Finite Group Databasebeta

← #416969

Subgroups — #416969

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic
4C2×C4×#106C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#13911C211yesmaximal, characteristic
7#11316C214yesmaximal
8#11231C214yesmaximal
9#10849C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×#391—21no—
16C2×#391—21no—
17C2×#384C2211yescharacteristic
18C2×#382—21no—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1990—24no—
23#1988—24no—
24#1939—24no—
25C4×#106—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32#1445—22no—
33#1442—22no—
34C4⋊#95—21no—
35C4⋊#95C2214yes—
36C4⋊#95C2214yes—
37C23×Q8—21no—
38C23×C8C2312yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8Q814yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8D414yes—
71#391—44no—
72#391—44no—
73#384—22no—
74#384—22no—
75#382—44no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4D414yes—
82C2×#95—21no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95—21no—
87C2×#95—24no—
88C2×#95D412yes—
89#340—24no—
90#340—24no—
91#340—28no—
92#340—24no—
93#340—28no—
94#336—24no—
95C22×Q8—24no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—42no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—24no—
123C22×C8C2×D412yes—
124C22×C8C2×D412yes—
125C22×C8#4114yes—
126C4⋊Q8—28no—
127C4⋊Q8—28no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
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—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—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4SD1614yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×Q814yes—
165C2×C4⋊C4#4114yes—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×Q812yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182#106—216no—
183C4⋊C8—48no—
184C4⋊C8—44no—
185C4×C8—28no—
186#95—24no—
187#95—24no—
188#95—48no—
189#95—48no—
190#95—42no—
191#95—48no—
192#95—84no—
193C24C22×D411yescharacteristic
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8—48no—
197C2×Q8—84no—
198C2×Q8—84no—
199C2×Q8—84no—
200C2×Q8—84no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
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—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4#12112yes—
239C22×C4#12112yes—
240C22×C4#12112yes—
241C22×C4#12112yes—
242C22×C4C22⋊Q812yes—
243C22×C4C22⋊Q812yes—
244C22×C4C2×SD1614yes—
245C22×C4C2×SD1612yes—
246C22×C4#13512yes—
247C22×C4AGL(1,ℤ/8)12yes—
248C22×C4AGL(1,ℤ/8)12yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C42−1+412yes—
258C22×C42−1+412yes—
259C2×C8—22no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—44no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8#12414yes—
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42—48no—
298C42—84no—
299C42#12114yes—
300C42C22⋊Q814yes—
301C42C22⋊Q814yes—
302C42#12614yes—
303C23C2×#12111yescharacteristic
304C23C2×#12111yescharacteristic
305C23C2×C22⋊Q811yescharacteristic
306C23#54712yes—
307C23#54812yes—
308C23C22×SD1611yescharacteristic
309C23C2×#13511yescharacteristic
310C23C2×AGL(1,ℤ/8)11yescharacteristic
311C23C2×AGL(1,ℤ/8)11yescharacteristic
312C23#57712yes—
313C23#57812yes—
314Q8—816no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#39614yes—
356C2×C4#39614yes—
357C2×C4#46014yes—
358C2×C4#46314yes—
359C2×C4#46514yes—
360C2×C4#46914yes—
361C2×C4#47614yes—
362C2×C4#47814yes—
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4#52512yes—
367C2×C4C2×#12411yescharacteristic
368C2×C4C2×#12611yescharacteristic
369C2×C4#52912yes—
370C2×C4#53211yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4#53611yescharacteristic
373C2×C4#53612yes—
374C2×C4#54112yes—
375C2×C4#55212yes—
376C2×C4#55612yes—
377C2×C4#55612yes—
378C2×C4#55612yes—
379C2×C4#55712yes—
380C2×C4#55712yes—
381C8—24no—
382C8—42no—
383C8—82no—
384C22C2×#39611yescharacteristic
385C22#241512yes—
386C22#245312yes—
387C22#254511yescharacteristic
388C22#254711yescharacteristic
389C22C2×#46311yescharacteristic
390C22C2×#46011yescharacteristic
391C22#304612yes—
392C22C2×#46511yescharacteristic
393C22C2×#46911yescharacteristic
394C22#305711yescharacteristic
395C22#305911yescharacteristic
396C22C2×#47611yescharacteristic
397C22C2×#47811yescharacteristic
398C22#308212yes—
399C22#333512yes—
400C22#334012yes—
401C22#335312yes—
402C22#335312yes—
403C22#335912yes—
404C22#336412yes—
405C22#336812yes—
406C22#337312yes—
407C4—24no—
408C4—44no—
409C4—82no—
410C4—84no—
411C4—84no—
412C4—84no—
413C4—162no—
414C4#206518yes—
415C4#251412yes—
416C4#256511yescharacteristic
417C4#256811yescharacteristic
418C4#258112yes—
419C4#262111yescharacteristic
420C4#285011yescharacteristic
421C2#1523711yescharacteristic
422C2C2×#206511yescharacteristic
423C2#2361311yescharacteristic
424C2#2467812yes—
425C2#2468112yes—
426C2#2650912yes—
427C2#2651012yes—
428C2#2863911yescharacteristic
429C2#2864811yescharacteristic
430C2#2870611yescharacteristic
431C2#2871211yescharacteristic
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×#19902
C2×#19882
C2×#19392
C2×C4×#1062
#141552
#139112
#113162
#112312
#108492