Finite Group Databasebeta

← #405007

Subgroups — #405007

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic
2C2×#1919C211yesmaximal, characteristic
3#15153C211yesmaximal, characteristic
4#13812C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11257C212yesmaximal
8#11255C211yesmaximal, characteristic
9#11227C211yesmaximal, characteristic
10#10908C212yesmaximal
11#10908C212yesmaximal
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C2×#400—21no—
16C2×#400—21no—
17C2×#389C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1948—24no—
22#1919—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1445—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39#1434—21no—
40#1434—22no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—22no—
56C22×C42—21no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67#400—44no—
68#400—44no—
69#389—22no—
70#389—22no—
71#384—22no—
72#384—22no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4D414yes—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95D411yescharacteristic
89C2×#95D412yes—
90C2×#95D411yescharacteristic
91#340—24no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#336—24no—
100#336—24no—
101#336—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2411yescharacteristic
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8#4112yes—
129C22×C8#4114yes—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
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⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4#4114yes—
168C2×C4⋊C4#4112yes—
169C2×C4⋊C4#4114yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—28no—
179C2×C42—24no—
180C2×C42—82no—
181C2×C42—82no—
182C2×C42—82no—
183C2×C42C2×D412yes—
184C2×C42#4112yes—
185#106—28no—
186#106—48no—
187C4⋊C8—28no—
188#95—28no—
189#95—24no—
190#95—22no—
191#95—42no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—42no—
196#95—44no—
197#95—44no—
198#95—42no—
199#95—81no—
200#95—82no—
201#95SD1614yes—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
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—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4#12112yes—
259C22×C4#12112yes—
260C22×C4C2×SD1614yes—
261C22×C4C2×SD1612yes—
262C22×C4AGL(1,ℤ/8)12yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4112yes—
270C22×C42−1+412yes—
271C22×C42−1+411yescharacteristic
272C22×C42−1+411yescharacteristic
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4#12414yes—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—84no—
323C42—88no—
324C23C2×#12111yescharacteristic
325C23#52512yes—
326C23#54712yes—
327C23#54711yescharacteristic
328C23#54811yescharacteristic
329C23C22×SD1611yescharacteristic
330C23C2×AGL(1,ℤ/8)11yescharacteristic
331C23#57512yes—
332C23#57712yes—
333C23#57711yescharacteristic
334C23#57811yescharacteristic
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
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—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4#46014yes—
395C2×C4#52511yescharacteristic
396C2×C4#52511yescharacteristic
397C2×C4C2×#12411yescharacteristic
398C2×C4#52912yes—
399C2×C4#52912yes—
400C2×C4#52912yes—
401C2×C4#53211yescharacteristic
402C2×C4#53611yescharacteristic
403C2×C4#53611yescharacteristic
404C2×C4#54112yes—
405C2×C4#54112yes—
406C2×C4#55212yes—
407C2×C4#55212yes—
408C2×C4#55212yes—
409C2×C4#55212yes—
410C2×C4#55211yescharacteristic
411C2×C4#55612yes—
412C2×C4#55611yescharacteristic
413C8—42no—
414C8—44no—
415C8—81no—
416C22#241311yescharacteristic
417C22#241511yescharacteristic
418C22#247512yes—
419C22#247812yes—
420C22#285011yescharacteristic
421C22C2×#46011yescharacteristic
422C22#304011yescharacteristic
423C22#304211yescharacteristic
424C22#304611yescharacteristic
425C22#310312yes—
426C22#311512yes—
427C22#312012yes—
428C22#312312yes—
429C22#333512yes—
430C22#333511yescharacteristic
431C22#334512yes—
432C22#334912yes—
433C22#335012yes—
434C22#335112yes—
435C22#335311yescharacteristic
436C22#335911yescharacteristic
437C22#337112yes—
438C22#337311yescharacteristic
439C4—24no—
440C4—44no—
441C4—42no—
442C4—44no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—84no—
447C4—162no—
448C4—161no—
449C4#239811yescharacteristic
450C4#252812yes—
451C4#253312yes—
452C4#260411yescharacteristic
453C4#261111yescharacteristic
454C4#264211yescharacteristic
455C2#1490811yescharacteristic
456C2#2457811yescharacteristic
457C2#2459311yescharacteristic
458C2#2467811yescharacteristic
459C2#2469711yescharacteristic
460C2#2736212yes—
461C2#2738612yes—
462C2#2745812yes—
463C2#2748212yes—
464C2#3302511yescharacteristic
465C2#3304011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#19482
C2×#19192
#151532
#138122
#113572
#113162
#112572
#112552
#112272
#109082