Finite Group Databasebeta

← #479296

Subgroups — #479296

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15125C211yesmaximal, characteristic
2#15125C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11384C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#11147C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#393—21no—
15C2×#393—21no—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1446—21no—
30#1446—21no—
31#1442—21no—
32#1442—21no—
33#1439—21no—
34#1439—21no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×Q8—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—41no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—42no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68#393—44no—
69#393—44no—
70#380—24no—
71#380—24no—
72#380—42no—
73#380—42no—
74C4×C22⋊C4—41no—
75C4×C22⋊C4—41no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D411yescharacteristic
87C2×#95D411yescharacteristic
88#340—24no—
89#340—24no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—42no—
97#336—42no—
98C25C2×D411yescharacteristic
99C22×Q8—44no—
100C22×Q8—44no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—82no—
108C22×Q8—82no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
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—22no—
140C22×C8—22no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—48no—
144C4⋊Q8—48no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
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—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4#4114yes—
174C2×C4⋊C4#4114yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4SD1614yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197#95—41no—
198#95—41no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—44no—
207#95—82no—
208#95—82no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—22no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24C2×SD1612yes—
220C24C22×D411yescharacteristic
221C2×Q8—48no—
222C2×Q8—48no—
223C2×Q8—88no—
224C2×Q8—88no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—88no—
229C2×Q8—84no—
230C2×Q8—84no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
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—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×SD1612yes—
297C22×C4C2×SD1612yes—
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4#13712yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+412yes—
307C22×C42−1+412yes—
308C2×C8—42no—
309C2×C8—42no—
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—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—48no—
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—
339C4⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—88no—
343C22⋊C4—88no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23#53411yescharacteristic
374C23#53611yescharacteristic
375C23#53611yescharacteristic
376C23#54612yes—
377C23#54612yes—
378C23C22×SD1611yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380C23C2×#13711yescharacteristic
381C23#57511yescharacteristic
382C23#57612yes—
383C23#57812yes—
384Q8—816no—
385Q8—816no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
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—82no—
443C2×C4#53812yes—
444C2×C4#53812yes—
445C2×C4#54212yes—
446C2×C4#54212yes—
447C2×C4#54212yes—
448C2×C4#54212yes—
449C2×C4#55111yescharacteristic
450C2×C4#55111yescharacteristic
451C2×C4#55612yes—
452C2×C4#55612yes—
453C2×C4H(3,𝔽4)11yescharacteristic
454C2×C4H(3,𝔽4)11yescharacteristic
455C2×C4#56212yes—
456C2×C4#56212yes—
457C2×C4#56312yes—
458C2×C4#56312yes—
459C8—82no—
460C8—82no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—44no—
465C22—44no—
466C22—82no—
467C22—81no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#268411yescharacteristic
473C22#268411yescharacteristic
474C22#270012yes—
475C22#270012yes—
476C22#281711yescharacteristic
477C22#318711yescharacteristic
478C22#319311yescharacteristic
479C22#319911yescharacteristic
480C22#320211yescharacteristic
481C22#322711yescharacteristic
482C22#322911yescharacteristic
483C22#323511yescharacteristic
484C22#323711yescharacteristic
485C22#324312yes—
486C22#324312yes—
487C22#329512yes—
488C22#329812yes—
489C22#330812yes—
490C22#331312yes—
491C22#331412yes—
492C22#331912yes—
493C22#332512yes—
494C22#332812yes—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—84no—
500C4—84no—
501C4—81no—
502C4—162no—
503C4—162no—
504C4#275912yes—
505C4#275912yes—
506C4#283212yes—
507C4#283212yes—
508C2—82no—
509C2#1719811yescharacteristic
510C2#3126111yescharacteristic
511C2#3126211yescharacteristic
512C2#3127011yescharacteristic
513C2#3127311yescharacteristic
514C2#3174012yes—
515C2#3174212yes—
516C2#3176312yes—
517C2#3176912yes—
518C2#3278811yescharacteristic
519C2#3279211yescharacteristic
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. #11384C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect

Maximal subgroups

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

subgroupindex
#151252
#113842
#113162
#113152
#112002
#111992
#111472