Finite Group Databasebeta

← #275272

Subgroups — #275272

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33107C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#22625C211yesmaximal, characteristic, complemented
4C2×#1947C211yesmaximal, characteristic, complemented
5#10658C212yesmaximal, complemented
6#10587C212yesmaximal, complemented
7#9613C212yesmaximal, complemented
8#9563C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic, complemented
11#8725C211yesmaximal, characteristic, complemented
12C2×#481—21no—
13C22×#107—21no—
14C22×C4⋊C8C2211yescharacteristic
15#2856C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2071—22no—
19#1982C2211yescharacteristic
20#1982C2211yescharacteristic
21#1957C2212yes—
22#1948C2211yescharacteristic
23#1948C2211yescharacteristic
24#1947—21no—
25#1947—21no—
26#1947C2212yes—
27#1947C2212yes—
28#1939C2212yes—
29#1926—21no—
30#1926—21no—
31#1916—22no—
32#1874—21no—
33#1874—21no—
34#1874—21no—
35#1834C2212yes—
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C8×C4⋊C4C2212yes—
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C2×#127C2311yescharacteristic
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#481—44no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—44no—
65C2×#106—21no—
66C2×#106—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×#102—21no—
75C2×#102—21no—
76C2×#102—21no—
77C2×#102—21no—
78C2×#102—22no—
79C2×#102—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8D411yescharacteristic
84C2×C4×C8—22no—
85C2×C4×C8C2312yes—
86#390—21no—
87#390—21no—
88#390—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2311yescharacteristic
93#384C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C2×#95—21no—
101#340—22no—
102#340—22no—
103#340—21no—
104#340—21no—
105#340—22no—
106#340—22no—
107#336—22no—
108#336—22no—
109#336—21no—
110#336—21no—
111#336D412yes—
112#334—24no—
113#331—24no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—41no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4#4111yescharacteristic
124C22×C8—24no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133#127—22no—
134#127—22no—
135#121—42no—
136#121—42no—
137#121—42no—
138#121—42no—
139#121—82no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4SD1614yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—82no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#107—44no—
178#107—44no—
179#106—42no—
180#106—42no—
181C4⋊C8—24no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—82no—
188C4⋊C8#4114yes—
189#102—42no—
190#102—42no—
191#102—42no—
192#102—42no—
193#102—82no—
194#102—82no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—82no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206C24—41no—
207C24#12111yescharacteristic
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—81no—
220C2×D4—81no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4C2×SD1612yes—
252C22×C4#13512yes—
253C22×C4#13512yes—
254C22×C4#13712yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—48no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—81no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4C2×SD1612yes—
298C4⋊C4C2×SD1612yes—
299C4⋊C4AGL(1,ℤ/8)12yes—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C42—21no—
304C42—21no—
305C42—22no—
306C42—22no—
307C42—42no—
308C42—42no—
309C42—81no—
310C42—81no—
311C42C2×#4112yes—
312C422−1+412yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23#46412yes—
325C23#46512yes—
326C23#47212yes—
327C23C2×#12111yescharacteristic
328D4—82no—
329D4—82no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334D4—162no—
335D4—164no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—164no—
373C2×C4#46412yes—
374C2×C4#46512yes—
375C2×C4#52511yescharacteristic
376C2×C4#53611yescharacteristic
377C2×C4#54712yes—
378C2×C4#55212yes—
379C2×C4C22×SD1611yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C2×C4C2×#13511yescharacteristic
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×#13711yescharacteristic
384C2×C4#57511yescharacteristic
385C2×C4#57612yes—
386C8—42no—
387C8—42no—
388C8—44no—
389C8—82no—
390C8—81no—
391C8—81no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22#167814yes—
406C22#241311yescharacteristic
407C22C2×#46411yescharacteristic
408C22#304111yescharacteristic
409C22C2×#46511yescharacteristic
410C22#305511yescharacteristic
411C22C2×#47211yescharacteristic
412C22#306911yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—41no—
418C4—42no—
419C4—42no—
420C4—81no—
421C4—162no—
422C4—161no—
423C4#310311yescharacteristic
424C4#310511yescharacteristic
425C4#322711yescharacteristic
426C4#323211yescharacteristic
427C4#333912yes—
428C4#334612yes—
429C2—24no—
430C2—321no—
431C2C2×#167811yescharacteristic
432C2#1917711yescharacteristic
433C2#1938511yescharacteristic
434C2#1945811yescharacteristic
435C2#2458311yescharacteristic
436C2#2460311yescharacteristic
437C2#2462011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22625C2semidirect
4. C2×#1947C2semidirect
5. #10658C2semidirect
6. #10587C2semidirect
7. #9613C2semidirect
8. #9563C2semidirect
10. #8830C2semidirect
11. #8725C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331072
#279732
#226252
C2×#19472
#106582
#105872
#96132
#95632
#91332
#88302
#87252