Finite Group Databasebeta

← #397972

Subgroups — #397972

Order 512.

1716 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1862C211yesmaximal, characteristic
3C2×#1804C212yesmaximal
4#15230C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11231C212yesmaximal, complemented
10#11227C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C2×#476—28no—
16C2×#476—24no—
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#382C2212yes—
26C22×#95—21no—
27#1862—24no—
28#1804—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1446—22no—
37#1441—21no—
38#1441—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—21no—
43C22⋊#95—22no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4D414yes—
65C2×C4×D4Q818yescomplemented
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77#476—232no—
78C2×#107—22no—
79C2×#107—42no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—42no—
88C2×#102—22no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102D418yes—
95C2×#102Q814yescomplemented
96#389—22no—
97#389—22no—
98#382—24no—
99#382—42no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—44no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—24no—
120C25—21no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—81no—
125C22×Q8—81no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4C2×D414yes—
132C22×D4C2×D412yes—
133C22×D4C2×Q814yes—
134C22×D4C2×Q812yes—
135C22×D4#4112yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—28no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D414yes—
174C22×C8C2×Q812yes—
175C4⋊Q8—28no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178C4×D4—28no—
179C4×D4—216no—
180C4×D4—44no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C2×D412yes—
228C2×C42C2×Q814yes—
229C2×C42#4112yes—
230C2×C42#4114yes—
231#107—44no—
232#106—216no—
233C4⋊C8—28no—
234C4⋊C8—48no—
235#102—216no—
236#102—28no—
237#102—44no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—82no—
247#95—81no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—24no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24C22×D411yescharacteristic
262C2×Q8—48no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8—88no—
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—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4#12014yes—
281C2×D4C22⋊Q814yes—
282C2×D4C22⋊Q818yes—
283C2×D4C4⋊Q814yes—
284C2×D4C4⋊Q818yes—
285C22×C4—22no—
286C22×C4—22no—
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—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—48no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—84no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q814yes—
350C22×C4#13512yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×Q812yes—
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+411yescharacteristic
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C22⋊C4—48no—
400C22⋊C4—416no—
401C42—22no—
402C42—24no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—28no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23C2×#12111yescharacteristic
432C23C2×C22⋊Q812yes—
433C23D4212yes—
434C23#54611yescharacteristic
435C23D4×Q811yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×AGL(1,ℤ/8)11yescharacteristic
438C23#57512yes—
439C23#57611yescharacteristic
440C23#57712yes—
441C23#57711yescharacteristic
442Q8—816no—
443D4—24no—
444D4—24no—
445D4—24no—
446D4—28no—
447D4—28no—
448D4#39518yes—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—816no—
485C2×C4—832no—
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—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4#46314yes—
506C2×C4C2×#12011yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×C22⋊Q812yes—
509C2×C4C2×C4⋊Q811yescharacteristic
510C2×C4C2×C4⋊Q812yes—
511C2×C4#53211yescharacteristic
512C2×C4#53312yes—
513C2×C4#53411yescharacteristic
514C2×C4#53711yescharacteristic
515C2×C4#54312yes—
516C2×C4#54412yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#54712yes—
520C2×C4#55112yes—
521C2×C4#55211yescharacteristic
522C2×C4#55412yes—
523C2×C4#55411yescharacteristic
524C8—42no—
525C8—44no—
526C8—81no—
527C22—28no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—48no—
532C22—48no—
533C22—44no—
534C22—44no—
535C22—48no—
536C22—48no—
537C22—44no—
538C22#241011yescharacteristic
539C22#241711yescharacteristic
540C22#243712yes—
541C22#244012yes—
542C22#285711yescharacteristic
543C22C2×#46311yescharacteristic
544C22#304011yescharacteristic
545C22#304211yescharacteristic
546C22#304511yescharacteristic
547C22#307612yes—
548C22#307712yes—
549C22#308112yes—
550C22#308412yes—
551C22#326512yes—
552C22#326712yes—
553C22#327312yes—
554C22#328212yes—
555C22#330311yescharacteristic
556C22#331112yes—
557C22#332511yescharacteristic
558C22#338011yescharacteristic
559C22#338411yescharacteristic
560C22#338812yes—
561C4—24no—
562C4—42no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—84no—
567C4—84no—
568C4—82no—
569C4—161no—
570C4C2×#39511yescharacteristic
571C4#243011yescharacteristic
572C4#243812yes—
573C4#254912yes—
574C4#256411yescharacteristic
575C4#265911yescharacteristic
576C2—48no—
577C2#1475811yescharacteristic
578C2#2435911yescharacteristic
579C2#2437311yescharacteristic
580C2#2481211yescharacteristic
581C2#2483411yescharacteristic
582C2#2596212yes—
583C2#2597412yes—
584C2#2605112yes—
585C2#2605312yes—
586C2#3320111yescharacteristic
587C2#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
4. #15230C2semidirect
6. #11384C2semidirect
7. #11325C2semidirect
8. #11322C2semidirect
9. #11231C2semidirect
10. #11227C2semidirect
65. C2×C4×D4Q8semidirect
95. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#18622
C2×#18042
#152302
D4×#952
#113842
#113252
#113222
#112312
#112272
#107882