Finite Group Databasebeta

← #375623

Subgroups — #375623

Order 512.

1177 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20364C211yesmaximal, characteristic, complemented
3#13755C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11075C212yesmaximal
6#10933C212yesmaximal
7#10924C212yesmaximal
8#10886C212yesmaximal, complemented
9#10879C212yesmaximal, complemented
10C22×#104C2214yes—
11C2×C4×M16C2212yes—
12#2863C412yescomplemented
13#2863C412yescomplemented
14#2860C412yescomplemented
15#2860C412yescomplemented
16#2852—22no—
17#2852—22no—
18C4×#126—22no—
19C4×#126—22no—
20#2297C2211yescharacteristic
21#2297C2211yescharacteristic
22#2297C2211yescharacteristic
23#2296C2211yescharacteristic
24#2296C2211yescharacteristic
25#2296C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2212yes—
28C2×C43C2211yescharacteristic
29#1745—22no—
30#1744—22no—
31C2×#339C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1728C2212yes—
35#1728C2212yes—
36C4⋊#95—22no—
37#1434—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C22×M16C2314yes—
41C2×#126—22no—
42C2×#126—22no—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42C2314yes—
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C2×#104—28no—
54C2×#104—28no—
55C2×#104—44no—
56C4×M16—28no—
57C2×#97—22no—
58C2×C4×C8—22no—
59#391C2×C412yes—
60#391C2×C411yescharacteristic
61#391C2×C411yescharacteristic
62#391C2×C412yes—
63#390C2×C412yes—
64#390C2×C411yescharacteristic
65#390C2×C411yescharacteristic
66#390C2×C412yes—
67#389C2×C412yes—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#386—21no—
86#386—22no—
87#386—21no—
88#386—22no—
89#385—21no—
90#385—22no—
91#385—22no—
92#385—21no—
93#383—22no—
94#383—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—28no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104#376—44no—
105C2×#95—22no—
106C2×#95—24no—
107C2×#95—24no—
108C43C2×C412yes—
109C43C2×C412yes—
110C43D412yes—
111C43D412yes—
112#339—28no—
113#339—28no—
114#339—44no—
115#336—24no—
116#336—24no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—22no—
120C2×#41—22no—
121C22×Q8C22×C411yescharacteristic
122C22×D4—21no—
123C22×D4C22×C411yescharacteristic
124C23×C4—24no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2411yescharacteristic
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16#4118yes—
137C22×C8—24no—
138#126—48no—
139#126—48no—
140#117—24no—
141#117—24no—
142#117—22no—
143#117—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—24no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4118yes—
196#104—416no—
197#97—28no—
198C4×C8—28no—
199#95—28no—
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#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—48no—
213#95—48no—
214C24—21no—
215C24C22×D411yescharacteristic
216#41—44no—
217#41—44no—
218#41—44no—
219#41—44no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—28no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4114yes—
263C22×C4C2×#4114yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+412yes—
266M16—28no—
267M16—28no—
268M16—44no—
269M16—44no—
270C2×C8—28no—
271C2×C8—44no—
272C2×C8—44no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—416no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—28no—
307C42—28no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42—44no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23C22×C22⋊C411yescharacteristic
325C23#52512yes—
326C23#53212yes—
327C23#53511yescharacteristic
328C23#53611yescharacteristic
329Q8—44no—
330D4—44no—
331D4—82no—
332D4—82no—
333C2×C4—28no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—88no—
371C2×C4—88no—
372C2×C4—84no—
373C2×C4#42114yes—
374C2×C4#42114yes—
375C2×C4C4×#4112yes—
376C2×C4C4×#4112yes—
377C2×C4#52012yes—
378C2×C4#52512yes—
379C2×C4#53212yes—
380C2×C4#53712yes—
381C2×C4#55114yes—
382C2×C4#55214yes—
383C2×C4#55314yes—
384C2×C4#55614yes—
385C8—48no—
386C22—24no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—81no—
392C22—82no—
393C22—82no—
394C22—81no—
395C22—81no—
396C22#229711yescharacteristic
397C22#229711yescharacteristic
398C22#231111yescharacteristic
399C22#260312yes—
400C22#260412yes—
401C22C2×#42112yes—
402C22C2×#42112yes—
403C4—24no—
404C4—24no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—88no—
410C4—81no—
411C4—82no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4#237314yes—
416C4#260114yes—
417C2—22no—
418C2—22no—
419C2—161no—
420C2#1416611yescharacteristic
421C2#2138312yes—
422C2#2141611yescharacteristic
423C2#2141711yescharacteristic
424C2#2190012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20364C2semidirect
3. #13755C2semidirect
4. #11226C2semidirect
8. #10886C2semidirect
9. #10879C2semidirect
12. #2863C4semidirect
13. #2863C4semidirect
14. #2860C4semidirect
15. #2860C4semidirect

Maximal subgroups

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

subgroupindex
#366412
#203642
#137552
#112262
#110752
#109332
#109242
#108862
#108792