Finite Group Databasebeta

← #453734

Subgroups — #453734

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13926C211yesmaximal, characteristic
2#13926C211yesmaximal, characteristic
3#11380C211yesmaximal, characteristic
4#11316C212yesmaximal
5#11315C212yesmaximal
6#11267C212yesmaximal
7#11267C212yesmaximal
8#10807C212yesmaximal
9#10806C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11C2×#391—21no—
12C2×#391—21no—
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1441—21no—
30#1441—21no—
31#1441—21no—
32#1441—21no—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8D411yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4×Q8—24no—
43C2×C4×Q8—24no—
44C2×C4×Q8—41no—
45C2×C4×Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#391—44no—
63#391—44no—
64#384—41no—
65#384—41no—
66#384—41no—
67#384—41no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—24no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—42no—
87#340—42no—
88#336—24no—
89#336—24no—
90#336—41no—
91#336—41no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8SD1614yes—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2411yescharacteristic
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C4×Q8—48no—
125C4×Q8—48no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
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—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
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—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4#4114yes—
168C2×C4⋊C4#4114yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42#4114yes—
186C2×C42#4114yes—
187#95—44no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197C24C22×D411yescharacteristic
198C2×Q8—28no—
199C2×Q8—28no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—24no—
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—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
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#12312yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4#12312yes—
264C22×C4C2×SD1612yes—
265C22×C4C2×SD1612yes—
266C22×C4C2×SD1612yes—
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4#13712yes—
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+412yes—
276C22×C42−1+412yes—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—84no—
321C42—84no—
322C23C2×#12311yescharacteristic
323C23C2×#12311yescharacteristic
324C23#53411yescharacteristic
325C23#54812yes—
326C23#54812yes—
327C23C22×SD1611yescharacteristic
328C23C2×AGL(1,ℤ/8)11yescharacteristic
329C23C2×#13711yescharacteristic
330C23#57511yescharacteristic
331C23#57612yes—
332C23#57812yes—
333Q8—416no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#48114yes—
397C2×C4#48314yes—
398C2×C4#52912yes—
399C2×C4#52912yes—
400C2×C4#54112yes—
401C2×C4#54112yes—
402C2×C4#54212yes—
403C2×C4#54212yes—
404C2×C4#55112yes—
405C2×C4#55112yes—
406C2×C4#55212yes—
407C2×C4#55212yes—
408C2×C4#55511yescharacteristic
409C2×C4#55511yescharacteristic
410C2×C4#55612yes—
411C2×C4#55612yes—
412C2×C4#56311yescharacteristic
413C2×C4#56311yescharacteristic
414C8—82no—
415C8—82no—
416C22#242112yes—
417C22#242112yes—
418C22#248311yescharacteristic
419C22#273111yescharacteristic
420C22#273111yescharacteristic
421C22C2×#48311yescharacteristic
422C22C2×#48111yescharacteristic
423C22#309411yescharacteristic
424C22#309511yescharacteristic
425C22#309812yes—
426C22#309812yes—
427C22#318711yescharacteristic
428C22#319311yescharacteristic
429C22#319911yescharacteristic
430C22#320211yescharacteristic
431C22#335912yes—
432C22#336212yes—
433C22#336612yes—
434C22#336712yes—
435C22#337012yes—
436C22#337412yes—
437C22#337512yes—
438C22#337712yes—
439C4—24no—
440C4—24no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—84no—
445C4—84no—
446C4—81no—
447C4—162no—
448C4—162no—
449C4#253212yes—
450C4#253212yes—
451C4#276212yes—
452C4#276212yes—
453C2#1611211yescharacteristic
454C2#2499212yes—
455C2#2499412yes—
456C2#2500712yes—
457C2#2500812yes—
458C2#2760111yescharacteristic
459C2#2760811yescharacteristic
460C2#3262911yescharacteristic
461C2#3263011yescharacteristic
462C2#3264411yescharacteristic
463C2#3264711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ 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
#139262
#113802
#113162
#113152
#112672
#108072
#108062