Finite Group Databasebeta

← #479720

Subgroups — #479720

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15195C211yesmaximal, characteristic
2#15150C211yesmaximal, characteristic
3#11323C212yesmaximal, complemented
4#11323C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11147C211yesmaximal, characteristic
8#11095C212yesmaximal
9#11095C212yesmaximal
10C22×#106C2212yes—
11C22×#106C2212yes—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#397—21no—
15C2×#397—21no—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1445—21no—
30#1445—21no—
31#1442—21no—
32#1442—21no—
33#1439—21no—
34#1439—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#106—42no—
57C2×#106—42no—
58C2×#106—42no—
59C2×#106—42no—
60C2×#106—42no—
61C2×#106—42no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66#397—44no—
67#397—44no—
68#384—41no—
69#384—41no—
70#384—41no—
71#384—41no—
72#380—42no—
73#380—42no—
74#380—42no—
75#380—42no—
76C4×C22⋊C4—41no—
77C4×C22⋊C4—41no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D411yescharacteristic
89C2×#95D411yescharacteristic
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—24no—
98C25C2×D411yescharacteristic
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8#4114yes—
130C22×C8#4114yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
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—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4SD1614yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185#106—48no—
186#106—48no—
187C22⋊C8—44no—
188C22⋊C8—44no—
189#95—41no—
190#95—41no—
191#95—42no—
192#95—44no—
193#95—44no—
194#95—42no—
195#95—42no—
196#95—44no—
197#95—42no—
198#95—44no—
199#95—82no—
200#95—82no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—22no—
205C24—22no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—22no—
210C24—22no—
211C24C2×SD1612yes—
212C24C22×D411yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
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—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×SD1612yes—
279C22×C4C2×SD1612yes—
280C22×C4AGL(1,ℤ/8)12yes—
281C22×C4#13712yes—
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)12yes—
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C42−1+412yes—
289C22×C42−1+412yes—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—88no—
327C22⋊C4—88no—
328C42—42no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C23—24no—
335C23—24no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23#53411yescharacteristic
358C23#53511yescharacteristic
359C23#53511yescharacteristic
360C23#54712yes—
361C23#54712yes—
362C23C22×SD1611yescharacteristic
363C23C2×AGL(1,ℤ/8)11yescharacteristic
364C23C2×#13711yescharacteristic
365C23#57511yescharacteristic
366C23#57712yes—
367C23#57712yes—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
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#53912yes—
426C2×C4#53912yes—
427C2×C4#54212yes—
428C2×C4#54212yes—
429C2×C4#54212yes—
430C2×C4#54212yes—
431C2×C4#55111yescharacteristic
432C2×C4#55111yescharacteristic
433C2×C4#55312yes—
434C2×C4#55312yes—
435C2×C4#56012yes—
436C2×C4#56012yes—
437C2×C4#56311yescharacteristic
438C2×C4#56311yescharacteristic
439C2×C4#56312yes—
440C2×C4#56312yes—
441C8—44no—
442C8—44no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—81no—
449C22—81no—
450C22—82no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22#271711yescharacteristic
455C22#271711yescharacteristic
456C22#271812yes—
457C22#271812yes—
458C22#281511yescharacteristic
459C22#318711yescharacteristic
460C22#319311yescharacteristic
461C22#320011yescharacteristic
462C22#320011yescharacteristic
463C22#320611yescharacteristic
464C22#320811yescharacteristic
465C22#321411yescharacteristic
466C22#321611yescharacteristic
467C22#322112yes—
468C22#322212yes—
469C22#333512yes—
470C22#333512yes—
471C22#334412yes—
472C22#334512yes—
473C22#334812yes—
474C22#335012yes—
475C22#335412yes—
476C22#335612yes—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—162no—
483C4—162no—
484C4—162no—
485C4—162no—
486C4#275712yes—
487C4#275712yes—
488C4#283812yes—
489C4#283812yes—
490C2—82no—
491C2#1722911yescharacteristic
492C2#3225911yescharacteristic
493C2#3226011yescharacteristic
494C2#3226911yescharacteristic
495C2#3227011yescharacteristic
496C2#3228012yes—
497C2#3228212yes—
498C2#3229712yes—
499C2#3229912yes—
500C2#3276011yescharacteristic
501C2#3276611yescharacteristic
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. #11323C2semidirect
4. #11323C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#151502
#113232
#113162
#113152
#111472
#110952