Finite Group Databasebeta

← #436725

Subgroups — #436725

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15216C211yesmaximal, characteristic
2#14129C211yesmaximal, characteristic
3#11385C212yesmaximal, complemented
4#11256C212yesmaximal, complemented
5#11219C212yesmaximal, complemented
6#11096C212yesmaximal
7#10922C211yesmaximal, characteristic
8#10881C212yesmaximal, complemented
9#10731C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#107C2212yes—
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C2×#389C2212yes—
16C2×#380C2211yescharacteristic
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28#1442—22no—
29#1439—22no—
30#1439—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8C2312yes—
37C23×C8C2312yes—
38C2×#125—24no—
39C2×#125—42no—
40C2×#125—41no—
41C2×#125—41no—
42C2×#117—24no—
43C2×#117—42no—
44C2×#117—41no—
45C2×#117—41no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D412yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—42no—
61C2×#107—42no—
62C2×C22⋊C8—42no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8Q818yes—
65C2×#97—41no—
66C2×#97—42no—
67#389—24no—
68#389—42no—
69#380—24no—
70#380—24no—
71#380—42no—
72C4×C22⋊C4—41no—
73C4×C22⋊C4—41no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D412yes—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—42no—
91#336—41no—
92C25C2×D411yescharacteristic
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×Q814yes—
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8C2×Q814yes—
122C22×C8C2×Q814yes—
123#125—44no—
124#125—48no—
125#125—44no—
126#117—44no—
127#117—48no—
128#117—44no—
129C2×C4⋊C4—22no—
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—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4#4114yes—
157C2×C4⋊C4#4114yes—
158C2×C4⋊C4#4114yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4—82no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42—42no—
179#107—48no—
180C22⋊C8—48no—
181#97—44no—
182#95—24no—
183#95—42no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—82no—
189#95—82no—
190#95—82no—
191C24—21no—
192C24—21no—
193C24—22no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—22no—
199C24C4⋊Q812yes—
200C24C22×D411yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
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—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C4⋊Q812yes—
264C22×C4C22×Q812yes—
265C22×C4C2×#4112yes—
266C22×C4C2×#4112yes—
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)12yes—
269C22×C4H(5,𝔽2)12yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—416no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—88no—
304C22⋊C4—48no—
305C22⋊C4—48no—
306C22⋊C4—84no—
307C22⋊C4—88no—
308C42—48no—
309C42—42no—
310C42—42no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23#52511yescharacteristic
337C23C2×C4⋊Q811yescharacteristic
338C23#53411yescharacteristic
339C23#54612yes—
340C23#54612yes—
341C23#57512yes—
342C23#57512yes—
343C23#57612yes—
344C23#57812yes—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#52912yes—
397C2×C4#53312yes—
398C2×C4#53812yes—
399C2×C4#53812yes—
400C2×C4#53812yes—
401C2×C4#54312yes—
402C2×C4#55111yescharacteristic
403C2×C4#55111yescharacteristic
404C2×C4#55412yes—
405C2×C4#55412yes—
406C2×C4#55412yes—
407C2×C4#55412yes—
408C2×C4#55512yes—
409C2×C4#55912yes—
410C2×C4#56011yescharacteristic
411C2×C4H(3,𝔽4)11yescharacteristic
412C8—44no—
413C8—44no—
414C22—44no—
415C22—44no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22#245912yes—
424C22#254912yes—
425C22#258311yescharacteristic
426C22#268411yescharacteristic
427C22#269711yescharacteristic
428C22#311112yes—
429C22#312211yescharacteristic
430C22#312711yescharacteristic
431C22#316211yescharacteristic
432C22#316412yes—
433C22#316611yescharacteristic
434C22#319712yes—
435C22#320312yes—
436C22#330912yes—
437C22#330912yes—
438C22#331312yes—
439C22#331512yes—
440C22#331512yes—
441C22#331912yes—
442C22#332212yes—
443C22#332712yes—
444C4—42no—
445C4—82no—
446C4—84no—
447C4—84no—
448C4—82no—
449C4—84no—
450C4—162no—
451C4#257412yes—
452C4#258912yes—
453C4#278412yes—
454C4#280412yes—
455C2—82no—
456C2#1556811yescharacteristic
457C2#2674712yes—
458C2#2674912yes—
459C2#2877812yes—
460C2#2878712yes—
461C2#2930511yescharacteristic
462C2#2930611yescharacteristic
463C2#3127412yes—
464C2#3168412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11385C2semidirect
4. #11256C2semidirect
5. #11219C2semidirect
8. #10881C2semidirect

Maximal subgroups

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

subgroupindex
#152162
#141292
#113852
#112562
#112192
#110962
#109222
#108812
#107312