Finite Group Databasebeta

← #449085

Subgroups — #449085

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11199C212yesmaximal
7#10922C211yesmaximal, characteristic
8#10821C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×#97C2211yescharacteristic
14C2×#393—21no—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28#1446—21no—
29#1445—21no—
30#1445—21no—
31#1442—21no—
32#1439—21no—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C23×Q8—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—42no—
67C2×#97—24no—
68C2×#97—41no—
69#393—44no—
70#380—24no—
71#380—24no—
72#380—42no—
73C4×C22⋊C4—41no—
74C4×C22⋊C4—41no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D411yescharacteristic
86C2×#95D411yescharacteristic
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#336—42no—
96#336—41no—
97C25C2×D411yescharacteristic
98C22×Q8—44no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—82no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—82no—
118C23×C4—82no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4114yes—
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8#4114yes—
140C4⋊Q8—44no—
141C4⋊Q8—48no—
142C4⋊Q8—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4#4114yes—
173C2×C4⋊C4#4114yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C22⋊C8—48no—
195#97—44no—
196#95—41no—
197#95—41no—
198#95—44no—
199#95—42no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—22no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24#12412yes—
217C24AGL(1,ℤ/8)12yes—
218C24#13712yes—
219C24C22×D411yescharacteristic
220C2×Q8—48no—
221C2×Q8—88no—
222C2×Q8—84no—
223C2×Q8—88no—
224C2×Q8—84no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—48no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4#12412yes—
297C22×C4#13712yes—
298C22×C4#13712yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)11yescharacteristic
306C22×C42−1+412yes—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—88no—
342C22⋊C4—84no—
343C42—42no—
344C42—42no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23#48914yes—
372C23#52511yescharacteristic
373C23C2×#12411yescharacteristic
374C23#53411yescharacteristic
375C23#54612yes—
376C23#54712yes—
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380C23C2×#13711yescharacteristic
381C23#57612yes—
382C23#57712yes—
383Q8—816no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—816no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#52912yes—
443C2×C4#52912yes—
444C2×C4#53812yes—
445C2×C4#53912yes—
446C2×C4#54212yes—
447C2×C4#54212yes—
448C2×C4#55111yescharacteristic
449C2×C4#55112yes—
450C2×C4#55112yes—
451C2×C4#55112yes—
452C2×C4#55111yescharacteristic
453C2×C4#55212yes—
454C2×C4#55512yes—
455C2×C4#55612yes—
456C2×C4#56011yescharacteristic
457C2×C4#56311yescharacteristic
458C8—44no—
459C8—82no—
460C22—28no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—44no—
470C22—44no—
471C22#247412yes—
472C22#248812yes—
473C22#256111yescharacteristic
474C22#269711yescharacteristic
475C22#271711yescharacteristic
476C22#311211yescharacteristic
477C22#311311yescharacteristic
478C22#311812yes—
479C22C2×#48911yescharacteristic
480C22#313612yes—
481C22#313911yescharacteristic
482C22#319311yescharacteristic
483C22#319511yescharacteristic
484C22#319911yescharacteristic
485C22#320011yescharacteristic
486C22#331412yes—
487C22#331612yes—
488C22#331812yes—
489C22#332012yes—
490C22#334512yes—
491C22#334612yes—
492C22#335612yes—
493C22#335712yes—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—84no—
500C4—84no—
501C4—162no—
502C4—162no—
503C4#252712yes—
504C4#257212yes—
505C4#273712yes—
506C4#274512yes—
507C2—44no—
508C2#1593011yescharacteristic
509C2#2734312yes—
510C2#2734512yes—
511C2#2771112yes—
512C2#2772112yes—
513C2#2911111yescharacteristic
514C2#2911611yescharacteristic
515C2#3168011yescharacteristic
516C2#3168211yescharacteristic
517C2#3226811yescharacteristic
518C2#3226911yescharacteristic
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
3. #11384C2semidirect
4. #11316C2semidirect
5. #11315C2semidirect
8. #10821C2semidirect

Maximal subgroups

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

subgroupindex
#151252
#139242
#113842
#113162
#113152
#111992
#109222
#108212
#107302