Finite Group Databasebeta

← #375622

Subgroups — #375622

Order 512.

1353 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36641C211yesmaximal, characteristic
2#20364C211yesmaximal, characteristic, complemented
3C42×C4⋊C4C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11075C212yesmaximal
6#10933C212yesmaximal
7#10923C212yesmaximal
8#10886C212yesmaximal, complemented
9#10878C212yesmaximal, 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⋊C4—22no—
27C2×C4×C4⋊C4C2212yes—
28C2×C43C2212yes—
29C2×C43C2211yescharacteristic
30#1745—22no—
31#1744—22no—
32C2×#339C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1728C2212yes—
36#1728C2212yes—
37C4×#95—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C22×M16C2314yes—
41C2×#126—22no—
42C2×#126—22no—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C2×#117C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C42—24no—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42C2314yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#104—28no—
55C2×#104—28no—
56C2×#104—44no—
57C4×M16—28no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#391C2×C412yes—
61#391C2×C411yescharacteristic
62#391C2×C411yescharacteristic
63#391C2×C412yes—
64#390C2×C412yes—
65#390C2×C411yescharacteristic
66#390C2×C411yescharacteristic
67#390C2×C412yes—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2×C412yes—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#386—21no—
87#386—22no—
88#386—22no—
89#386—21no—
90#385—21no—
91#385—22no—
92#385—22no—
93#385—21no—
94#383—22no—
95#383—22no—
96C4×C4⋊C4—216no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—28no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C2×#95—22no—
107C2×#95—24no—
108C43—44no—
109C43C2×C412yes—
110C43C2×C412yes—
111C43D412yes—
112C43D412yes—
113#339—28no—
114#339—28no—
115#339—44no—
116#336—24no—
117#336—24no—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—22no—
121C2×#41—22no—
122C22×Q8C22×C411yescharacteristic
123C22×D4—21no—
124C22×D4C22×C411yescharacteristic
125C23×C4—24no—
126C23×C4—21no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16#4118yes—
138C22×C8—24no—
139#126—48no—
140#126—48no—
141#117—24no—
142#117—24no—
143#117—22no—
144#117—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—216no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—28no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—216no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—44no—
181C2×C42—48no—
182C2×C42—48no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4118yes—
200#104—416no—
201#97—28no—
202C4×C8—28no—
203#95—216no—
204#95—28no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217C24—21no—
218C24C22×D411yescharacteristic
219#41—44no—
220#41—44no—
221#41—44no—
222#41—44no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—216no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—48no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4114yes—
266C22×C4C2×#4114yes—
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269M16—28no—
270M16—28no—
271M16—44no—
272M16—44no—
273C2×C8—28no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—232no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C42—24no—
298C42—24no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—28no—
311C42—28no—
312C42—42no—
313C42—42no—
314C42—432no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—44no—
319C42—44no—
320C42#10418yescomplemented
321C23—22no—
322C23—22no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23C22×C22⋊C411yescharacteristic
329C23#52512yes—
330C23#53212yes—
331C23#53511yescharacteristic
332C23#53611yescharacteristic
333Q8—44no—
334D4—44no—
335D4—82no—
336D4—82no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—28no—
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—216no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—48no—
356C2×C4—416no—
357C2×C4—416no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—84no—
375C2×C4C2×#10414yes—
376C2×C4C2×#10414yes—
377C2×C4C2×#10414yes—
378C2×C4#42114yes—
379C2×C4C4×#4112yes—
380C2×C4C4×#4112yes—
381C2×C4#52012yes—
382C2×C4#52512yes—
383C2×C4#53212yes—
384C2×C4#53712yes—
385C2×C4#55114yes—
386C2×C4#55214yes—
387C2×C4#55314yes—
388C2×C4#55614yes—
389C8—48no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—81no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22#229711yescharacteristic
401C22#229711yescharacteristic
402C22#231111yescharacteristic
403C22#260312yes—
404C22#260412yes—
405C22C22×#10412yes—
406C22C2×#42112yes—
407C4—24no—
408C4—24no—
409C4—24no—
410C4—24no—
411C4—416no—
412C4—42no—
413C4—42no—
414C4—81no—
415C4—82no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4#237314yes—
420C4#260114yes—
421C2—22no—
422C2—22no—
423C2—161no—
424C2#1416611yescharacteristic
425C2#2138212yes—
426C2#2141511yescharacteristic
427C2#2141811yescharacteristic
428C2#2189912yes—
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. C42×C4⋊C4C2semidirect
4. #11226C2semidirect
8. #10886C2semidirect
9. #10878C2semidirect
12. #2863C4semidirect
13. #2863C4semidirect
14. #2860C4semidirect
15. #2860C4semidirect
320. C42#104semidirect

Maximal subgroups

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

subgroupindex
#366412
#203642
C42×C4⋊C42
#112262
#110752
#109332
#109232
#108862
#108782