Finite Group Databasebeta

← #274255

Subgroups — #274255

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33149C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3#22629C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10588C212yesmaximal, complemented
6#10541C212yesmaximal, complemented
7#9745C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9127C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15#2857C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#2043—22no—
18#1988—21no—
19#1988—21no—
20#1988C2212yes—
21#1988C2212yes—
22#1987C2211yescharacteristic
23#1987C2211yescharacteristic
24#1956C2212yes—
25#1947C2211yescharacteristic
26#1947C2211yescharacteristic
27#1939C2212yes—
28#1921—21no—
29#1921—21no—
30#1917—22no—
31#1852—21no—
32#1824—21no—
33#1797—21no—
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1787—21no—
37#1787—21no—
38#1770C2212yes—
39#1749—21no—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#487—24no—
66#487—42no—
67C2×#107—21no—
68C2×#107—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8D411yescharacteristic
84C2×#97—22no—
85C2×C4×C8C2312yes—
86#391—21no—
87#389—21no—
88#389—21no—
89#384C2311yescharacteristic
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4C2311yescharacteristic
97C4×C4⋊C4C2311yescharacteristic
98C2×#95—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336D412yes—
114#334—42no—
115#333—42no—
116#332—24no—
117#332—42no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—44no—
121C22×Q8—41no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—82no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C4⋊Q8—24no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—42no—
149C4⋊Q8#4114yes—
150C22⋊Q8—82no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—82no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#107—42no—
187#107—42no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—42no—
195C4⋊C8—82no—
196#103—44no—
197#103—44no—
198#103—82no—
199#103—82no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204#97—82no—
205C4×C8—22no—
206C4×C8—22no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—41no—
212#95—41no—
213C24#12111yescharacteristic
214C2×Q8—24no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—88no—
223C2×Q8—88no—
224C2×Q8—82no—
225C2×Q8—81no—
226C2×Q8—81no—
227C2×Q8—82no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#13512yes—
257C22×C4#13712yes—
258C22×C4#13712yes—
259C22×C4#13712yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—84no—
282C4⋊C4—21no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—82no—
305C4⋊C4#13712yes—
306C22⋊C4—82no—
307C42—21no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—42no—
312C42—42no—
313C42—81no—
314C42—81no—
315C42C2×#4112yes—
316C42H(5,𝔽2)12yes—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23#46412yes—
322C23#47012yes—
323C23#47112yes—
324C23C2×#12111yescharacteristic
325Q8—44no—
326Q8—44no—
327Q8—88no—
328Q8—82no—
329Q8—82no—
330Q8—162no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—88no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4#46412yes—
372C2×C4#47012yes—
373C2×C4#52511yescharacteristic
374C2×C4#53511yescharacteristic
375C2×C4#54612yes—
376C2×C4#55112yes—
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4C2×#13711yescharacteristic
381C2×C4C2×#13711yescharacteristic
382C2×C4#57511yescharacteristic
383C2×C4#57612yes—
384C8—42no—
385C8—42no—
386C8—81no—
387C8—82no—
388C8—81no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22#170414yes—
397C22#241111yescharacteristic
398C22C2×#46411yescharacteristic
399C22#304111yescharacteristic
400C22C2×#47011yescharacteristic
401C22#305611yescharacteristic
402C22C2×#47111yescharacteristic
403C22#306911yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—42no—
408C4—41no—
409C4—81no—
410C4—84no—
411C4—81no—
412C4—81no—
413C4—161no—
414C4—161no—
415C4#311311yescharacteristic
416C4#311711yescharacteristic
417C4#321511yescharacteristic
418C4#321811yescharacteristic
419C4#329712yes—
420C4#331412yes—
421C2—24no—
422C2C2×#170411yescharacteristic
423C2#1917511yescharacteristic
424C2#1942311yescharacteristic
425C2#1945711yescharacteristic
426C2#2447911yescharacteristic
427C2#2450811yescharacteristic
428C2#2452311yescharacteristic
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. #10588C2semidirect
6. #10541C2semidirect
7. #9745C2semidirect
8. #9569C2semidirect

Maximal subgroups

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

subgroupindex
#331492
#258562
#226292
C2×#19882
#105882
#105412
#97452
#95692
#91272
#88302
#83402