Finite Group Databasebeta

← #468343

Subgroups — #468343

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3#14283C211yesmaximal, characteristic
4#11324C211yesmaximal, characteristic
5#11323C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11220C212yesmaximal
9#11219C212yesmaximal
10#11207C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15#2647—28no—
16C2×#400—22no—
17C2×#398—22no—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#2059—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1441—22no—
36#1441—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—42no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—42no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69#400—216no—
70#398—216no—
71#389—24no—
72#389—24no—
73#389—24no—
74#389D418yes—
75#384—24no—
76#384—42no—
77#382—24no—
78#376—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#336—24no—
95#336—42no—
96#336Q814yes—
97#336Q814yes—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2411yescharacteristic
109C22×C8—22no—
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×C8C2×Q814yes—
122C22×C8C2×Q814yes—
123#125—48no—
124#125—48no—
125C2×C4⋊C4—22no—
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—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4#4112yes—
171C2×C4⋊C4#4112yes—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42C2×D414yes—
182C2×C42C2×Q812yes—
183C2×C42C2×Q812yes—
184#107—48no—
185#106—48no—
186#95—28no—
187#95—28no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195C24C22×D411yescharacteristic
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4#12012yes—
252C22×C4#12012yes—
253C22×C4C4⋊Q814yes—
254C22×C4C22×D412yes—
255C22×C4C22×Q811yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4H(5,𝔽2)11yescharacteristic
260C22×C4H(5,𝔽2)12yes—
261C22×C42−1+411yescharacteristic
262C22×C42−1+412yes—
263C22×C42−1+412yes—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—88no—
297C4⋊C4—88no—
298C4⋊C4—88no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—42no—
306C42—42no—
307C23C2×#12011yescharacteristic
308C23#52511yescharacteristic
309C23C2×C4⋊Q811yescharacteristic
310C23#53511yescharacteristic
311C23#53611yescharacteristic
312C23#53611yescharacteristic
313C23#53711yescharacteristic
314C23#57612yes—
315C23#57712yes—
316C23#57712yes—
317C23#57812yes—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4#53612yes—
377C2×C4#53712yes—
378C2×C4#54112yes—
379C2×C4#54112yes—
380C2×C4#54412yes—
381C2×C4#54412yes—
382C2×C4#54612yes—
383C2×C4#54612yes—
384C2×C4D4×Q812yes—
385C2×C4D4×Q812yes—
386C2×C4#55411yescharacteristic
387C2×C4#55411yescharacteristic
388C2×C4#55711yescharacteristic
389C2×C4#55711yescharacteristic
390C2×C4#56212yes—
391C2×C4#56312yes—
392C8—44no—
393C8—44no—
394C22#240411yescharacteristic
395C22#243411yescharacteristic
396C22#258411yescharacteristic
397C22#261611yescharacteristic
398C22#263411yescharacteristic
399C22#266011yescharacteristic
400C22#282011yescharacteristic
401C22#301212yes—
402C22#301312yes—
403C22#311912yes—
404C22#312012yes—
405C22#316312yes—
406C22#316412yes—
407C22#322011yescharacteristic
408C22#322111yescharacteristic
409C22#322211yescharacteristic
410C22#322311yescharacteristic
411C22#324112yes—
412C22#324212yes—
413C22#324312yes—
414C22#324412yes—
415C22#325712yes—
416C22#325812yes—
417C4—42no—
418C4—42no—
419C4—44no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4—84no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4#270412yes—
428C4#270412yes—
429C4#271312yes—
430C4#271312yes—
431C2#1660811yescharacteristic
432C2#2394111yescharacteristic
433C2#2394411yescharacteristic
434C2#2582212yes—
435C2#2932412yes—
436C2#2970212yes—
437C2#3010312yes—
438C2#3031311yescharacteristic
439C2#3031411yescharacteristic
440C2#3283111yescharacteristic
441C2#3283211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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
C2×#26472
C2×#20592
#142832
#113242
#113232
#112682
#112202
#112192
#112072
#107962