Finite Group Databasebeta

← #237818

Subgroups — #237818

Order 512.

1090 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 190 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
1#35052C211yesmaximal, characteristic
2#21768C212yesmaximal
3#14281C211yesmaximal, characteristic
4#10347C212yesmaximal, complemented
5#10345C212yesmaximal, complemented
6#10343C212yesmaximal, complemented
7#10341C212yesmaximal, complemented
8#8016C212yesmaximal
9#7909C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2212yes—
12#2848—22no—
13#2659C2212yes—
14#2646C2212yes—
15#2569—22no—
16#2549—22no—
17#2438—24no—
18#2329—22no—
19#2280C2212yes—
20C2×#380C2211yescharacteristic
21C22×#95C2211yescharacteristic
22#1894—22no—
23#1894—22no—
24#1873—24no—
25#1824C2214yes—
26#1787C2214yes—
27#1446C2211yescharacteristic
28#1446C2211yescharacteristic
29#1445C2211yescharacteristic
30#1445C2211yescharacteristic
31#1441—22no—
32C22⋊#95—22no—
33#1294C2214yes—
34#1294C2214yes—
35#1280C2212yes—
36#1280C2212yes—
37C23×Q8C2311yescharacteristic
38C24×C4D412yes—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8Q814yescomplemented
49C2×#117D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C2×#103—24no—
56C2×#103—24no—
57#407—24no—
58#407—24no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2314yes—
61#400—24no—
62#398—24no—
63#398—24no—
64#395—24no—
65#391—24no—
66#389—24no—
67#387—24no—
68#387—24no—
69#385—24no—
70#385—24no—
71#385—24no—
72#381—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—24no—
77#380D412yes—
78#380D412yes—
79#380C2314yes—
80#379—22no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95C2311yescharacteristic
91C2×#95C2311yescharacteristic
92C2×#95C2311yescharacteristic
93C2×#95C2311yescharacteristic
94#340—24no—
95#340—24no—
96C25C2×D411yescharacteristic
97C22×Q8—22no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—82no—
102C22×Q8C2×Q812yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—42no—
111C23×C4—82no—
112C23×C4—82no—
113C23×C4—82no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×Q812yes—
120C23×C4#4114yes—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
123C2×M16—44no—
124C2×M16—84no—
125C22×C8—24no—
126C22⋊Q8—28no—
127C22⋊Q8—28no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—84no—
133C22⋊Q8—84no—
134#117—84no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D414yes—
151C2×C4⋊C4C2×Q812yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C42—24no—
163#103—84no—
164#103—84no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—44no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—24no—
177#95—22no—
178#95—22no—
179#95—82no—
180#95—82no—
181#95—82no—
182#95—82no—
183C24—22no—
184C24—22no—
185C24—41no—
186C24—42no—
187C24—41no—
188C24#12011yescharacteristic
189C24#12011yescharacteristic
190C24C22⋊Q812yes—
191C24#12412yes—
192C24C22×D411yescharacteristic
193C2×Q8—24no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—84no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—88no—
248C22×C4—88no—
249C22×C4#12012yes—
250C22×C4C22⋊Q814yes—
251C22×C4C22⋊Q814yes—
252C22×C4C22⋊Q812yes—
253C22×C4#12414yes—
254C22×C4#12414yes—
255C22×C4#12412yes—
256C22×C4#13714yes—
257C22×C4C22×D412yes—
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4111yescharacteristic
261M16—88no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—48no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C22⋊C4—44no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—84no—
285C42—84no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—24no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23#39312yes—
309C23#39312yes—
310C23C2×#12011yescharacteristic
311C23C2×C22⋊Q811yescharacteristic
312C23C2×#12411yescharacteristic
313C23C2×#13712yes—
314C23#57512yes—
315Q8—48no—
316Q8—82no—
317Q8—82no—
318Q8—164no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—88no—
351C2×C4—88no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#39314yes—
369C2×C4C2×C22⋊Q812yes—
370C2×C4C2×#12412yes—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—48no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22C2×#39311yescharacteristic
394C22#300712yes—
395C22#301911yescharacteristic
396C22#302011yescharacteristic
397C22#302612yes—
398C22#307812yes—
399C22#313911yescharacteristic
400C22#314011yescharacteristic
401C4—22no—
402C4—22no—
403C4—24no—
404C4—44no—
405C4—82no—
406C4—82no—
407C4—84no—
408C4—82no—
409C4—84no—
410C4—161no—
411C4—161no—
412C2—21no—
413C2—22no—
414C2—21no—
415C2—42no—
416C2—42no—
417C2#1834712yes—
418C2#1834912yes—
419C2#2345511yescharacteristic
420C2#2346211yescharacteristic
421C2#2346811yescharacteristic
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. #10347C2semidirect
5. #10345C2semidirect
6. #10343C2semidirect
7. #10341C2semidirect
48. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350522
#217682
#142812
#103472
#103452
#103432
#103412
#80162
#79092