Finite Group Databasebeta

← #232400

Subgroups — #232400

Order 512.

1568 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 258 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×#3189C212yesmaximal
2#35011C211yesmaximal, characteristic
3C22×#328C212yesmaximal
4#14159C211yesmaximal, characteristic
5#9972C214yesmaximal, complemented
6#9967C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C22×C22⋊Q8C2211yescharacteristic
10#3189—28no—
11#3189—28no—
12C2×#484—22no—
13C2×#484C2214yes—
14C2×#467—24no—
15C2×#451—22no—
16C2×#451C2214yes—
17C22×C22⋊C8C2212yes—
18#2857—22no—
19#2659C2214yes—
20#2583—24no—
21#2502—22no—
22#2434—22no—
23#2315—22no—
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26C2×#328—28no—
27C2×#328C2218yes—
28#1446C2212yes—
29#1445C2212yes—
30#1436—22no—
31C22⋊#95—22no—
32#1280C2214yes—
33C23×Q8D412yes—
34C23×Q8C2311yescharacteristic
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C22×Q16—24no—
38C23×C8—22no—
39C2×C4⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8D414yes—
51C2×C22⋊Q8D414yes—
52C2×C22⋊Q8C2314yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D412yes—
60C22×C42D412yes—
61#484—216no—
62#484—28no—
63#484—28no—
64#467—232no—
65#451—216no—
66#451—28no—
67#451—28no—
68C2×#107—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8D418yes—
74C2×C22⋊C8C2314yes—
75#398—28no—
76#395—28no—
77#391—24no—
78#389—28no—
79#385—28no—
80#385—28no—
81#384—22no—
82#380—28no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95C2312yes—
90C2×#95C2312yes—
91#328—216no—
92C25C2×D411yescharacteristic
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—82no—
100C22×Q8—84no—
101C22×Q8C2×D414yes—
102C22×Q8C2×D412yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—44no—
109C23×C4—42no—
110C23×C4—44no—
111C23×C4—82no—
112C23×C4—82no—
113C23×C4C2×D414yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D414yes—
119C23×C4#4112yes—
120C23×C4C2411yescharacteristic
121C2×Q16—216no—
122C2×Q16—88no—
123C22×C8—24no—
124C22×C8—28no—
125C22×C8—84no—
126C4⋊Q8—84no—
127C22⋊Q8—28no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C22⋊Q8—28no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4Q1618yescomplemented
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—82no—
172#107—216no—
173#103—216no—
174#103—216no—
175C22⋊C8—28no—
176C22⋊C8—24no—
177C22⋊C8C2×D418yes—
178#95—216no—
179#95—24no—
180#95—24no—
181#95—48no—
182#95—42no—
183#95—84no—
184C24—42no—
185C24—42no—
186C24#12012yes—
187C24#12112yes—
188C24#12712yes—
189C24C2×Q1614yes—
190C24#13714yes—
191C24C22×D411yescharacteristic
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—48no—
196C2×Q8—48no—
197C2×Q8—48no—
198C2×Q8—42no—
199C2×Q8—44no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—88no—
203C2×Q8—84no—
204C2×Q8—88no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
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—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—48no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—816no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12014yes—
242C22×C4#12014yes—
243C22×C4#12114yes—
244C22×C4#12114yes—
245C22×C4#12112yes—
246C22×C4#12712yes—
247C22×C4C2×Q1618yes—
248C22×C4C22×D412yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C2×#4111yescharacteristic
252Q16—816no—
253C2×C8—28no—
254C2×C8—48no—
255C2×C8—88no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—88no—
267C4⋊C4—88no—
268C22⋊C4—216no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C42—44no—
274C42—82no—
275C23—22no—
276C23—24no—
277C23—22no—
278C23—24no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—48no—
283C23—42no—
284C23—48no—
285C23#39214yes—
286C23#45118yes—
287C23#46218yes—
288C23#49414yes—
289C23#49714yes—
290C23C2×#12011yescharacteristic
291C23C2×#12111yescharacteristic
292C23C2×#12711yescharacteristic
293C23C22×Q1612yes—
294C23C2×#13712yes—
295Q8—48no—
296Q8—48no—
297Q8—84no—
298Q8—84no—
299Q8—88no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—28no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4—816no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#39212yes—
338C2×C4#39212yes—
339C2×C4C2×#12012yes—
340C2×C4C2×#12112yes—
341C8—48no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—216no—
351C22—44no—
352C22—48no—
353C22—44no—
354C22#201618yes—
355C22C2×#39211yescharacteristic
356C22C2×#45112yes—
357C22#301812yes—
358C22#302612yes—
359C22C2×#46212yes—
360C22C2×#49411yescharacteristic
361C22C2×#49711yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—48no—
367C4—84no—
368C4—82no—
369C4—84no—
370C4—82no—
371C4—84no—
372C2—21no—
373C2—21no—
374C2—22no—
375C2—28no—
376C2#1838314yes—
377C2#2340611yescharacteristic
378C2C2×#201611yescharacteristic
379C2#2344411yescharacteristic
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
5. #9972C2semidirect
6. #9967C2semidirect
168. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#350112
C22×#3282
#141592
#99722
#99672
#79072