Finite Group Databasebeta

← #399598

Subgroups — #399598

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#2088C212yesmaximal, complemented
3C2×#1850C211yesmaximal, characteristic
4#14129C212yesmaximal
5#11378C212yesmaximal, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11328C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11232C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2212yes—
14C2×#427—28no—
15C2×#427—24no—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#2088—28no—
27#1850—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1439—22no—
38#1439—22no—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C23×C8—24no—
45C23×C8—24no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8D414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#427—232no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106D418yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#389—42no—
91#380—24no—
92#380Q818yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—24no—
105#340—22no—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113C25C2×D411yescharacteristic
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×Q814yes—
131C23×C4#4112yes—
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C22×C8—216no—
135C22×C8—216no—
136C22×C8—28no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—28no—
143C22×C8—28no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D414yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8#4112yes—
158C22×C8#4112yes—
159#117—216no—
160#117—48no—
161#117—48no—
162#117—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×Q814yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—44no—
217#107—28no—
218#107—28no—
219#106—216no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222C22⋊C8—44no—
223#95—22no—
224#95—22no—
225#95—28no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—82no—
231#95—82no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—21no—
239C24—22no—
240C24—21no—
241C24#12112yes—
242C24C22⋊Q814yes—
243C24C22×D411yescharacteristic
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4#12112yes—
311C22×C4C22⋊Q814yes—
312C22×C4#12314yes—
313C22×C4#12518yes—
314C22×C4C4⋊Q814yes—
315C22×C4#13512yes—
316C22×C4#13512yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C22×Q812yes—
320C22×C4C2×#4112yes—
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C42−1+411yescharacteristic
326C22×C42−1+411yescharacteristic
327C2×C8—232no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8#12014yes—
347C2×C8#12314yes—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C22⋊C4—28no—
367C22⋊C4—44no—
368C22⋊C4—416no—
369C22⋊C4—88no—
370C42—416no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—41no—
377C42—41no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23#39814yes—
401C23C2×#12111yescharacteristic
402C23C2×C22⋊Q812yes—
403C23D4212yes—
404C23#54711yescharacteristic
405C23#54711yescharacteristic
406C23C2×#13511yescharacteristic
407C23C2×#13511yescharacteristic
408C23#57512yes—
409C23#57611yescharacteristic
410C23#57712yes—
411C23#57811yescharacteristic
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4#39814yes—
473C2×C4C2×#12011yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C2×C4#52512yes—
477C2×C4#52512yes—
478C2×C4C2×#12512yes—
479C2×C4C2×C4⋊Q811yescharacteristic
480C2×C4#53511yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#53912yes—
483C2×C4#53912yes—
484C2×C4#54612yes—
485C2×C4D4×Q812yes—
486C2×C4D4×Q812yes—
487C2×C4#55211yescharacteristic
488C2×C4#55211yescharacteristic
489C2×C4#55412yes—
490C2×C4#55412yes—
491C8—24no—
492C8—24no—
493C8—81no—
494C8—81no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22C2×#39811yescharacteristic
507C22#240811yescharacteristic
508C22#244912yes—
509C22#244912yes—
510C22#248611yescharacteristic
511C22#305011yescharacteristic
512C22#305411yescharacteristic
513C22#306011yescharacteristic
514C22#306111yescharacteristic
515C22#307612yes—
516C22#307612yes—
517C22#308112yes—
518C22#308212yes—
519C22#326812yes—
520C22#326912yes—
521C22#328212yes—
522C22#328412yes—
523C22#333611yescharacteristic
524C22#333911yescharacteristic
525C22#334111yescharacteristic
526C22#334211yescharacteristic
527C22#334812yes—
528C22#335012yes—
529C4—24no—
530C4—42no—
531C4—42no—
532C4—81no—
533C4—84no—
534C4—81no—
535C4—84no—
536C4—84no—
537C4—84no—
538C4#243411yescharacteristic
539C4#247312yes—
540C4#247312yes—
541C4#250211yescharacteristic
542C4#262311yescharacteristic
543C4#284911yescharacteristic
544C2—44no—
545C2#1478611yescharacteristic
546C2#2366611yescharacteristic
547C2#2366911yescharacteristic
548C2#2426511yescharacteristic
549C2#2427011yescharacteristic
550C2#2633712yes—
551C2#2634312yes—
552C2#2634512yes—
553C2#2635112yes—
554C2#2766011yescharacteristic
555C2#2766511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2088C2semidirect
5. #11378C2semidirect
6. #11328C2semidirect
7. #11328C2semidirect
8. #11232C2semidirect
9. #11232C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#20882
C2×#18502
#141292
#113782
#113282
#112322
#107312
#107212