Finite Group Databasebeta

← #401681

Subgroups — #401681

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2086C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5#14697C212yesmaximal
6#11231C211yesmaximal, characteristic, complemented
7#11231C211yesmaximal, characteristic, complemented
8#11228C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#10722C212yesmaximal
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#423—24no—
18C2×#423—24no—
19C2×#423—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2087—24no—
30#2086—24no—
31#1809—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1436—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—22no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—42no—
56C22×M16—42no—
57C22×M16D414yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—42no—
85#423—232no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8D414yes—
97C2×C4⋊C8D414yes—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#385—48no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380Q818yes—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—22no—
119#340—24no—
120#340—42no—
121#340—42no—
122#340—42no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×Q814yes—
151C23×C4#4112yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
154C2×M16—28no—
155C2×M16—28no—
156C2×M16—28no—
157C2×M16—44no—
158C2×M16—48no—
159C2×M16—48no—
160C2×M16—48no—
161C2×M16—48no—
162C2×M16—44no—
163C2×M16—44no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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—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—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
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×Q814yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C42—28no—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—28no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242#107—44no—
243#106—44no—
244C4⋊C8—28no—
245C4⋊C8—28no—
246C4⋊C8—28no—
247C4⋊C8—28no—
248#95—28no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24#12112yes—
264C24C22⋊Q814yes—
265C24C22×D411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
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—42no—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4C22⋊Q814yes—
342C22×C4#12314yes—
343C22×C4#12518yes—
344C22×C4C4⋊Q814yes—
345C22×C4AGL(1,ℤ/8)14yes—
346C22×C4#13714yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q812yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357M16—416no—
358M16—416no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8#12014yes—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C42—28no—
400C42—28no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23#39814yes—
430C23C2×#12111yescharacteristic
431C23C2×#12111yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×C22⋊Q812yes—
434C23#53712yes—
435C23C2×AGL(1,ℤ/8)12yes—
436C23C2×#13712yes—
437C23#57512yes—
438C23#57512yes—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—416no—
462C2×C4—416no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#39814yes—
498C2×C4#46914yes—
499C2×C4#47014yes—
500C2×C4C2×#12011yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4#52511yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4C2×#12512yes—
506C2×C4C2×C4⋊Q811yescharacteristic
507C2×C4#53411yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53912yes—
510C2×C4#54612yes—
511C2×C4#54612yes—
512C2×C4D4×Q812yes—
513C2×C4D4×Q812yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C2×C4#55412yes—
517C2×C4#55412yes—
518C8—24no—
519C8—42no—
520C8—42no—
521C8—42no—
522C22—28no—
523C22—24no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22C2×#39811yescharacteristic
532C22#241712yes—
533C22#241712yes—
534C22#261811yescharacteristic
535C22#284711yescharacteristic
536C22C2×#46911yescharacteristic
537C22C2×#47011yescharacteristic
538C22#305411yescharacteristic
539C22#305411yescharacteristic
540C22#305512yes—
541C22#305512yes—
542C22#305612yes—
543C22#305612yes—
544C22#307712yes—
545C22#307812yes—
546C22#307912yes—
547C22#308012yes—
548C22#325112yes—
549C22#325212yes—
550C22#325412yes—
551C22#325612yes—
552C4—24no—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—84no—
558C4—84no—
559C4—162no—
560C4—162no—
561C4#240511yescharacteristic
562C4#243411yescharacteristic
563C4#247312yes—
564C4#247312yes—
565C4#248411yescharacteristic
566C4#258311yescharacteristic
567C2—44no—
568C2#1482111yescharacteristic
569C2#2367012yes—
570C2#2484112yes—
571C2#2484212yes—
572C2#2484412yes—
573C2#2484612yes—
574C2#2978412yes—
575C2#3293611yescharacteristic
576C2#3293711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2087C2semidirect
3. C2×#2086C2semidirect
6. #11231C2semidirect
7. #11231C2semidirect
8. #11228C2semidirect
9. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
C2×#20872
C2×#20862
C2×#18092
#146972
#112312
#112282
#112272
#107222