Finite Group Databasebeta

← #474833

Subgroups — #474833

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2042C212yesmaximal
2#14708C211yesmaximal, characteristic
3#14707C211yesmaximal, characteristic, complemented
4#11208C212yesmaximal, complemented
5#11190C212yesmaximal, complemented
6#11189C212yesmaximal
7#11052C211yesmaximal, characteristic, complemented
8#11052C211yesmaximal, characteristic, complemented
9#11051C211yesmaximal, characteristic
10#11051C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C43C2211yescharacteristic
26#2042—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1437—22no—
36#1437—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C23×Q8C2311yescharacteristic
41C23×D4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#124—22no—
45C2×#124—42no—
46C2×#124—42no—
47C2×#124D414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74#391—22no—
75#391—22no—
76#391—42no—
77#390—22no—
78#390—22no—
79#390—42no—
80#389—24no—
81#389—24no—
82#382—24no—
83#382—42no—
84#382—42no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C43—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336—24no—
102#336—22no—
103#336—22no—
104#336—42no—
105C25—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8C2×D412yes—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4C2×D412yes—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150#124—48no—
151#124—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
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—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4#4114yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×D412yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203#103—48no—
204#103—48no—
205#102—48no—
206#102—48no—
207#95—24no—
208#95—24no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—44no—
218C24—22no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24C22×D411yescharacteristic
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
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—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4#12114yes—
314C22×C4#12412yes—
315C22×C4#12412yes—
316C22×C4#13514yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C42−1+411yescharacteristic
326C22×C42−1+411yescharacteristic
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C22⋊C4—416no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C22⋊C4—88no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—48no—
363C42—48no—
364C42—42no—
365C42—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—82no—
373C23—81no—
374C23—82no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23C2×#12111yescharacteristic
390C23#52511yescharacteristic
391C23C2×#12411yescharacteristic
392C23#53411yescharacteristic
393C23#53511yescharacteristic
394C23#53511yescharacteristic
395C23#53611yescharacteristic
396C23C2×#13512yes—
397C23#57512yes—
398C23#57512yes—
399C23#57512yes—
400Q8—44no—
401Q8—44no—
402Q8—82no—
403Q8—82no—
404D4—44no—
405D4—44no—
406D4—82no—
407D4—82no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#53412yes—
464C2×C4#53812yes—
465C2×C4#54012yes—
466C2×C4#54012yes—
467C2×C4#54212yes—
468C2×C4#54212yes—
469C2×C4#54612yes—
470C2×C4#54711yescharacteristic
471C2×C4#54711yescharacteristic
472C2×C4#55312yes—
473C2×C4#55312yes—
474C2×C4#55312yes—
475C2×C4#55611yescharacteristic
476C2×C4#55611yescharacteristic
477C2×C4#56212yes—
478C2×C4#56212yes—
479C8—82no—
480C8—82no—
481C22—84no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—162no—
487C22—161no—
488C22—161no—
489C22—162no—
490C22—161no—
491C22—161no—
492C22#253511yescharacteristic
493C22#254711yescharacteristic
494C22#259111yescharacteristic
495C22#261711yescharacteristic
496C22#263211yescharacteristic
497C22#264111yescharacteristic
498C22#281611yescharacteristic
499C22#303512yes—
500C22#304012yes—
501C22#310212yes—
502C22#311112yes—
503C22#313111yescharacteristic
504C22#313511yescharacteristic
505C22#314011yescharacteristic
506C22#314011yescharacteristic
507C22#319112yes—
508C22#319712yes—
509C22#321112yes—
510C22#321212yes—
511C22#321912yes—
512C22#321912yes—
513C22#323212yes—
514C22#324012yes—
515C4—44no—
516C4—44no—
517C4—42no—
518C4—42no—
519C4—84no—
520C4—82no—
521C4—161no—
522C4—161no—
523C4—162no—
524C4#270112yes—
525C4#270112yes—
526C4#280012yes—
527C4#280012yes—
528C2—162no—
529C2#1701111yescharacteristic
530C2#2816312yes—
531C2#2872011yescharacteristic
532C2#2872711yescharacteristic
533C2#2933311yescharacteristic
534C2#2933911yescharacteristic
535C2#2972012yes—
536C2#3006512yes—
537C2#3022111yescharacteristic
538C2#3022511yescharacteristic
539C2#3277812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14707C2semidirect
4. #11208C2semidirect
5. #11190C2semidirect
7. #11052C2semidirect
8. #11052C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20422
#147082
#147072
#112082
#111902
#111892
#110522
#110512
#107952