Finite Group Databasebeta

← #241387

Subgroups — #241387

Order 512.

956 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#35167C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5#10582C214yesmaximal, complemented
6#10574C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#544—21no—
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#106—21no—
14C22×C4⋊C8C2211yescharacteristic
15#2857—22no—
16#2659—22no—
17#2552—21no—
18#2552C2212yes—
19#2551—21no—
20#2551C2212yes—
21#2550—22no—
22#2550—22no—
23#2316—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1948—22no—
29#1948—22no—
30#1948C2214yes—
31#1948C2214yes—
32#1853—22no—
33#1853—22no—
34#1844—21no—
35#1844C2212yes—
36#1834—21no—
37#1834C2212yes—
38C2×#340—21no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×C8—21no—
44#544—44no—
45#544—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×C22⋊Q8—24no—
52C2×#117—21no—
53C2×#117—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#431—22no—
62#431—22no—
63#431Q814yescomplemented
64#431Q814yescomplemented
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—44no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8Q814yescomplemented
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#398—24no—
80#398—24no—
81#395—24no—
82#395—24no—
83#391—24no—
84#389—22no—
85#389—22no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#384—22no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#384C2312yes—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#382—21no—
101#382—21no—
102#380—24no—
103#380—21no—
104#380—21no—
105#380D412yes—
106#380D412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2312yes—
115C2×#95—21no—
116#340—24no—
117#340—24no—
118#340—24no—
119C22×Q8—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—24no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—42no—
131C22×C8—42no—
132C4⋊Q8—44no—
133C4⋊Q8—48no—
134C4⋊Q8—44no—
135C4⋊Q8—84no—
136#125—44no—
137#125—48no—
138#125—44no—
139C22⋊Q8—416no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4SD1614yes—
172C2×C4⋊C4SD1614yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—82no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q812yes—
190C2×C42C2×Q812yes—
191C2×C42C2×Q812yes—
192C2×C42#4112yes—
193C2×C42C2411yescharacteristic
194#106—48no—
195#106—84no—
196#106—84no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8C2×Q814yes—
202C4⋊C8C2×Q814yes—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C4×C8—24no—
206C4×C8—22no—
207C4×C8C2×Q814yes—
208#95—24no—
209#95—24no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216C24#12011yescharacteristic
217C2×Q8—48no—
218C2×Q8—84no—
219C2×Q8—84no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4C22⋊Q812yes—
244C22×C4C22⋊Q812yes—
245C22×C4C22⋊Q812yes—
246C22×C4C4⋊Q812yes—
247C22×C4C4⋊Q812yes—
248C22×C4C4⋊Q812yes—
249C22×C4C2×SD1612yes—
250C22×C4C2×SD1612yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13712yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—48no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—82no—
296C4⋊C4C2×SD1614yes—
297C4⋊C4C2×SD1614yes—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C42—42no—
301C42—42no—
302C42—44no—
303C42—44no—
304C42—82no—
305C42C22⋊Q812yes—
306C42C22⋊Q812yes—
307C42C22⋊Q812yes—
308C42C4⋊Q812yes—
309C42C4⋊Q812yes—
310C42C4⋊Q812yes—
311C42C22×Q811yescharacteristic
312C42C22×Q811yescharacteristic
313C42C22×Q811yescharacteristic
314C42C2×#4111yescharacteristic
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#39512yes—
320C23#44812yes—
321C23#45012yes—
322C23C2×#12011yescharacteristic
323Q8—168no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—164no—
358C2×C4#39512yes—
359C2×C4#39512yes—
360C2×C4#39512yes—
361C2×C4#44812yes—
362C2×C4#45012yes—
363C2×C4#47514yes—
364C2×C4#47614yes—
365C2×C4#49814yes—
366C2×C4#50114yes—
367C2×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×C22⋊Q811yescharacteristic
369C2×C4C2×C22⋊Q811yescharacteristic
370C2×C4C2×C4⋊Q811yescharacteristic
371C2×C4C2×C4⋊Q811yescharacteristic
372C2×C4C2×C4⋊Q811yescharacteristic
373C2×C4C22×SD1611yescharacteristic
374C2×C4C22×SD1611yescharacteristic
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C2×#13711yescharacteristic
377C2×C4#57712yes—
378C2×C4#57712yes—
379C8—44no—
380C8—44no—
381C8—44no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22#205814yes—
389C22C2×#39511yescharacteristic
390C22C2×#45011yescharacteristic
391C22C2×#44811yescharacteristic
392C22#300911yescharacteristic
393C22#301011yescharacteristic
394C22#302111yescharacteristic
395C22#302211yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—44no—
401C4—44no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—162no—
407C4#205814yes—
408C4C2×#47611yescharacteristic
409C4C2×#47511yescharacteristic
410C4#308412yes—
411C4C2×#49811yescharacteristic
412C4C2×#50111yescharacteristic
413C4#316312yes—
414C2—24no—
415C2#1889311yescharacteristic
416C2#1890411yescharacteristic
417C2#1891411yescharacteristic
418C2#1892311yescharacteristic
419C2C2×#205811yescharacteristic
420C2#2355611yescharacteristic
421C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10582C2semidirect
6. #10574C2semidirect
63. #431Q8semidirect
64. #431Q8semidirect
72. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219812
C2×#19482
#105822
#105742
#80392
#78932
#78922