Finite Group Databasebeta

← #358622

Subgroups — #358622

Order 512.

1034 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
1C2×#1768C211yesmaximal, characteristic
2C2×#1745C211yesmaximal, characteristic
3#13759C211yesmaximal, characteristic
4#11346C212yesmaximal
5#11298C212yesmaximal
6#11255C212yesmaximal
7#11202C212yesmaximal
8#10694C211yesmaximal, characteristic
9#10694C211yesmaximal, characteristic
10#10694C211yesmaximal, characteristic
11C4×#336C211yesmaximal, characteristic
12C22×C4⋊C8C2212yes—
13C22×#97C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376C2211yescharacteristic
20C2×#376C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#1768—24no—
23#1745—24no—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1436C412yes—
36#1436C412yes—
37#1436C412yes—
38#1436C412yes—
39C4×#95C2211yescharacteristic
40C4×#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C42C2×C412yes—
48C22×C42C2×C412yes—
49C22×C42C2311yescharacteristic
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—24no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—24no—
59C2×#97—22no—
60C2×#97—22no—
61C2×#97—22no—
62C2×#97—22no—
63C2×#97D414yes—
64C2×#97Q814yes—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73#376—22no—
74#376—22no—
75#376—22no—
76#376—22no—
77C2×#95C2×C412yes—
78C2×#95C2×C412yes—
79C2×#95C2×C411yescharacteristic
80C2×#95C2×C411yescharacteristic
81C2×#95C2×C411yescharacteristic
82C2×#95C2×C411yescharacteristic
83C2×#95C2×C412yes—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C412yes—
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89C43—22no—
90C43—22no—
91#336—24no—
92#336—24no—
93#336—24no—
94#336—24no—
95#336—24no—
96#336—24no—
97#336—24no—
98#336—24no—
99#336—24no—
100#336—24no—
101#336—24no—
102C23×C4C22×C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C2411yescharacteristic
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×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C22×C8C2×Q814yes—
139C22×C8#4114yes—
140C22×C8#4114yes—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C2×D412yes—
176C2×C42C2×Q812yes—
177C2×C42#4114yes—
178C2×C42#4114yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183C4⋊C8—216no—
184#97—28no—
185#97—28no—
186C4×C8—44no—
187C4×C8—44no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—42no—
199#95—42no—
200C24C23×C411yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
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—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4#11712yes—
249C22×C4#11714yes—
250C22×C4#11712yes—
251C22×C4#11712yes—
252C22×C4#11712yes—
253C22×C4C4×D412yes—
254C22×C4C4×D412yes—
255C22×C4C4×Q812yes—
256C22×C4C4×Q812yes—
257C22×C4#13114yes—
258C22×C4#13114yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×Q811yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4112yes—
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
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—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—48no—
304C42—48no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42C22⋊Q814yes—
314C42#12314yes—
315C42#12514yes—
316C42#12614yes—
317C23C2×#11711yescharacteristic
318C23C2×#11711yescharacteristic
319C23C2×#11711yescharacteristic
320C23C2×C4×D411yescharacteristic
321C23C2×C4×Q811yescharacteristic
322C23C4×#4111yescharacteristic
323C23C4×#4111yescharacteristic
324C23C2×#13112yes—
325C23C2×#13112yes—
326C23#56812yes—
327C23#56812yes—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4#38714yes—
385C2×C4#38714yes—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×#12311yescharacteristic
388C2×C4#52512yes—
389C2×C4C2×#12511yescharacteristic
390C2×C4C2×#12611yescharacteristic
391C2×C4#52912yes—
392C2×C4#52912yes—
393C2×C4#52912yes—
394C2×C4#52912yes—
395C2×C4#52912yes—
396C2×C4#52912yes—
397C2×C4#52912yes—
398C2×C4#52912yes—
399C2×C4#53212yes—
400C2×C4#53312yes—
401C2×C4#53312yes—
402C8—44no—
403C8—44no—
404C8—44no—
405C8—44no—
406C22#228311yescharacteristic
407C22C2×#38711yescharacteristic
408C22#229011yescharacteristic
409C22#229311yescharacteristic
410C22#229511yescharacteristic
411C22#229511yescharacteristic
412C22#229811yescharacteristic
413C22#292211yescharacteristic
414C22#292211yescharacteristic
415C22#292311yescharacteristic
416C22#292311yescharacteristic
417C22#292412yes—
418C22#292412yes—
419C22#292512yes—
420C22#292512yes—
421C22#293312yes—
422C22#293312yes—
423C22#296212yes—
424C22#296212yes—
425C22#296612yes—
426C22#296612yes—
427C22#296712yes—
428C22#296712yes—
429C4—24no—
430C4—24no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—44no—
435C4—44no—
436C4—82no—
437C4—81no—
438C4—81no—
439C4#239512yes—
440C4#239612yes—
441C4#240112yes—
442C4#240212yes—
443C2#1384611yescharacteristic
444C2#2078011yescharacteristic
445C2#2078511yescharacteristic
446C2#2102211yescharacteristic
447C2#2102211yescharacteristic
448C2#2111212yes—
449C2#2127312yes—
450C2#2134812yes—
451C2#2134812yes—
452C2#2143211yescharacteristic
453C2#2143311yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socle, prime radicalC24C23×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×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
C2×#17682
C2×#17452
#137592
#113462
#112982
#112552
#112022
#106942
C4×#3362