Finite Group Databasebeta

← #468006

Subgroups — #468006

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15152C211yesmaximal, characteristic
2#14155C211yesmaximal, characteristic
3#11202C211yesmaximal, characteristic
4#11124C212yesmaximal
5#11124C212yesmaximal
6#11077C212yesmaximal
7#11050C212yesmaximal
8#11010C212yesmaximal
9#10907C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#400—21no—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29#1445—21no—
30#1445—21no—
31#1445—21no—
32#1442—21no—
33#1441—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×Q8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8Q814yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—24no—
62C2×#107—42no—
63C2×#107—42no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8—42no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78#400—44no—
79#391—24no—
80#391—24no—
81#391—42no—
82#391—42no—
83#384—41no—
84#384—41no—
85C4×C4⋊C4—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C4×C4⋊C4—41no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102#340—24no—
103#340—42no—
104#340—42no—
105#336—24no—
106#336—41no—
107#336—42no—
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—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8C2×Q814yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
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#4114yes—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4Q1614yes—
185C2×C4⋊C4#4114yes—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C2×Q812yes—
202C2×C42#4112yes—
203#107—48no—
204C4⋊C8—44no—
205#103—48no—
206#103—48no—
207#103—48no—
208#103—48no—
209#95—41no—
210#95—41no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—44no—
219#95—44no—
220#95—82no—
221#95—82no—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
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—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
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—24no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
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×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C2×Q1612yes—
307C22×C4C2×Q1612yes—
308C22×C4C2×Q1612yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
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—42no—
325C2×C8—42no—
326C2×C8—42no—
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—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
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—84no—
352C4⋊C4—84no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—42no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—84no—
364C23C2×C22⋊Q811yescharacteristic
365C23#53411yescharacteristic
366C23#53511yescharacteristic
367C23#54612yes—
368C23#55012yes—
369C23C22×Q1611yescharacteristic
370C23C2×AGL(1,ℤ/8)11yescharacteristic
371C23C2×#13711yescharacteristic
372C23#57511yescharacteristic
373C23#57712yes—
374C23#57812yes—
375Q8—44no—
376Q8—44no—
377Q8—44no—
378Q8—44no—
379Q8—82no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
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#47714yes—
443C2×C4#52912yes—
444C2×C4#53811yescharacteristic
445C2×C4#53811yescharacteristic
446C2×C4#53812yes—
447C2×C4#53812yes—
448C2×C4#53812yes—
449C2×C4#54012yes—
450C2×C4#54012yes—
451C2×C4#54311yescharacteristic
452C2×C4#54311yescharacteristic
453C2×C4#55312yes—
454C2×C4#55412yes—
455C2×C4#55412yes—
456C2×C4#55511yescharacteristic
457C2×C4#55911yescharacteristic
458C2×C4#56012yes—
459C2×C4#56011yescharacteristic
460C2×C4H(3,𝔽4)11yescharacteristic
461C8—44no—
462C8—82no—
463C22#245412yes—
464C22#262611yescharacteristic
465C22#270911yescharacteristic
466C22#271412yes—
467C22#273311yescharacteristic
468C22C2×#47711yescharacteristic
469C22#307811yescharacteristic
470C22#308212yes—
471C22#318911yescharacteristic
472C22#319311yescharacteristic
473C22#320011yescharacteristic
474C22#320211yescharacteristic
475C22#320911yescharacteristic
476C22#321311yescharacteristic
477C22#322512yes—
478C22#329912yes—
479C22#330712yes—
480C22#331212yes—
481C22#333012yes—
482C22#339512yes—
483C22#339712yes—
484C22#340012yes—
485C22#340212yes—
486C4—24no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—84no—
493C4—84no—
494C4—162no—
495C4—162no—
496C4#259311yescharacteristic
497C4#264812yes—
498C4#278012yes—
499C4#278411yescharacteristic
500C4#280411yescharacteristic
501C4#282811yescharacteristic
502C2#1659911yescharacteristic
503C2#2653212yes—
504C2#2653912yes—
505C2#2995311yescharacteristic
506C2#2996011yescharacteristic
507C2#3202911yescharacteristic
508C2#3203911yescharacteristic
509C2#3216812yes—
510C2#3218912yes—
511C2#3267011yescharacteristic
512C2#3268011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#151522
#141552
#112022
#111242
#110772
#110502
#110102
#109072