Finite Group Databasebeta

← #461656

Subgroups — #461656

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2071C212yesmaximal
2C2×#1862C211yesmaximal, characteristic
3C2×#1802C211yesmaximal, characteristic
4#15145C211yesmaximal, characteristic, complemented
5#13920C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11227C211yesmaximal, characteristic, complemented
9#11219C212yesmaximal, complemented
10#11207C212yesmaximal, complemented
11#11189C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C2×#398—21no—
19C2×#398—21no—
20C2×#390C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C2×#380—21no—
26#2071—28no—
27#1862—24no—
28#1802—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1444—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—21no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4—42no—
52C2×C4×D4D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#107—22no—
70C2×#107—24no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—22no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102D414yes—
81#398—44no—
82#398—44no—
83#390—24no—
84#390—42no—
85#389—24no—
86#389—22no—
87#389—22no—
88#389—42no—
89#384—24no—
90#384—22no—
91#384—22no—
92#384—42no—
93#380—44no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95D411yescharacteristic
101C2×#95D412yes—
102C2×#95D411yescharacteristic
103#340—24no—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—22no—
111#336—42no—
112C25—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4C2×D412yes—
121C22×D4C2×D412yes—
122C22×D4#4112yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—44no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—82no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8C2×D412yes—
155C4×D4—28no—
156C4×D4—44no—
157C4×D4—48no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220#107—44no—
221C4⋊C8—44no—
222#102—28no—
223#102—44no—
224#102—48no—
225#95—22no—
226#95—24no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—44no—
232#95—81no—
233#95D814yes—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—42no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24C22×D411yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4#12114yes—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—48no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—84no—
316C22×C4—88no—
317C22×C4—88no—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4C2×D814yes—
321C22×C4C2×D812yes—
322C22×C4AGL(1,ℤ/8)12yes—
323C22×C4#13714yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)12yes—
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+412yes—
332C22×C42−1+412yes—
333C22×C42−1+411yescharacteristic
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4#12314yes—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C42—22no—
379C42—22no—
380C42—24no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C23—24no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23C2×#12111yescharacteristic
409C23#53612yes—
410C23D4211yescharacteristic
411C23#54612yes—
412C23#54811yescharacteristic
413C23C22×D811yescharacteristic
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×#13712yes—
416C23#57611yescharacteristic
417C23#57711yescharacteristic
418C23#57712yes—
419D4—24no—
420D4—24no—
421D4—42no—
422D4—44no—
423D4—44no—
424D4—42no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—816no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4#45914yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4#53411yescharacteristic
484C2×C4#53611yescharacteristic
485C2×C4#53611yescharacteristic
486C2×C4#53712yes—
487C2×C4#53711yescharacteristic
488C2×C4#53912yes—
489C2×C4#54112yes—
490C2×C4#54112yes—
491C2×C4#54212yes—
492C2×C4#54711yescharacteristic
493C2×C4#54712yes—
494C2×C4#55112yes—
495C2×C4#55212yes—
496C2×C4#55311yescharacteristic
497C2×C4#56212yes—
498C2×C4#56312yes—
499C8—42no—
500C8—82no—
501C8—81no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—84no—
506C22—82no—
507C22—82no—
508C22—84no—
509C22—82no—
510C22—84no—
511C22—82no—
512C22—84no—
513C22#241211yescharacteristic
514C22#241611yescharacteristic
515C22#262111yescharacteristic
516C22#267212yes—
517C22#269012yes—
518C22C2×#45911yescharacteristic
519C22#303911yescharacteristic
520C22#304211yescharacteristic
521C22#304511yescharacteristic
522C22#322612yes—
523C22#323912yes—
524C22#324112yes—
525C22#324412yes—
526C22#326612yes—
527C22#327211yescharacteristic
528C22D4×D811yescharacteristic
529C22#329412yes—
530C22#330612yes—
531C22#331612yes—
532C22#333312yes—
533C22#335811yescharacteristic
534C22#336611yescharacteristic
535C22#337212yes—
536C4—24no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—42no—
541C4—82no—
542C4—162no—
543C4—162no—
544C4—161no—
545C4#248611yescharacteristic
546C4#261811yescharacteristic
547C4#262511yescharacteristic
548C4#272212yes—
549C4#274712yes—
550C4#281411yescharacteristic
551C2—84no—
552C2#1639911yescharacteristic
553C2#2452611yescharacteristic
554C2#2454611yescharacteristic
555C2#2472611yescharacteristic
556C2#2474811yescharacteristic
557C2#2986411yescharacteristic
558C2#2987411yescharacteristic
559C2#3062912yes—
560C2#3065512yes—
561C2#3145412yes—
562C2#3147412yes—
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. #15145C2semidirect
6. #11373C2semidirect
7. #11255C2semidirect
8. #11227C2semidirect
9. #11219C2semidirect
10. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20712
C2×#18622
C2×#18022
#151452
#139202
#113732
#112552
#112272
#112192
#112072
#111892