Finite Group Databasebeta

← #474454

Subgroups — #474454

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14276C211yesmaximal, characteristic
2#14261C211yesmaximal, characteristic
3#11323C211yesmaximal, characteristic
4#11293C212yesmaximal
5#11268C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11122C212yesmaximal
8#11067C212yesmaximal
9#11052C211yesmaximal, characteristic
10#11050C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1443—22no—
33#1442—22no—
34#1441—22no—
35#1441—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×Q8C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C2×#106—24no—
54C2×#106—42no—
55C2×#106—42no—
56C2×#103—24no—
57C2×#103—24no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62#391—22no—
63#391—42no—
64#391—42no—
65#391Q814yes—
66#389—42no—
67#389—42no—
68#389—42no—
69#384—24no—
70#382—42no—
71#376—42no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D412yes—
84C2×#95D412yes—
85#340—24no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—24no—
91#336—22no—
92#336—22no—
93#336—42no—
94C22×Q8—21no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8C2×Q812yes—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
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×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8#4114yes—
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—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
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—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4C2×Q812yes—
160C2×C4⋊C4#4114yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42C2×Q812yes—
172C2×C42#4112yes—
173#106—48no—
174#103—48no—
175#103—48no—
176#95—28no—
177#95—42no—
178#95—42no—
179#95—44no—
180#95—42no—
181#95—44no—
182#95—44no—
183#95—42no—
184#95—42no—
185#95—82no—
186#95—82no—
187#95—82no—
188#95—82no—
189C24C22×D411yescharacteristic
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
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—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
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×C4C22⋊Q814yes—
266C22×C4C22×Q811yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274C22×C42−1+411yescharacteristic
275C22×C42−1+411yescharacteristic
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
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—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
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—42no—
304C4⋊C4—88no—
305C4⋊C4—84no—
306C4⋊C4—88no—
307C42—24no—
308C42—24no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C23C2×C22⋊Q811yescharacteristic
314C23#52511yescharacteristic
315C23#53211yescharacteristic
316C23#53411yescharacteristic
317C23#53511yescharacteristic
318C23#53511yescharacteristic
319C23#53611yescharacteristic
320C23#57712yes—
321C23#57712yes—
322C23#57712yes—
323C23#57812yes—
324Q8—44no—
325Q8—44no—
326Q8—82no—
327Q8—82no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#53812yes—
389C2×C4#53812yes—
390C2×C4#53812yes—
391C2×C4#54012yes—
392C2×C4#54211yescharacteristic
393C2×C4#54211yescharacteristic
394C2×C4#54411yescharacteristic
395C2×C4#54411yescharacteristic
396C2×C4#55212yes—
397C2×C4#55312yes—
398C2×C4#55312yes—
399C2×C4#55412yes—
400C2×C4#55412yes—
401C2×C4#55612yes—
402C2×C4#56212yes—
403C2×C4#56212yes—
404C8—44no—
405C8—82no—
406C22#256611yescharacteristic
407C22#257911yescharacteristic
408C22#260411yescharacteristic
409C22#262811yescharacteristic
410C22#263111yescharacteristic
411C22#266111yescharacteristic
412C22#281611yescharacteristic
413C22#308212yes—
414C22#308412yes—
415C22#312112yes—
416C22#312312yes—
417C22#317312yes—
418C22#317812yes—
419C22#320112yes—
420C22#320412yes—
421C22#322111yescharacteristic
422C22#322111yescharacteristic
423C22#322211yescharacteristic
424C22#322212yes—
425C22#322311yescharacteristic
426C22#322512yes—
427C22#324212yes—
428C22#324412yes—
429C4—44no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—82no—
434C4—84no—
435C4—82no—
436C4—161no—
437C4—161no—
438C4—162no—
439C4#276812yes—
440C4#277912yes—
441C4#278812yes—
442C4#280012yes—
443C2#1697811yescharacteristic
444C2#2919911yescharacteristic
445C2#2920511yescharacteristic
446C2#2925012yes—
447C2#2952512yes—
448C2#2999312yes—
449C2#3004911yescharacteristic
450C2#3005011yescharacteristic
451C2#3033412yes—
452C2#3278411yescharacteristic
453C2#3278511yescharacteristic
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
#142762
#142612
#113232
#112932
#112682
#112202
#111222
#110672
#110522
#110502
#107962