Finite Group Databasebeta

← #398462

Subgroups — #398462

Order 512.

1556 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#1947C212yesmaximal, complemented
3C2×#1862C211yesmaximal, characteristic
4#15143C211yesmaximal, characteristic, complemented
5#13820C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11076C212yesmaximal
10#11049C212yesmaximal
11#10907C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#474—28no—
15C2×#474—24no—
16C22×#107C2211yescharacteristic
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C2×#395—21no—
24C2×#395—21no—
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#384C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4—21no—
30#1947—28no—
31#1862—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1444—22no—
40#1437—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⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2312yes—
74#474—232no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—22no—
78C2×#107—24no—
79C2×#107D418yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—24no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—22no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—42no—
93#395—44no—
94#395—44no—
95#390—24no—
96#390Q818yescomplemented
97#389—22no—
98#389—22no—
99#384—24no—
100#384—42no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—42no—
103C4×C22⋊C4—44no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95D411yescharacteristic
111C2×#95D412yes—
112C2×#95D411yescharacteristic
113#340—24no—
114#340—22no—
115#340—24no—
116#340—24no—
117#340—42no—
118#336—22no—
119#336—24no—
120C25—21no—
121C22×Q8—44no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—82no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×D412yes—
135C22×D4C2×Q814yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D414yes—
167C22×C8#4112yes—
168C4⋊Q8—28no—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171C4×D4—28no—
172C4×D4—44no—
173C4×D4—48no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
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—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q814yes—
210C2×C4⋊C4#4112yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42—82no—
228C2×C42C2×D412yes—
229C2×C42C2×Q814yes—
230C2×C42#4112yes—
231C2×C42#4114yes—
232#107—216no—
233#107—44no—
234C4⋊C8—216no—
235C4⋊C8—28no—
236#102—28no—
237#102—216no—
238#102—44no—
239#95—22no—
240#95—24no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—44no—
246#95—81no—
247#95D814yes—
248C24—22no—
249C24—24no—
250C24—42no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4C4⋊Q814yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—48no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—48no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—84no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4C22⋊Q814yes—
338C22×C4C22⋊Q814yes—
339C22×C4C2×D812yes—
340C22×C4C2×D814yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13714yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×Q812yes—
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+411yescharacteristic
352C22×C42−1+411yescharacteristic
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4#12014yes—
392C22⋊C4—48no—
393C22⋊C4—416no—
394C22⋊C4—88no—
395C42—22no—
396C42—24no—
397C42—22no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—88no—
403C23—24no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—81no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23C2×#12111yescharacteristic
427C23C2×C22⋊Q812yes—
428C23D4212yes—
429C23#54711yescharacteristic
430C23#54811yescharacteristic
431C23C22×D811yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23C2×#13712yes—
434C23#57612yes—
435C23#57711yescharacteristic
436C23#57811yescharacteristic
437Q8—816no—
438D4—24no—
439D4—24no—
440D4—42no—
441D4—42no—
442D4—44no—
443D4—44no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—816no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4#45914yes—
499C2×C4#47418yescomplemented
500C2×C4C2×#12011yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#52512yes—
503C2×C4C2×C4⋊Q811yescharacteristic
504C2×C4#53211yescharacteristic
505C2×C4#53212yes—
506C2×C4#53312yes—
507C2×C4#53411yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53912yes—
510C2×C4#54212yes—
511C2×C4#54612yes—
512C2×C4D4×Q812yes—
513C2×C4D4×Q812yes—
514C2×C4#55211yescharacteristic
515C2×C4#55412yes—
516C2×C4#55412yes—
517C2×C4#55611yescharacteristic
518C8—44no—
519C8—42no—
520C8—81no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—84no—
527C22—82no—
528C22—84no—
529C22—82no—
530C22—84no—
531C22—84no—
532C22#241011yescharacteristic
533C22#241511yescharacteristic
534C22#244912yes—
535C22#245212yes—
536C22#285711yescharacteristic
537C22C2×#45911yescharacteristic
538C22#303911yescharacteristic
539C22#304111yescharacteristic
540C22#304611yescharacteristic
541C22C2×#47412yes—
542C22#307912yes—
543C22#308112yes—
544C22#308312yes—
545C22#326412yes—
546C22#327012yes—
547C22#327812yes—
548C22D4×D812yes—
549C22#333411yescharacteristic
550C22#334612yes—
551C22#335411yescharacteristic
552C22#335811yescharacteristic
553C22#336711yescharacteristic
554C22#337512yes—
555C4—24no—
556C4—28no—
557C4—44no—
558C4—42no—
559C4—44no—
560C4—84no—
561C4—84no—
562C4—82no—
563C4—161no—
564C4#238811yescharacteristic
565C4#243411yescharacteristic
566C4#246512yes—
567C4#255612yes—
568C4#258311yescharacteristic
569C4#266311yescharacteristic
570C2—84no—
571C2#1476311yescharacteristic
572C2#2435411yescharacteristic
573C2#2437411yescharacteristic
574C2#2467611yescharacteristic
575C2#2469611yescharacteristic
576C2#2633612yes—
577C2#2635712yes—
578C2#2646012yes—
579C2#2647812yes—
580C2#3319511yescharacteristic
581C2#3320311yescharacteristic
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×#1947C2semidirect
4. #15143C2semidirect
6. #11384C2semidirect
7. #11322C2semidirect
8. #11227C2semidirect
11. #10907C2semidirect
96. #390Q8semidirect
499. C2×C4#474semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#19472
C2×#18622
#151432
#138202
#113842
#113222
#112272
#110762
#110492
#109072