Finite Group Databasebeta

← #470387

Subgroups — #470387

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11293C212yesmaximal
9#11268C212yesmaximal
10#11208C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13#2571—24no—
14#2571—24no—
15C2×#400—22no—
16C2×#398—21no—
17C2×#389—21no—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#382C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2099—24no—
27#1919—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—22no—
42#1441—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67#400—216no—
68#398—44no—
69#389—28no—
70#389—24no—
71#389—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389—42no—
76#384—22no—
77#384D414yes—
78#382—28no—
79#382—24no—
80#382—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yes—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95D412yes—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—22no—
108#336—42no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8#4112yes—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4#4114yes—
187C2×C4⋊C4#4112yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200#106—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—24no—
205#95—24no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—42no—
210#95—42no—
211#95—81no—
212C24C22×D411yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
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—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4#12012yes—
270C22×C4#12012yes—
271C22×C4#13512yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C42−1+412yes—
280C22×C42−1+412yes—
281C22×C42−1+412yes—
282C22×C42−1+411yescharacteristic
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—84no—
326C4⋊C4—88no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—48no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C23C2×#12011yescharacteristic
338C23#53612yes—
339C23#54611yescharacteristic
340C23#54711yescharacteristic
341C23#54812yes—
342C23C2×#13511yescharacteristic
343C23C2×AGL(1,ℤ/8)11yescharacteristic
344C23#57512yes—
345C23#57711yescharacteristic
346C23#57712yes—
347C23#57811yescharacteristic
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#44914yes—
408C2×C4#52511yescharacteristic
409C2×C4#53211yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4#53612yes—
414C2×C4#53711yescharacteristic
415C2×C4#53912yes—
416C2×C4#54112yes—
417C2×C4#54112yes—
418C2×C4#54112yes—
419C2×C4#54712yes—
420C2×C4#54811yescharacteristic
421C2×C4#55011yescharacteristic
422C2×C4#55212yes—
423C2×C4#55212yes—
424C2×C4#56312yes—
425C2×C4#56412yes—
426C8—42no—
427C8—81no—
428C8—82no—
429C22#240711yescharacteristic
430C22#243211yescharacteristic
431C22#261411yescharacteristic
432C22#269012yes—
433C22#271912yes—
434C22C2×#44911yescharacteristic
435C22#300511yescharacteristic
436C22#300911yescharacteristic
437C22#301311yescharacteristic
438C22#323212yes—
439C22#323412yes—
440C22#324212yes—
441C22#324412yes—
442C22#330211yescharacteristic
443C22#332112yes—
444C22#332411yescharacteristic
445C22#334011yescharacteristic
446C22#334812yes—
447C22#335311yescharacteristic
448C22#336512yes—
449C22#337312yes—
450C22#337612yes—
451C22#337712yes—
452C4—24no—
453C4—42no—
454C4—84no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—84no—
461C4—161no—
462C4#261111yescharacteristic
463C4#263411yescharacteristic
464C4#266411yescharacteristic
465C4#271912yes—
466C4#272112yes—
467C4#282211yescharacteristic
468C2#1668311yescharacteristic
469C2#2411911yescharacteristic
470C2#2413311yescharacteristic
471C2#2565211yescharacteristic
472C2#2565611yescharacteristic
473C2#2964011yescharacteristic
474C2#2964811yescharacteristic
475C2#3146912yes—
476C2#3148312yes—
477C2#3232512yes—
478C2#3233912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊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×#25712
C2×#20992
C2×#19192
#151492
#113572
#113462
#113272
#112932
#112682
#112082
#108072