Finite Group Databasebeta

← #242819

Subgroups — #242819

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21887C212yesmaximal
2#21375C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10626C212yesmaximal, complemented
5#10625C212yesmaximal, complemented
6#10618C211yesmaximal, characteristic, complemented
7#10617C211yesmaximal, characteristic, complemented
8#10616C211yesmaximal, characteristic, complemented
9#10615C211yesmaximal, characteristic, complemented
10#8044C212yesmaximal, complemented
11C2×#1277C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2285C2212yes—
17#1989—22no—
18#1984C2211yescharacteristic
19#1984C2211yescharacteristic
20#1983—22no—
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1893—22no—
24#1892—22no—
25#1869—21no—
26#1869—21no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1834C2212yes—
30#1808—21no—
31#1808—21no—
32#1803C2212yes—
33#1803C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36C4×#107—21no—
37C4×#107—21no—
38#1750—21no—
39#1750—21no—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C23×C8—22no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50#529—24no—
51#529—24no—
52C2×#126—21no—
53C2×#125D411yescharacteristic
54C2×#124D411yescharacteristic
55C2×#123—21no—
56C2×C22⋊Q8D411yescharacteristic
57C2×#121D411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4—21no—
60C2×C4×D4C2311yescharacteristic
61C2×#117—21no—
62C2×#117—22no—
63C22×C42C2311yescharacteristic
64#432—24no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—21no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×#97—21no—
86C2×C4×C8—24no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#384C2312yes—
92#384C2312yes—
93#382—22no—
94#380—22no—
95#380—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—21no—
105#336—21no—
106#336—21no—
107#336—21no—
108C22×Q8C2×D411yescharacteristic
109C22×D4—21no—
110C22×D4C2×D411yescharacteristic
111C23×C4—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—44no—
126C22×C8—82no—
127C22×C8—82no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132#126—44no—
133#125—24no—
134#124—24no—
135#123—44no—
136C22⋊Q8—24no—
137#121—24no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—42no—
143C4×D4—42no—
144#117—42no—
145#117—42no—
146#117—84no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—42no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4114yes—
170C2×C42#4114yes—
171C2×C42#4111yescharacteristic
172C2×C42C2411yescharacteristic
173#107—42no—
174#107—42no—
175#107—42no—
176#107—42no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—44no—
184#103—82no—
185#103—82no—
186#102—82no—
187#102—82no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193#97—42no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198#95—42no—
199#95—42no—
200C24—21no—
201C24#12111yescharacteristic
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12312yes—
237C22×C4#12314yes—
238C22×C4#12412yes—
239C22×C4#12614yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—88no—
268C2×C8#13512yes—
269C2×C8#13512yes—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—84no—
292C42—41no—
293C42—41no—
294C42—84no—
295C42#12312yes—
296C42#12314yes—
297C42#12412yes—
298C42#12614yes—
299C42C2×#4112yes—
300C42C2×#4112yes—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23#39712yes—
313C23#47212yes—
314C23#47312yes—
315C23C2×#12111yescharacteristic
316Q8—42no—
317Q8—42no—
318D4—42no—
319D4—42no—
320D4—84no—
321C2×C4—28no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#39712yes—
356C2×C4#39712yes—
357C2×C4#39712yes—
358C2×C4C2×#12311yescharacteristic
359C2×C4C2×#12312yes—
360C2×C4C2×#12411yescharacteristic
361C2×C4C2×#12612yes—
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4#57511yescharacteristic
365C2×C4#57511yescharacteristic
366C2×C4#57611yescharacteristic
367C2×C4#57712yes—
368C2×C4#57811yescharacteristic
369C8—22no—
370C8—22no—
371C8—41no—
372C8—41no—
373C8—82no—
374C8—82no—
375C8—81no—
376C8—81no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#207814yes—
391C22C2×#39711yescharacteristic
392C22#303511yescharacteristic
393C22#303511yescharacteristic
394C22#306011yescharacteristic
395C22#306111yescharacteristic
396C22C2×#47211yescharacteristic
397C22C2×#47311yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—162no—
406C4—162no—
407C4#309111yescharacteristic
408C4#309111yescharacteristic
409C4#309612yes—
410C4#309712yes—
411C4#313111yescharacteristic
412C4#313511yescharacteristic
413C2—24no—
414C2—161no—
415C2#1922211yescharacteristic
416C2#1922311yescharacteristic
417C2#1922711yescharacteristic
418C2#1922811yescharacteristic
419C2#2362711yescharacteristic
420C2#2364111yescharacteristic
421C2C2×#207811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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. #20817C2semidirect
4. #10626C2semidirect
5. #10625C2semidirect
6. #10618C2semidirect
7. #10617C2semidirect
8. #10616C2semidirect
9. #10615C2semidirect
10. #8044C2semidirect
11. C2×#1277C2semidirect

Maximal subgroups

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

subgroupindex
#218872
#213752
#208172
#106262
#106252
#106182
#106172
#106162
#106152
#80442
C2×#12772