Finite Group Databasebeta

← #415315

Subgroups — #415315

Order 512.

2468 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2468 conjugacy classes of subgroups — too many to draw. Only its 314 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#35011C211yesmaximal, characteristic
2C2×#2434C212yesmaximal
3C2×#1996C212yesmaximal
4#13922C211yesmaximal, characteristic
5#11344C218yesmaximal
6C4×#342C211yesmaximal, characteristic
7C24×Q8C2211yescharacteristic
8C22×C4⋊Q8—22no—
9C22×C22⋊Q8C2212yes—
10#2857—22no—
11#2659—22no—
12#2583—24no—
13#2502—22no—
14#2434—28no—
15#2434—28no—
16#2434—22no—
17C2×#395—24no—
18#2315C2214yes—
19C2×#391—22no—
20C2×#391—22no—
21C2×#385C2218yes—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1996—28no—
26C2×#342C2218yes—
27C2×#342C2214yes—
28#1446—22no—
29C22⋊#95C2214yes—
30C23×Q8—22no—
31C23×Q8—21no—
32C23×Q8—22no—
33C23×Q8—24no—
34C23×Q8—21no—
35C23×Q8—22no—
36C23×Q8—48no—
37C23×Q8D414yes—
38C23×Q8D412yes—
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×C4⋊Q8—216no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—416no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C22×C4⋊C4—28no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4D412yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2312yes—
59C22×C22⋊C4C2314yes—
60C22×C42—22no—
61C22×C42D412yes—
62#398—28no—
63#395—28no—
64#395—232no—
65#391—24no—
66#391—216no—
67#391—48no—
68#389—28no—
69#385—28no—
70#385—28no—
71#385—216no—
72#385—216no—
73#384—22no—
74#380—28no—
75#380—24no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—24no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4D418yes—
80C2×#95—22no—
81C2×#95—28no—
82C2×#95—28no—
83C2×#95—21no—
84C2×#95D412yes—
85#342—216no—
86#342Q8116yescomplemented
87C25C2411yescharacteristic
88C22×Q8—216no—
89C22×Q8—28no—
90C22×Q8—28no—
91C22×Q8—24no—
92C22×Q8—22no—
93C22×Q8—28no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—44no—
99C22×Q8—48no—
100C22×Q8—48no—
101C22×Q8—416no—
102C22×Q8—48no—
103C22×Q8—416no—
104C22×Q8—432no—
105C22×Q8—416no—
106C22×Q8—42no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8—42no—
112C22×Q8—42no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—48no—
120C23×C4—44no—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D414yes—
124C23×C4C2×D414yes—
125C23×C4C2×D414yes—
126C23×C4C2×D414yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D414yes—
131C23×C4#4112yes—
132C4⋊Q8—464no—
133C4⋊Q8—84no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C2×C4⋊C4—232no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—432no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4C2×D418yes—
169C2×C22⋊C4C2×Q818yes—
170C2×C22⋊C4C2×Q8116yes—
171C2×C42—28no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—48no—
175C2×C42—44no—
176#95—232no—
177#95—216no—
178#95—28no—
179#95—24no—
180#95—416no—
181#95—42no—
182C24—42no—
183C24—42no—
184C24C22×D412yes—
185C24C22×D411yescharacteristic
186C24C22×D411yescharacteristic
187C24C22×D412yes—
188C24C22×Q814yes—
189C24C2×#4111yescharacteristic
190C242−1+414yes—
191C2×Q8—232no—
192C2×Q8—216no—
193C2×Q8—48no—
194C2×Q8—416no—
195C2×Q8—416no—
196C2×Q8—416no—
197C2×Q8—48no—
198C2×Q8—432no—
199C2×Q8—432no—
200C2×Q8—416no—
201C2×Q8—416no—
202C2×Q8—432no—
203C2×Q8—416no—
204C2×Q8—44no—
205C2×Q8—48no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×Q8—84no—
212C2×Q8—88no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—216no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—216no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—416no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4#12014yes—
253C22×C4#12012yes—
254C22×C4#12011yescharacteristic
255C22×C4#12014yes—
256C22×C4#12011yescharacteristic
257C22×C4#12114yes—
258C22×C4#12112yes—
259C22×C4#12114yes—
260C22×C4#12112yes—
261C22×C4#12712yes—
262C22×C4#12712yes—
263C22×C4C4⋊Q818yescomplemented
264C4⋊C4—464no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—48no—
269C4⋊C4—84no—
270C4⋊C4—88no—
271C4⋊C4—88no—
272C22⋊C4—28no—
273C22⋊C4—216no—
274C22⋊C4—216no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—48no—
280C22⋊C4C4⋊Q8116yescomplemented
281C42—416no—
282C42—44no—
283C42—82no—
284C23—24no—
285C23—22no—
286C23—24no—
287C23—22no—
288C23—41no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—48no—
294C23—41no—
295C23—48no—
296C23C2×#12012yes—
297C23C2×#12011yescharacteristic
298C23C2×#12111yescharacteristic
299C23C2×#12112yes—
300C23C2×#12711yescharacteristic
301C23C2×C4⋊Q812yes—
302C23C2×C4⋊Q814yes—
303C23#53612yes—
304C23#53614yes—
305C23D4×Q818yes—
306C23D4×Q814yes—
307C23#55714yes—
308Q8—416no—
309Q8—432no—
310Q8—416no—
311Q8—88no—
312Q8—88no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—216no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—216no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—432no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—416no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4#39212yes—
353C2×C4#39214yes—
354C2×C4#39212yes—
355C2×C4B(4,𝔽2)14yes—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—216no—
366C22—24no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22C2×#39211yescharacteristic
372C22#243414yes—
373C22#243412yes—
374C22#254612yes—
375C22#254614yes—
376C22#265811yescharacteristic
377C22#286811yescharacteristic
378C22C2×B(4,𝔽2)12yes—
379C22#302812yes—
380C4—22no—
381C4—24no—
382C4—416no—
383C4—48no—
384C4—48no—
385C4—84no—
386C4—84no—
387C4—162no—
388C4#202114yes—
389C2—24no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—24no—
394C2#1519011yescharacteristic
395C2C2×#202111yescharacteristic
396C2#2344611yescharacteristic
397C2#2585214yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
86. #342Q8semidirect
263. C22×C4C4⋊Q8semidirect
280. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#24342
C2×#19962
#139222
#113442
C4×#3422