Finite Group Databasebeta

← #436454

Subgroups — #436454

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2035C212yesmaximal
2C2×#1872C212yesmaximal
3#15178C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11122C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10926C212yesmaximal, complemented
9#10921C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2211yescharacteristic
20C2×#391C2212yes—
21C2×#381—22no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#2035—28no—
25#1872—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1445—22no—
32#1444—22no—
33#1439—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C23×Q8C2312yes—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—42no—
51C2×C22⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—41no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—42no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8D418yes—
85C2×C4×C8—41no—
86C2×C4×C8—42no—
87#391—24no—
88#391—42no—
89#381—48no—
90#380—24no—
91C4×C22⋊C4—41no—
92C4×C22⋊C4—41no—
93C2×#95—22no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95—21no—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—42no—
103#336—41no—
104C25C2×D411yescharacteristic
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8#4114yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D414yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2411yescharacteristic
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—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C2×D414yes—
153C22×C8C2×D414yes—
154C4⋊Q8—44no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C22⋊Q8—216no—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
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—24no—
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—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4#4114yes—
188C2×C4⋊C4#4114yes—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4SD1618yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205#106—48no—
206#103—48no—
207#103—48no—
208#103—48no—
209#103—48no—
210C22⋊C8—48no—
211C4×C8—44no—
212#95—42no—
213#95—48no—
214#95—44no—
215#95—42no—
216#95—48no—
217#95—48no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—24no—
224C24#12712yes—
225C24C2×SD1614yes—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
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×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—48no—
287C22×C4—416no—
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—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—84no—
308C22×C4—88no—
309C22×C4—84no—
310C22×C4#12712yes—
311C22×C4C2×SD1614yes—
312C22×C4C2×SD1614yes—
313C22×C4C22×D412yes—
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—416no—
341C4⋊C4—416no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—48no—
351C22⋊C4—416no—
352C22⋊C4—84no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—24no—
363C23—28no—
364C23—28no—
365C23—24no—
366C23—22no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—44no—
373C23—44no—
374C23#49214yes—
375C23#52511yescharacteristic
376C23C2×#12711yescharacteristic
377C23#53411yescharacteristic
378C23#54612yes—
379C23#54612yes—
380C23C22×SD1612yes—
381C23#57512yes—
382C23#57814yes—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—44no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—816no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4#52512yes—
434C2×C4#52912yes—
435C2×C4#53412yes—
436C2×C4#53812yes—
437C2×C4#54012yes—
438C2×C4#54012yes—
439C2×C4#54612yes—
440C2×C4#54612yes—
441C2×C4#54612yes—
442C2×C4#54612yes—
443C2×C4#55111yescharacteristic
444C2×C4#55312yes—
445C2×C4#55512yes—
446C2×C4#55911yescharacteristic
447C2×C4#56011yescharacteristic
448C2×C4#56011yescharacteristic
449C8—44no—
450C8—44no—
451C22—24no—
452C22—28no—
453C22—48no—
454C22—44no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22#247612yes—
460C22#253911yescharacteristic
461C22#253912yes—
462C22#269711yescharacteristic
463C22#270911yescharacteristic
464C22#310112yes—
465C22#312712yes—
466C22C2×#49211yescharacteristic
467C22#315311yescharacteristic
468C22#315612yes—
469C22#318712yes—
470C22#320212yes—
471C22#329814yes—
472C22#329814yes—
473C22#331514yes—
474C22#332714yes—
475C4—44no—
476C4—44no—
477C4—42no—
478C4—84no—
479C4—84no—
480C4—84no—
481C4—162no—
482C4#245712yes—
483C4#245912yes—
484C4#268812yes—
485C4#269912yes—
486C2—44no—
487C2#1556111yescharacteristic
488C2#2739214yes—
489C2#2830211yescharacteristic
490C2#2832211yescharacteristic
491C2#2833612yes—
492C2#2835612yes—
493C2#3166812yes—
494C2#3202712yes—
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
5. #11378C2semidirect
6. #11122C2semidirect
7. #11010C2semidirect
8. #10926C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20352
C2×#18722
#151782
#139042
#113782
#111222
#110102
#109262
#109212