Finite Group Databasebeta

← #455421

Subgroups — #455421

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2083C212yesmaximal, complemented
3C2×#1862C211yesmaximal, characteristic
4#15143C211yesmaximal, characteristic, complemented
5#13920C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11219C212yesmaximal, complemented
10#11189C212yesmaximal
11#11049C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#474—28no—
15C2×#474—24no—
16C22×#107C2211yescharacteristic
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C2×#395—21no—
23C2×#395—21no—
24C2×#390C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#384C2212yes—
28C2×#380—21no—
29#2083—28no—
30#1862—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4×D4—24no—
56C2×C4×D4—22no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#474—232no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#107—22no—
79C2×#107D418yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—22no—
88C2×#102—42no—
89C2×#102Q814yescomplemented
90#395—44no—
91#395—44no—
92#390—24no—
93#390—42no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—42no—
98#384—24no—
99#384—42no—
100#380—44no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95D411yescharacteristic
108C2×#95D412yes—
109C2×#95D411yescharacteristic
110#340—24no—
111#340—24no—
112#340—22no—
113#340—42no—
114#340—42no—
115#336—24no—
116#336—22no—
117#336—42no—
118C25—21no—
119C22×Q8—44no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—82no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D412yes—
132C22×D4C2×Q812yes—
133C22×D4#4112yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8C2×D414yes—
165C22×C8C2×Q812yes—
166C4⋊Q8—28no—
167C4⋊Q8—48no—
168C4⋊Q8—44no—
169C4×D4—28no—
170C4×D4—44no—
171C4×D4—48no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C2×D412yes—
229C2×C42#4112yes—
230#107—216no—
231#107—44no—
232C4⋊C8—28no—
233#102—216no—
234#102—28no—
235#102—44no—
236#95—22no—
237#95—24no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—44no—
243#95—81no—
244#95D814yes—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—42no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—42no—
254C24—41no—
255C24—41no—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—88no—
259C2×Q8—88no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4C22⋊Q814yes—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—48no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—84no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4C2×D814yes—
337C22×C4C2×D812yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13714yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C42−1+412yes—
348C22×C42−1+411yescharacteristic
349C22×C42−1+412yes—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—48no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C4⋊C4#12014yes—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23C2×#12111yescharacteristic
424C23#53612yes—
425C23D4212yes—
426C23#54611yescharacteristic
427C23D4×Q811yescharacteristic
428C23C22×D811yescharacteristic
429C23C2×AGL(1,ℤ/8)11yescharacteristic
430C23C2×#13712yes—
431C23#57612yes—
432C23#57711yescharacteristic
433C23#57811yescharacteristic
434Q8—816no—
435D4—24no—
436D4—24no—
437D4—42no—
438D4—44no—
439D4—44no—
440D4—42no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—816no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4#45914yes—
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#53612yes—
502C2×C4#53611yescharacteristic
503C2×C4#53712yes—
504C2×C4#53711yescharacteristic
505C2×C4#53912yes—
506C2×C4#54112yes—
507C2×C4#54412yes—
508C2×C4#54612yes—
509C2×C4#54812yes—
510C2×C4D4×Q812yes—
511C2×C4#55211yescharacteristic
512C2×C4#55411yescharacteristic
513C2×C4#56212yes—
514C2×C4#56312yes—
515C8—44no—
516C8—42no—
517C8—81no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22#241011yescharacteristic
530C22#241711yescharacteristic
531C22#262111yescharacteristic
532C22#267712yes—
533C22#268112yes—
534C22C2×#45911yescharacteristic
535C22#303911yescharacteristic
536C22#304111yescharacteristic
537C22#304611yescharacteristic
538C22#322612yes—
539C22#323912yes—
540C22#324112yes—
541C22#324312yes—
542C22#326412yes—
543C22#327012yes—
544C22#327812yes—
545C22D4×D812yes—
546C22#329411yescharacteristic
547C22#331412yes—
548C22#333011yescharacteristic
549C22Q8×D811yescharacteristic
550C22#338411yescharacteristic
551C22#338512yes—
552C4—24no—
553C4—44no—
554C4—42no—
555C4—44no—
556C4—44no—
557C4—84no—
558C4—82no—
559C4—161no—
560C4—162no—
561C4#240911yescharacteristic
562C4#261611yescharacteristic
563C4#262711yescharacteristic
564C4#269012yes—
565C4#270412yes—
566C4#281411yescharacteristic
567C2—84no—
568C2#1626411yescharacteristic
569C2#2435411yescharacteristic
570C2#2437411yescharacteristic
571C2#2481111yescharacteristic
572C2#2483311yescharacteristic
573C2#2986411yescharacteristic
574C2#2987211yescharacteristic
575C2#3094612yes—
576C2#3096212yes—
577C2#3111612yes—
578C2#3113812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
4. #15143C2semidirect
6. #11384C2semidirect
7. #11255C2semidirect
8. #11227C2semidirect
9. #11219C2semidirect
89. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#20832
C2×#18622
#151432
#139202
#113842
#112552
#112272
#112192
#111892
#110492