Finite Group Databasebeta

← #275257

Subgroups — #275257

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33106C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic, complemented
4C2×#1948C211yesmaximal, characteristic, complemented
5#10655C212yesmaximal, complemented
6#10599C212yesmaximal, complemented
7#9616C212yesmaximal, complemented
8#9564C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12C2×#480—21no—
13C2×#431—21no—
14C22×#106—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2856C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2071—22no—
20#1982C2211yescharacteristic
21#1982C2211yescharacteristic
22#1957C2212yes—
23#1948—21no—
24#1948—21no—
25#1948C2212yes—
26#1948C2212yes—
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1939C2212yes—
30#1925—21no—
31#1925—21no—
32#1916—22no—
33#1874—21no—
34#1874—21no—
35#1874—21no—
36#1834C2212yes—
37#1817—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C8×C4⋊C4C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#335C2212yes—
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#480—44no—
62#431—44no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—44no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×#102—21no—
76C2×#102—21no—
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—22no—
80C2×#102—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8D411yescharacteristic
84C2×C22⋊C8D411yescharacteristic
85C2×C4×C8—22no—
86C2×C4×C8C2312yes—
87#390—21no—
88#390—21no—
89#390—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384C2311yescharacteristic
94#384C2312yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C2×#95—21no—
102#340—22no—
103#340—22no—
104#340—21no—
105#340—21no—
106#340—22no—
107#340—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336D412yes—
113#335—24no—
114#331—24no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—41no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—41no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C22×C8—24no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134#127—22no—
135#127—22no—
136#121—42no—
137#121—42no—
138#121—42no—
139#121—42no—
140#121—82no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4D814yes—
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42C2411yescharacteristic
179#107—42no—
180#107—42no—
181#106—44no—
182#106—44no—
183C4⋊C8—24no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—44no—
190C4⋊C8#4114yes—
191#102—42no—
192#102—42no—
193#102—42no—
194#102—42no—
195#102—82no—
196#102—82no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—44no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208C24—41no—
209C24#12111yescharacteristic
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4—82no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C2×D812yes—
254C22×C4#13512yes—
255C22×C4#13512yes—
256C22×C4AGL(1,ℤ/8)12yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C2×C8—24no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—48no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—82no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4C2×D812yes—
300C4⋊C4C2×D812yes—
301C4⋊C4#13712yes—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—22no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42—42no—
314C42—42no—
315C42C2×#4112yes—
316C422−1+412yes—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23#46312yes—
329C23#46612yes—
330C23#47112yes—
331C23C2×#12111yescharacteristic
332D4—82no—
333D4—82no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—162no—
339D4—164no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—164no—
377C2×C4#46312yes—
378C2×C4#46612yes—
379C2×C4#52511yescharacteristic
380C2×C4#53611yescharacteristic
381C2×C4#54712yes—
382C2×C4#55212yes—
383C2×C4C22×D811yescharacteristic
384C2×C4C2×#13511yescharacteristic
385C2×C4C2×#13511yescharacteristic
386C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4#57511yescharacteristic
389C2×C4#57712yes—
390C8—42no—
391C8—42no—
392C8—44no—
393C8—82no—
394C8—81no—
395C8—81no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22#168114yes—
410C22#241311yescharacteristic
411C22C2×#46311yescharacteristic
412C22#304311yescharacteristic
413C22C2×#46611yescharacteristic
414C22#305611yescharacteristic
415C22C2×#47111yescharacteristic
416C22#306811yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—81no—
425C4—161no—
426C4—162no—
427C4#310011yescharacteristic
428C4#310811yescharacteristic
429C4#322611yescharacteristic
430C4#323211yescharacteristic
431C4#333712yes—
432C4#334512yes—
433C2—24no—
434C2—321no—
435C2C2×#168111yescharacteristic
436C2#1916811yescharacteristic
437C2#1939811yescharacteristic
438C2#1946911yescharacteristic
439C2#2458111yescharacteristic
440C2#2460011yescharacteristic
441C2#2462311yescharacteristic
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. #22599C2semidirect
4. C2×#1948C2semidirect
5. #10655C2semidirect
6. #10599C2semidirect
7. #9616C2semidirect
8. #9564C2semidirect
10. #8829C2semidirect
11. #8724C2semidirect

Maximal subgroups

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

subgroupindex
#331062
#279732
#225992
C2×#19482
#106552
#105992
#96162
#95642
#91332
#88292
#87242