Finite Group Databasebeta

← #241144

Subgroups — #241144

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#21938C212yesmaximal
3#14275C211yesmaximal, characteristic
4#10570C214yesmaximal, complemented
5#10568C214yesmaximal, complemented
6#8023C212yesmaximal
7C2×#1280C211yesmaximal, characteristic
8C22×C22⋊Q8—21no—
9C22×C22⋊C8C2212yes—
10#2659—24no—
11#2659—22no—
12#2659C2214yes—
13C2×#398—22no—
14C2×#395—22no—
15C2×#385—21no—
16C2×#380—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1853—22no—
22#1853—22no—
23#1853C2214yes—
24#1853C2214yes—
25#1807—24no—
26#1444C2212yes—
27#1442C2212yes—
28#1441—22no—
29C22⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8—21no—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8—22no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—44no—
40C2×C22⋊Q8—44no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C2×#106—24no—
55C2×#106—24no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8Q818yescomplemented
59C2×C22⋊C8C2314yes—
60#398—28no—
61#398—28no—
62#395—28no—
63#395—28no—
64#385—44no—
65#380—24no—
66#380—22no—
67#380—22no—
68#380—24no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380D414yes—
73#380Q814yescomplemented
74#380C2314yes—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D412yes—
84C2×#95C2312yes—
85C2×#95C2312yes—
86#340—24no—
87#340—24no—
88C25C2×D411yescharacteristic
89C22×Q8—24no—
90C22×Q8—44no—
91C22×Q8—82no—
92C22×Q8—82no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4—81no—
107C23×C4—81no—
108C23×C4—81no—
109C23×C4—81no—
110C23×C4SD1614yes—
111C23×C4SD1614yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×Q814yes—
118C23×C4C2×Q812yes—
119C23×C4#4112yes—
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C22×C8—28no—
123C22×C8—82no—
124C22×C8—82no—
125C22⋊Q8—48no—
126C22⋊Q8—416no—
127C22⋊Q8—48no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—81no—
141C2×C4⋊C4—81no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×D414yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163#106—84no—
164#106—84no—
165C22⋊C8—28no—
166C22⋊C8—24no—
167C22⋊C8C2×Q818yescomplemented
168#95—28no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—42no—
175#95—44no—
176#95—82no—
177#95—82no—
178#95C2×Q814yescomplemented
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24#12011yescharacteristic
184C24#12011yescharacteristic
185C24C22⋊Q812yes—
186C24C4⋊Q812yes—
187C24C2×SD1612yes—
188C24C2×SD1612yes—
189C24AGL(1,ℤ/8)12yes—
190C24#13712yes—
191C24C22×D411yescharacteristic
192C2×Q8—48no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—816no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4#12012yes—
257C22×C4C22⋊Q814yes—
258C22×C4C22⋊Q814yes—
259C22×C4C22⋊Q812yes—
260C22×C4C4⋊Q814yes—
261C22×C4C4⋊Q814yes—
262C22×C4C4⋊Q812yes—
263C22×C4C2×SD1614yes—
264C22×C4C2×SD1614yes—
265C22×C4C22×D412yes—
266C22×C4C22×Q812yes—
267C22×C4C22×Q811yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C2×C8—28no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—88no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—416no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4—84no—
284C4⋊C4—88no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23#39512yes—
316C23#39512yes—
317C23#44812yes—
318C23#44812yes—
319C23#45012yes—
320C23#45012yes—
321C23#47514yes—
322C23#47614yes—
323C23#49814yes—
324C23#50114yes—
325C23C2×#12011yescharacteristic
326C23C2×C22⋊Q811yescharacteristic
327C23C2×C4⋊Q811yescharacteristic
328C23C22×SD1611yescharacteristic
329C23C22×SD1611yescharacteristic
330C23C2×AGL(1,ℤ/8)11yescharacteristic
331C23C2×#13711yescharacteristic
332Q8—816no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#39514yes—
386C2×C4C2×C22⋊Q812yes—
387C2×C4C2×C4⋊Q812yes—
388C8—48no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—22no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22—44no—
417C22#205814yes—
418C22#205814yes—
419C22C2×#39511yescharacteristic
420C22C2×#45011yescharacteristic
421C22C2×#44811yescharacteristic
422C22#301911yescharacteristic
423C22#302011yescharacteristic
424C22#302611yescharacteristic
425C22#302611yescharacteristic
426C22C2×#47611yescharacteristic
427C22C2×#47511yescharacteristic
428C22C2×#49811yescharacteristic
429C22C2×#50111yescharacteristic
430C4—22no—
431C4—22no—
432C4—24no—
433C4—44no—
434C4—44no—
435C4—84no—
436C4—84no—
437C4—84no—
438C4—161no—
439C4—161no—
440C2—21no—
441C2—21no—
442C2—24no—
443C2—24no—
444C2—21no—
445C2—21no—
446C2#1828511yescharacteristic
447C2#1828611yescharacteristic
448C2#1829311yescharacteristic
449C2#1829411yescharacteristic
450C2C2×#205811yescharacteristic
451C2#2356111yescharacteristic
452C2#2356511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10570C2semidirect
5. #10568C2semidirect
58. C2×C22⋊C8Q8semidirect
73. #380Q8semidirect
167. C22⋊C8C2×Q8semidirect
178. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#219382
#142752
#105702
#105682
#80232
C2×#12802