Finite Group Databasebeta

← #163889

Subgroups — #163889

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34254C211yesmaximal, characteristic
2#22589C212yesmaximal
3#7596C212yesmaximal, complemented
4#7595C211yesmaximal, characteristic
5#7581C212yesmaximal, complemented
6#7579C212yesmaximal
7#7577C212yesmaximal, complemented
8#7562C212yesmaximal, complemented
9C2×#1272C211yesmaximal, characteristic
10C2×#519C411yescharacteristic
11#2356—22no—
12#2355—22no—
13#2343—22no—
14#2343—22no—
15#2320C412yes—
16#2320C412yes—
17#2317C411yescharacteristic
18#2310C411yescharacteristic
19#2296C2211yescharacteristic
20#2295C2212yes—
21#2295C2212yes—
22C2×#391C2211yescharacteristic
23#2274C411yescharacteristic
24C4×#117C2211yescharacteristic
25#1916—22no—
26#1916—22no—
27#1850—22no—
28#1839C2212yes—
29#1831C2212yes—
30#1817C2212yes—
31C4×C4⋊C8C2212yes—
32C8×C22⋊C4—22no—
33#1744C2212yes—
34C2×#336C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2211yescharacteristic
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×C8—22no—
47#519—24no—
48#519C818yescomplemented
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#391—21no—
66#391—21no—
67#391—22no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C411yescharacteristic
71#391C2×C411yescharacteristic
72#391C2311yescharacteristic
73#391C2311yescharacteristic
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2312yes—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#384—21no—
89#383—22no—
90#383—22no—
91#382—21no—
92#382—22no—
93#382—22no—
94#381—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4D412yes—
106C4×C22⋊C4D412yes—
107#376—22no—
108#376—22no—
109#376C2×C412yes—
110C2×#95—22no—
111C43C2312yes—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8C2×C814yes—
127C22×Q8C22×C411yescharacteristic
128C22×Q8C22×C411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D412yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—82no—
140C4×Q8—216no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8C2×C818yescomplemented
144C4×Q8C2×C818yescomplemented
145#117—22no—
146#117—28no—
147#117—28no—
148#117—42no—
149#117—42no—
150#117C2×C814yes—
151#117C2×C814yes—
152#117M1614yes—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—44no—
190C4⋊C8—82no—
191C22⋊C8—44no—
192C4×C8—44no—
193C4×C8—82no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24C2×C22⋊C411yescharacteristic
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—24no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—48no—
219C2×Q8—48no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8C22×C812yes—
225C2×Q8C22×C812yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C22×C812yes—
250C22×C4C2×M1612yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—88no—
266C4⋊C4—22no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4C22⋊C818yes—
281C4⋊C4C22×C814yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4C22⋊C814yes—
286C22⋊C4C22⋊C814yes—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—28no—
292C42—28no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42—41no—
297C42—42no—
298C42—41no—
299C42—42no—
300C42—42no—
301C42—82no—
302C42—82no—
303C42C22×C814yes—
304C42C2×M1614yes—
305C23—22no—
306C23—22no—
307C23C2×C22⋊C812yes—
308C23#41012yes—
309C23#41312yes—
310C23C22×C22⋊C411yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—48no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—162no—
357C2×C4C2×C22⋊C814yes—
358C2×C4C2×C22⋊C814yes—
359C2×C4C2×C22⋊C812yes—
360C2×C4C23×C811yescharacteristic
361C2×C4C22×M1611yescharacteristic
362C2×C4#57712yes—
363C2×C4#57812yes—
364C8—44no—
365C8—82no—
366C8—82no—
367C8—82no—
368C22—22no—
369C22—24no—
370C22—22no—
371C22—24no—
372C22#145814yes—
373C22C22×C22⋊C811yescharacteristic
374C22C2×#41011yescharacteristic
375C22C2×#41311yescharacteristic
376C22#289812yes—
377C22#290112yes—
378C4—24no—
379C4—24no—
380C4—21no—
381C4—21no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—42no—
386C4—44no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—82no—
394C2—24no—
395C2C2×#145811yescharacteristic
396C2#1743912yes—
397C2#1760912yes—
398C2#1768611yescharacteristic
399C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7596C2semidirect
5. #7581C2semidirect
7. #7577C2semidirect
8. #7562C2semidirect
48. #519C8semidirect
143. C4×Q8C2×C8semidirect
144. C4×Q8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#342542
#225892
#75962
#75952
#75812
#75792
#75772
#75622
C2×#12722