Finite Group Databasebeta

← #398468

Subgroups — #398468

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3C2×#1862C211yesmaximal, characteristic
4#15230C211yesmaximal, characteristic, complemented
5#13820C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11076C212yesmaximal
10#11051C212yesmaximal
11#10908C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—28no—
15C2×#476—24no—
16C22×#107C2211yescharacteristic
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389C2211yescharacteristic
25C2×#382C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28#1949—28no—
29#1862—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1446—22no—
38#1441—21no—
39#1441—21no—
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⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72#476—232no—
73C2×#107—22no—
74C2×#107—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106D418yes—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—24no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—42no—
91#390—24no—
92#390Q818yescomplemented
93#389—22no—
94#389—22no—
95#382—24no—
96#382—42no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C4×C22⋊C4—44no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110C2×#95D412yes—
111#340—24no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—22no—
117#336—24no—
118C25—21no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—42no—
122C22×Q8—81no—
123C22×Q8—81no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D412yes—
133C22×D4C2×Q814yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—24no—
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—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D414yes—
165C22×C8#4112yes—
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—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4#4112yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—44no—
218C2×C42—44no—
219C2×C42—82no—
220C2×C42C2×D412yes—
221C2×C42C2×Q814yes—
222C2×C42#4112yes—
223C2×C42#4114yes—
224#107—44no—
225#106—216no—
226C4⋊C8—216no—
227C4⋊C8—28no—
228#102—28no—
229#102—216no—
230#102—44no—
231#95—22no—
232#95—22no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—82no—
240#95—81no—
241#95—82no—
242C24—22no—
243C24—24no—
244C24—42no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—42no—
250C24—41no—
251C24—42no—
252C24—41no—
253C24C22×D411yescharacteristic
254C2×Q8—48no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—88no—
258C2×Q8—88no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4C4⋊Q814yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
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—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—84no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4C22⋊Q814yes—
334C22×C4C22⋊Q814yes—
335C22×C4#13512yes—
336C22×C4AGL(1,ℤ/8)12yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×Q812yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4#12014yes—
384C22⋊C4—48no—
385C22⋊C4—416no—
386C22⋊C4—88no—
387C42—22no—
388C42—24no—
389C42—22no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—88no—
395C23—24no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—81no—
409C23—82no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23C2×#12111yescharacteristic
419C23C2×C22⋊Q812yes—
420C23D4212yes—
421C23#54711yescharacteristic
422C23#54811yescharacteristic
423C23C2×#13511yescharacteristic
424C23C2×AGL(1,ℤ/8)11yescharacteristic
425C23#57512yes—
426C23#57611yescharacteristic
427C23#57712yes—
428C23#57711yescharacteristic
429Q8—816no—
430D4—24no—
431D4—24no—
432D4—42no—
433D4—42no—
434D4—44no—
435D4—44no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
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—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—816no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4#46314yes—
492C2×C4C2×#12011yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52512yes—
495C2×C4C2×C4⋊Q811yescharacteristic
496C2×C4#53211yescharacteristic
497C2×C4#53212yes—
498C2×C4#53312yes—
499C2×C4#53411yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#53912yes—
502C2×C4#54212yes—
503C2×C4#54612yes—
504C2×C4D4×Q812yes—
505C2×C4D4×Q812yes—
506C2×C4#55211yescharacteristic
507C2×C4#55412yes—
508C2×C4#55412yes—
509C2×C4#55611yescharacteristic
510C8—44no—
511C8—42no—
512C8—81no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—84no—
520C22—84no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22#241011yescharacteristic
525C22#241511yescharacteristic
526C22#244912yes—
527C22#245212yes—
528C22#285711yescharacteristic
529C22C2×#46311yescharacteristic
530C22#304011yescharacteristic
531C22#304211yescharacteristic
532C22#304511yescharacteristic
533C22#307612yes—
534C22#307712yes—
535C22#308112yes—
536C22#308412yes—
537C22#326512yes—
538C22#326812yes—
539C22#327512yes—
540C22#328212yes—
541C22#334111yescharacteristic
542C22#335012yes—
543C22#335411yescharacteristic
544C22#336011yescharacteristic
545C22#336611yescharacteristic
546C22#337112yes—
547C4—24no—
548C4—44no—
549C4—42no—
550C4—44no—
551C4—44no—
552C4—84no—
553C4—82no—
554C4—84no—
555C4—161no—
556C4#238811yescharacteristic
557C4#243411yescharacteristic
558C4#246512yes—
559C4#255612yes—
560C4#258311yescharacteristic
561C4#266311yescharacteristic
562C2—84no—
563C2#1476311yescharacteristic
564C2#2435911yescharacteristic
565C2#2437111yescharacteristic
566C2#2467711yescharacteristic
567C2#2469711yescharacteristic
568C2#2634312yes—
569C2#2635412yes—
570C2#2646912yes—
571C2#2647212yes—
572C2#3320111yescharacteristic
573C2#3320511yescharacteristic
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×#1949C2semidirect
4. #15230C2semidirect
6. #11384C2semidirect
7. #11322C2semidirect
8. #11227C2semidirect
11. #10908C2semidirect
92. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19492
C2×#18622
#152302
#138202
#113842
#113222
#112272
#110762
#110512
#109082