440D4—816no—
441D4—82no—
442D4—82no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—816no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4#44718yes—
488C2×C4C2×#12312yes—
489C2×C4#52511yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53612yes—
493C2×C4#53611yescharacteristic
494C2×C4#53912yes—
495C2×C4#54212yes—
496C2×C4D4212yes—
497C2×C4D4212yes—
498C2×C4#54712yes—
499C2×C4#54712yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4#55312yes—
503C2×C4#56312yes—
504C8—28no—
505C8—44no—
506C22—28no—
507C22—24no—
508C22—48no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—82no—
515C22—84no—
516C22—84no—
517C22—82no—
518C22—82no—
519C22—164no—
520C22—162no—
521C22—162no—
522C22#240611yescharacteristic
523C22#243211yescharacteristic
524C22#243311yescharacteristic
525C22#250112yes—
526C22#271812yes—
527C22C2×#44712yes—
528C22#301112yes—
529C22C2×#49311yescharacteristic
530C22#315111yescharacteristic
531C22#315612yes—
532C22#320512yes—
533C22#322312yes—
534C22#333414yes—
535C22#333414yes—
536C22#334314yes—
537C22#335314yes—
538C4—28no—
539C4—42no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—162no—
545C4—162no—
546C4#242212yes—
547C4#244312yes—
548C4#267412yes—
549C4#268012yes—
550C2—44no—
551C2—162no—
552C2#1538411yescharacteristic
553C2#2408812yes—
554C2#2570312yes—
555C2#2572111yescharacteristic
556C2#2573511yescharacteristic
557C2#2792912yes—
558C2#2794412yes—
559C2#3227814yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×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
2. C2×#2093C2semidirect
4. C2×#1852C2semidirect
5. C2×#1845C2semidirect
6. #15150C2semidirect
7. #11373C2semidirect
8. #11324C2semidirect
9. #11123C2semidirect
10. #10925C2semidirect
11. #10921C2semidirect
368. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#20932
C2×#20332
C2×#18522
C2×#18452
#151502
#113732
#113242
#111232
#109252
#109212