Finite Group Databasebeta

← #274694

Subgroups — #274694

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32954C211yesmaximal, characteristic
2#27972C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#10626C212yesmaximal, complemented
6#10486C212yesmaximal, complemented
7#9220C212yesmaximal, complemented
8#9129C211yesmaximal, characteristic
9#8830C211yesmaximal, characteristic
10#8724C211yesmaximal, characteristic
11#8647C212yesmaximal, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#475C2211yescharacteristic
14C2×#475C2211yescharacteristic
15C2×C8×Q8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103—21no—
18#2857C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2072—22no—
21#1984—22no—
22#1984C2212yes—
23#1956C2212yes—
24#1872—21no—
25#1872—21no—
26#1865—21no—
27#1865—21no—
28#1853—21no—
29#1834C2212yes—
30#1803—21no—
31#1803—21no—
32#1803C2212yes—
33#1803C2211yescharacteristic
34#1803C2212yes—
35#1803C2211yescharacteristic
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1787—21no—
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C23×C8—21no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#475—24no—
69#475—42no—
70C8×Q8—24no—
71C8×Q8—42no—
72C2×#106—21no—
73C2×#106—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—44no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8—22no—
95C2×C4×C8C2312yes—
96#391—21no—
97#389—21no—
98#389—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2311yescharacteristic
103#384C2312yes—
104#380—21no—
105#380—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—21no—
111#340—21no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336D412yes—
117#334—42no—
118#332—42no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8SD1614yes—
127C22×Q8C2×D411yescharacteristic
128C22×Q8C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—44no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—44no—
157C4×Q8#4114yes—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—41no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—44no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#106—42no—
197#106—42no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—44no—
205C4⋊C8—42no—
206#103—42no—
207#103—42no—
208#103—42no—
209#103—44no—
210#103—44no—
211#103—42no—
212#103—82no—
213#103—82no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—44no—
221#95—41no—
222#95—41no—
223C24#12111yescharacteristic
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—24no—
229C2×Q8—28no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8#12312yes—
244C2×Q8C2×SD1612yes—
245C2×Q8C2×SD1612yes—
246C2×Q8#13712yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4C2×SD1612yes—
276C22×C4#13512yes—
277C22×C4#13512yes—
278C22×C4#13712yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—84no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4#12312yes—
327C22⋊C4—82no—
328C42—21no—
329C42—21no—
330C42—24no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42C2×#4112yes—
337C42H(5,𝔽2)12yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#46412yes—
343C23#46512yes—
344C23#47312yes—
345C23C2×#12111yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—22no—
349Q8—22no—
350Q8—22no—
351Q8—48no—
352Q8—41no—
353Q8—41no—
354Q8—82no—
355Q8—82no—
356Q8#48314yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—84no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—164no—
397C2×C4#46412yes—
398C2×C4#46512yes—
399C2×C4C2×#12311yescharacteristic
400C2×C4#53411yescharacteristic
401C2×C4#54612yes—
402C2×C4#55112yes—
403C2×C4C22×SD1611yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57511yescharacteristic
409C2×C4#57712yes—
410C8—42no—
411C8—42no—
412C8—81no—
413C8—82no—
414C8—81no—
415C8—82no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22#168014yes—
423C22#241211yescharacteristic
424C22C2×#46411yescharacteristic
425C22#304311yescharacteristic
426C22C2×#46511yescharacteristic
427C22#305611yescharacteristic
428C22C2×#47311yescharacteristic
429C22#306811yescharacteristic
430C4—22no—
431C4—22no—
432C4—24no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—81no—
438C4—162no—
439C4—161no—
440C4—161no—
441C4C2×#48311yescharacteristic
442C4#309111yescharacteristic
443C4#318711yescharacteristic
444C4#319011yescharacteristic
445C4#330112yes—
446C4#332012yes—
447C2—24no—
448C2C2×#168011yescharacteristic
449C2#1917611yescharacteristic
450C2#1939611yescharacteristic
451C2#1947111yescharacteristic
452C2#2453311yescharacteristic
453C2#2455111yescharacteristic
454C2#2457211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10626C2semidirect
6. #10486C2semidirect
7. #9220C2semidirect
11. #8647C2semidirect

Maximal subgroups

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

subgroupindex
#329542
#279722
Q8×C22⋊C82
C2×#18032
#106262
#104862
#92202
#91292
#88302
#87242
#86472