Finite Group Databasebeta

← #361124

Subgroups — #361124

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#1918C212yesmaximal
3#13935C211yesmaximal, characteristic
4#13817C211yesmaximal, characteristic
5#11067C212yesmaximal
6#11024C212yesmaximal
7#10932C212yesmaximal
8#10820C211yesmaximal, characteristic
9#10797C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×C4×Q16—28no—
16C2×C4×Q16C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×C4×C4⋊C4C2212yes—
27#1918—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1441—22no—
36#1436—22no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×Q8C2311yescharacteristic
46C22×Q16—22no—
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—24no—
50C22×Q16C2×C414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C4×Q16—232no—
75C2×#107C2×C414yes—
76C2×#107D418yes—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103C2×C414yes—
86C2×#103C2×C414yes—
87C2×C4×C8—24no—
88C2×C4×C8C2×C414yes—
89#391—24no—
90#391—24no—
91#391—24no—
92#391—42no—
93#389—24no—
94#389—42no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109C2×#95D412yes—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—24no—
115#336—22no—
116#336—42no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C22×Q8C22×C412yes—
134C22×Q8#4112yes—
135C22×Q8#4114yes—
136C22×Q8#4112yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C2×Q16—216no—
149C2×Q16—216no—
150C2×Q16—28no—
151C2×Q16—28no—
152C2×Q16—28no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C22×C412yes—
168C22×C8C22×C412yes—
169C22×C8C2×D414yes—
170C4×Q8—28no—
171C4×Q8—28no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8—48no—
175C4×Q8—48no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
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—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4#4114yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—82no—
218C2×C42C22×C412yes—
219C2×C42#4112yes—
220#107C22⋊C4116yes—
221#103—28no—
222#103—28no—
223#103—48no—
224#103—48no—
225C4×C8—28no—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—81no—
238#95—81no—
239C24C22×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8#11714yes—
275C2×Q8#11714yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
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×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4#13514yes—
332C22×C4#13712yes—
333C22×C4#13712yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344Q16—232no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8C2×C22⋊C414yes—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4C2×C22⋊C414yes—
378C4⋊C4C2×C22⋊C414yes—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—48no—
383C42—48no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—88no—
389C23C2×C4×D411yescharacteristic
390C23#52512yes—
391C23D4212yes—
392C23#54611yescharacteristic
393C23#54811yescharacteristic
394C23C2×#13512yes—
395C23C2×#13711yescharacteristic
396C23C2×#13711yescharacteristic
397C23#57611yescharacteristic
398C23#57811yescharacteristic
399C23#57812yes—
400Q8—24no—
401Q8—24no—
402Q8—24no—
403Q8—24no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
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#44114yes—
470C2×C4C22×C22⋊C411yescharacteristic
471C2×C4C2×#11711yescharacteristic
472C2×C4C2×#11711yescharacteristic
473C2×C4C4×#4111yescharacteristic
474C2×C4#51811yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4#52512yes—
477C2×C4#52512yes—
478C2×C4#52912yes—
479C2×C4#53812yes—
480C2×C4#53912yes—
481C2×C4#54612yes—
482C2×C4#54812yes—
483C2×C4#55012yes—
484C2×C4#55112yes—
485C2×C4#55211yescharacteristic
486C2×C4#55312yes—
487C2×C4#55311yescharacteristic
488C8—24no—
489C8—42no—
490C8—82no—
491C22#229011yescharacteristic
492C22D4×C22⋊C411yescharacteristic
493C22#235211yescharacteristic
494C22#245612yes—
495C22#246312yes—
496C22C4×#13511yescharacteristic
497C22C2×#44111yescharacteristic
498C22#295011yescharacteristic
499C22#295111yescharacteristic
500C22#310712yes—
501C22#310812yes—
502C22#312412yes—
503C22#312712yes—
504C22#327012yes—
505C22#327012yes—
506C22#328012yes—
507C22#328312yes—
508C22#330012yes—
509C22#332811yescharacteristic
510C22#333311yescharacteristic
511C22#336412yes—
512C22#337411yescharacteristic
513C22#337511yescharacteristic
514C4—24no—
515C4—44no—
516C4—44no—
517C4—42no—
518C4—44no—
519C4—44no—
520C4—44no—
521C4—84no—
522C4—82no—
523C4—162no—
524C4#228011yescharacteristic
525C4#231811yescharacteristic
526C4#234111yescharacteristic
527C4#235311yescharacteristic
528C4#246012yes—
529C4#246712yes—
530C2#1389011yescharacteristic
531C2#2111611yescharacteristic
532C2#2113511yescharacteristic
533C2#2244011yescharacteristic
534C2#2244811yescharacteristic
535C2#2276311yescharacteristic
536C2#2276911yescharacteristic
537C2#2662012yes—
538C2#2662212yes—
539C2#2689312yes—
540C2#2689512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
16. C2×C4×Q16C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#19182
#139352
#138172
#110672
#110242
#109322
#108202
#107972
#107892