Finite Group Databasebeta

← #403181

Subgroups — #403181

Order 512.

1326 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×#2083C212yesmaximal, complemented
2C2×#2067C212yesmaximal
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1925C211yesmaximal, characteristic
5#15155C211yesmaximal, characteristic
6#13821C211yesmaximal, characteristic, complemented
7#11357C211yesmaximal, characteristic, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11076C211yesmaximal, characteristic
10#10931C212yesmaximal
11#10849C212yesmaximal, complemented
12C22×#107C2212yes—
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#399—21no—
20C2×#399—22no—
21C2×#399—21no—
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2083—28no—
31#2067—28no—
32#1947—24no—
33#1925—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—22no—
71C2×#107—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—42no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—42no—
84C2×#102Q814yes—
85#399—44no—
86#399—48no—
87#399—44no—
88#390—24no—
89#390—22no—
90#390—22no—
91#390—24no—
92#389—24no—
93#389D418yes—
94#384—24no—
95#384—22no—
96#384—22no—
97#384—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—42no—
101C4×C4⋊C4—44no—
102C4×C4⋊C4D414yes—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95D411yescharacteristic
109C2×#95D412yes—
110C2×#95D411yescharacteristic
111#340—24no—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—24no—
116#336—24no—
117#336—22no—
118C25—21no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4C2×Q812yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×Q812yes—
155C22×C8#4114yes—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×Q812yes—
195C2×C4⋊C4#4112yes—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—82no—
219C2×C42C2×D412yes—
220C2×C42C2×D414yes—
221C2×C42#4112yes—
222C2×C42#4114yes—
223#107—48no—
224#107—44no—
225C4⋊C8—28no—
226C4⋊C8—48no—
227#102—28no—
228#102—48no—
229#102—44no—
230#95—24no—
231#95—28no—
232#95—22no—
233#95—24no—
234#95—44no—
235#95—42no—
236#95—81no—
237#95D814yes—
238C24—22no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24C22×D411yescharacteristic
250C2×D4—22no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#12112yes—
316C22×C4#12114yes—
317C22×C4#12112yes—
318C22×C4#12114yes—
319C22×C4C2×D812yes—
320C22×C4C2×D814yes—
321C22×C4AGL(1,ℤ/8)12yes—
322C22×C4AGL(1,ℤ/8)14yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+412yes—
332C22×C42−1+411yescharacteristic
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—84no—
368C4⋊C4—88no—
369C4⋊C4—84no—
370C4⋊C4—88no—
371C4⋊C4C22⋊Q814yes—
372C4⋊C4#12714yes—
373C22⋊C4—84no—
374C22⋊C4—88no—
375C22⋊C4—84no—
376C22⋊C4—88no—
377C42—22no—
378C42—24no—
379C42—22no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—48no—
385C42—88no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—44no—
391C23—81no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—82no—
399C23—82no—
400C23—81no—
401C23—82no—
402C23—81no—
403C23—81no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—82no—
409C23C2×#12111yescharacteristic
410C23C2×#12112yes—
411C23#54611yescharacteristic
412C23#54712yes—
413C23D4×Q811yescharacteristic
414C23C22×D811yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23C2×AGL(1,ℤ/8)12yes—
417C23#57711yescharacteristic
418C23#57811yescharacteristic
419C23#57812yes—
420D4—42no—
421D4—42no—
422D4—44no—
423D4—81no—
424D4—82no—
425D4—81no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#45914yes—
480C2×C4#45918yes—
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4#52912yes—
484C2×C4C2×#12711yescharacteristic
485C2×C4#53211yescharacteristic
486C2×C4#53212yes—
487C2×C4#53312yes—
488C2×C4#53611yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#53912yes—
491C2×C4#54112yes—
492C2×C4#54612yes—
493C2×C4#54712yes—
494C2×C4#54812yes—
495C2×C4D4×Q812yes—
496C2×C4#55211yescharacteristic
497C2×C4#55212yes—
498C2×C4#55411yescharacteristic
499C8—44no—
500C8—42no—
501C8—81no—
502C22—82no—
503C22—84no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—161no—
508C22—161no—
509C22—162no—
510C22—162no—
511C22—161no—
512C22—161no—
513C22#241311yescharacteristic
514C22#241711yescharacteristic
515C22#242512yes—
516C22#242912yes—
517C22#285611yescharacteristic
518C22C2×#45911yescharacteristic
519C22C2×#45912yes—
520C22#303811yescharacteristic
521C22#303812yes—
522C22#304411yescharacteristic
523C22#304412yes—
524C22#304611yescharacteristic
525C22#304612yes—
526C22#329411yescharacteristic
527C22#330812yes—
528C22#333011yescharacteristic
529C22#333412yes—
530C22#334312yes—
531C22#334512yes—
532C22#335312yes—
533C22Q8×D811yescharacteristic
534C22#338411yescharacteristic
535C22#338612yes—
536C4—24no—
537C4—28no—
538C4—42no—
539C4—44no—
540C4—44no—
541C4—44no—
542C4—84no—
543C4—161no—
544C4—162no—
545C4#239611yescharacteristic
546C4#251912yes—
547C4#254011yescharacteristic
548C4#254211yescharacteristic
549C4#255612yes—
550C4#266411yescharacteristic
551C2—162no—
552C2#1487011yescharacteristic
553C2#2457611yescharacteristic
554C2#2459211yescharacteristic
555C2#2481111yescharacteristic
556C2#2482711yescharacteristic
557C2#2520012yes—
558C2#2521712yes—
559C2#2544912yes—
560C2#2546512yes—
561C2#3316011yescharacteristic
562C2#3316911yescharacteristic
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
1. C2×#2083C2semidirect
3. C2×#1947C2semidirect
6. #13821C2semidirect
7. #11357C2semidirect
8. #11322C2semidirect
11. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20672
C2×#19472
C2×#19252
#151552
#138212
#113572
#113222
#110762
#109312
#108492