Finite Group Databasebeta

← #403717

Subgroups — #403717

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#2059C212yesmaximal
3#14283C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11323C211yesmaximal, characteristic
8#11207C212yesmaximal
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16#2330—28no—
17C2×#389—22no—
18C2×#389C2212yes—
19C2×#387—22no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×#376C2211yescharacteristic
24#2059—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1441—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—24no—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125—42no—
50C2×#125Q814yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74#389—216no—
75#389—24no—
76#389D418yes—
77#387—216no—
78#384—24no—
79#384—42no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82#376—24no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95D412yes—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—42no—
96#340—42no—
97#336—24no—
98#336—22no—
99#336—22no—
100#336Q818yes—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
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—
124C22×C8C2×Q814yes—
125C22×C8C2×Q814yes—
126#125—216no—
127#125—216no—
128#125—48no—
129#125—48no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—28no—
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—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
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—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4C2×Q814yes—
174C2×C4⋊C4#4112yes—
175C2×C4⋊C4#4112yes—
176C2×C4⋊C4#4114yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—28no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42C2×D414yes—
189C2×C42C2×Q812yes—
190C2×C42C2×Q814yes—
191C2×C42#4112yes—
192#107—48no—
193#106—48no—
194#95—28no—
195#95—28no—
196#95—24no—
197#95—24no—
198#95—82no—
199#95—82no—
200#95—82no—
201#95—82no—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
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—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
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—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4#12012yes—
258C22×C4#12012yes—
259C22×C4C22⋊Q814yes—
260C22×C4C4⋊Q814yes—
261C22×C4C22×D412yes—
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q812yes—
264C22×C4C2×#4112yes—
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4H(5,𝔽2)12yes—
269C22×C42−1+411yescharacteristic
270C22×C42−1+411yescharacteristic
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—232no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—88no—
304C4⋊C4—88no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4C22⋊Q818yes—
308C42—24no—
309C42—24no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—88no—
315C23C2×#12011yescharacteristic
316C23C2×C22⋊Q811yescharacteristic
317C23#52511yescharacteristic
318C23#52511yescharacteristic
319C23C2×C4⋊Q811yescharacteristic
320C23#53511yescharacteristic
321C23#53611yescharacteristic
322C23#57612yes—
323C23#57712yes—
324C23#57712yes—
325C23#57812yes—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4C2×C22⋊Q812yes—
385C2×C4#52512yes—
386C2×C4#53312yes—
387C2×C4#53312yes—
388C2×C4#54212yes—
389C2×C4#54212yes—
390C2×C4#54612yes—
391C2×C4#54612yes—
392C2×C4D4×Q812yes—
393C2×C4D4×Q812yes—
394C2×C4#55412yes—
395C2×C4#55611yescharacteristic
396C2×C4#55611yescharacteristic
397C2×C4#55711yescharacteristic
398C2×C4#55711yescharacteristic
399C2×C4Q8212yes—
400C8—44no—
401C8—44no—
402C22#239911yescharacteristic
403C22#240411yescharacteristic
404C22#243411yescharacteristic
405C22#256711yescharacteristic
406C22#258111yescharacteristic
407C22#263211yescharacteristic
408C22#284811yescharacteristic
409C22#301212yes—
410C22#301312yes—
411C22#308312yes—
412C22#308412yes—
413C22#311912yes—
414C22#311912yes—
415C22#312012yes—
416C22#312012yes—
417C22#316312yes—
418C22#316412yes—
419C22#322011yescharacteristic
420C22#322111yescharacteristic
421C22#322211yescharacteristic
422C22#322311yescharacteristic
423C22#324112yes—
424C22#324212yes—
425C4—42no—
426C4—42no—
427C4—44no—
428C4—48no—
429C4—44no—
430C4—82no—
431C4—84no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4#245112yes—
436C4#245112yes—
437C4#246512yes—
438C4#246512yes—
439C2#1488111yescharacteristic
440C2#2384312yes—
441C2#2394111yescharacteristic
442C2#2394411yescharacteristic
443C2#2582212yes—
444C2#2921511yescharacteristic
445C2#2921611yescharacteristic
446C2#2928512yes—
447C2#3007111yescharacteristic
448C2#3007311yescharacteristic
449C2#3296812yes—
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×#23302
C2×#20592
#142832
#113852
#113242
#113232
#112072
#109082
#109072
#107962