Finite Group Databasebeta

← #241154

Subgroups — #241154

Order 512.

1208 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#21942C212yesmaximal
3#13966C211yesmaximal, characteristic
4#10569C214yesmaximal, complemented
5#10568C214yesmaximal, complemented
6#8019C212yesmaximal
7#7909C211yesmaximal, characteristic
8C22×C22⋊Q8—21no—
9C2×#407C2212yes—
10#2659—24no—
11#2659—22no—
12#2659C2214yes—
13C2×#398—22no—
14C2×#395—22no—
15C2×#380—21no—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20#1853—22no—
21#1853C2214yes—
22#1852—22no—
23#1852C2214yes—
24#1809—24no—
25#1442C2212yes—
26#1442C2212yes—
27#1434—22no—
28C22⋊#95—21no—
29C22⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8—21no—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×C22⋊Q8—24no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—48no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42—21no—
52C2×#107—24no—
53C2×#106—24no—
54#407—24no—
55#407Q818yescomplemented
56C2×C22⋊C8—22no—
57C2×C22⋊C8C2314yes—
58#398—28no—
59#398—28no—
60#395—28no—
61#395—28no—
62#380—24no—
63#380—22no—
64#380—22no—
65#380—24no—
66#380—22no—
67#380—22no—
68#380D414yes—
69#380C2314yes—
70C4×C22⋊C4—22no—
71C4×C22⋊C4Q814yescomplemented
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95D412yes—
82C2×#95C2312yes—
83C2×#95C2312yes—
84#340—24no—
85#340—24no—
86C25C2×D411yescharacteristic
87C22×Q8—24no—
88C22×Q8—44no—
89C22×Q8—82no—
90C22×Q8—82no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4—81no—
107C23×C4—81no—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×Q814yes—
114C23×C4C2×Q812yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C2×M16—28no—
118C2×M16—82no—
119C2×M16—82no—
120C22×C8—24no—
121C22⋊Q8—416no—
122C22⋊Q8—48no—
123C22⋊Q8—48no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—81no—
135C2×C4⋊C4—81no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×D414yes—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—48no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—24no—
152C2×C42—81no—
153C2×C42—81no—
154#107—84no—
155#106—84no—
156C22⋊C8—28no—
157C22⋊C8—24no—
158C22⋊C8C2×Q818yescomplemented
159#95—28no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—22no—
165#95—42no—
166#95—44no—
167#95—44no—
168#95—82no—
169#95—82no—
170#95C2×Q814yescomplemented
171C24—22no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—41no—
176C24—41no—
177C24#12011yescharacteristic
178C24#12011yescharacteristic
179C24C22⋊Q812yes—
180C24C4⋊Q812yes—
181C24C22×D411yescharacteristic
182C2×Q8—48no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×Q8—816no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—82no—
236C22×C4—88no—
237C22×C4—88no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4#12012yes—
243C22×C4C22⋊Q814yes—
244C22×C4C22⋊Q814yes—
245C22×C4C22⋊Q812yes—
246C22×C4C4⋊Q814yes—
247C22×C4C4⋊Q814yes—
248C22×C4C4⋊Q812yes—
249C22×C4AGL(1,ℤ/8)14yes—
250C22×C4#13714yes—
251C22×C4C22×D412yes—
252C22×C4C22×Q812yes—
253C22×C4C22×Q811yescharacteristic
254C22×C4C2×#4111yescharacteristic
255M16—88no—
256C2×C8—28no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—416no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C42—84no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—42no—
294C23—42no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23#39512yes—
300C23#39512yes—
301C23#45412yes—
302C23#45512yes—
303C23C2×#12011yescharacteristic
304C23C2×C22⋊Q811yescharacteristic
305C23C2×C4⋊Q811yescharacteristic
306C23C2×AGL(1,ℤ/8)12yes—
307C23C2×#13712yes—
308Q8—816no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#39514yes—
360C2×C4C2×C22⋊Q812yes—
361C2×C4C2×C4⋊Q812yes—
362C8—48no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—48no—
381C22—48no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22#206114yes—
387C22C2×#39511yescharacteristic
388C22#300311yescharacteristic
389C22#300411yescharacteristic
390C22C2×#45511yescharacteristic
391C22C2×#45411yescharacteristic
392C22#302611yescharacteristic
393C22#302611yescharacteristic
394C22#307912yes—
395C22#316512yes—
396C4—22no—
397C4—22no—
398C4—24no—
399C4—44no—
400C4—44no—
401C4—84no—
402C4—84no—
403C4—84no—
404C4—161no—
405C4—161no—
406C2—22no—
407C2—24no—
408C2—22no—
409C2—42no—
410C2#1831711yescharacteristic
411C2#1831911yescharacteristic
412C2#1832411yescharacteristic
413C2#1832611yescharacteristic
414C2#2355211yescharacteristic
415C2C2×#206111yescharacteristic
416C2#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. #10569C2semidirect
5. #10568C2semidirect
55. #407Q8semidirect
71. C4×C22⋊C4Q8semidirect
158. C22⋊C8C2×Q8semidirect
170. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#219422
#139662
#105692
#105682
#80192
#79092