Finite Group Databasebeta

← #375624

Subgroups — #375624

Order 512.

1129 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34639C211yesmaximal, characteristic
2#20365C211yesmaximal, characteristic, complemented
3#13759C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11075C212yesmaximal
6#10933C212yesmaximal
7#10924C211yesmaximal, characteristic
8#10923C211yesmaximal, characteristic
9#10886C212yesmaximal, complemented
10#10879C211yesmaximal, characteristic, complemented
11#10878C211yesmaximal, characteristic, complemented
12C22×#104C2212yes—
13C22×#104C2212yes—
14C2×C4×M16C2212yes—
15#2645C414yescomplemented
16#2644C414yescomplemented
17#2643—24no—
18#2401—24no—
19#2297C2211yescharacteristic
20#2297C2211yescharacteristic
21#2297C2211yescharacteristic
22#2296C2211yescharacteristic
23#2296C2211yescharacteristic
24#2296C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27C2×#376C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1746—24no—
30C2×#339C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36#1728C2212yes—
37#1728C2211yescharacteristic
38#1728C2211yescharacteristic
39#1436—24no—
40C4×#95C2212yes—
41C22×#41C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×#117C2311yescharacteristic
45C2×#117C2311yescharacteristic
46C2×#117C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C42—22no—
49C22×C42C2312yes—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#104—24no—
55C2×#104—24no—
56C2×#104—24no—
57C2×#104—24no—
58C2×#104—42no—
59C2×#104—42no—
60C4×M16—24no—
61C4×M16—24no—
62C2×#97—22no—
63C2×C4×C8—22no—
64#401—24no—
65#400—24no—
66#400—24no—
67#400—24no—
68#397—24no—
69#397—24no—
70#397—24no—
71#396—24no—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2×C414yes—
81#386—22no—
82#386—22no—
83#386—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95#376—42no—
96#376C2×C414yes—
97#376D414yes—
98C2×#95—22no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95D414yes—
102C43—42no—
103#339—24no—
104#339—24no—
105#339—24no—
106#339—24no—
107#339—42no—
108#339—42no—
109#336—24no—
110#336—22no—
111#336—22no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C22×Q8C22×C411yescharacteristic
119C22×D4—21no—
120C22×D4C22×C411yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16#4114yes—
139C2×M16#4114yes—
140C22×C8—22no—
141C22×C8—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—44no—
182C2×C42—48no—
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C22×C412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42#4114yes—
190C2×C42#4114yes—
191#104—48no—
192#104—48no—
193#97—28no—
194C4×C8—28no—
195#95—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—48no—
203#95—44no—
204#95C22⋊C414yes—
205#95C22⋊C414yes—
206#95C22⋊C414yes—
207#95C22⋊C414yes—
208C24—21no—
209C24C22×D411yescharacteristic
210#41—42no—
211#41—42no—
212#41—42no—
213#41—42no—
214#41—42no—
215#41—42no—
216#41—42no—
217#41—42no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—28no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C414yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4112yes—
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4C2×#4112yes—
265C22×C4H(5,𝔽2)12yes—
266C22×C42−1+412yes—
267M16—24no—
268M16—24no—
269M16—24no—
270M16—24no—
271M16—42no—
272M16—42no—
273M16—42no—
274M16—42no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—416no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—84no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—416no—
310C42—42no—
311C42—41no—
312C42—41no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C23—24no—
319C23—41no—
320C23—44no—
321C23—41no—
322C23—41no—
323C23C22×C22⋊C411yescharacteristic
324C23#52512yes—
325C23#53212yes—
326C23#53511yescharacteristic
327C23#53611yescharacteristic
328Q8—42no—
329Q8—42no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—82no—
334C2×C4—28no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4C4×#4111yescharacteristic
370C2×C4C4×#4111yescharacteristic
371C2×C4C4×#4112yes—
372C2×C4#52012yes—
373C2×C4#52512yes—
374C2×C4#53211yescharacteristic
375C2×C4#53211yescharacteristic
376C2×C4#53712yes—
377C2×C4#55112yes—
378C2×C4#55112yes—
379C2×C4#55212yes—
380C2×C4#55212yes—
381C2×C4#55312yes—
382C2×C4#55312yes—
383C2×C4#55612yes—
384C2×C4#55612yes—
385C8—44no—
386C8—44no—
387C22—24no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—81no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22#229711yescharacteristic
397C22#229711yescharacteristic
398C22#231111yescharacteristic
399C22#260312yes—
400C22#260412yes—
401C22#290114yes—
402C4—48no—
403C4—44no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—81no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4#237312yes—
412C4#237312yes—
413C4#260112yes—
414C4#260112yes—
415C2—22no—
416C2—22no—
417C2—161no—
418C2#1416611yescharacteristic
419C2#2138412yes—
420C2#2141912yes—
421C2#2190112yes—
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×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20365C2semidirect
3. #13759C2semidirect
4. #11226C2semidirect
9. #10886C2semidirect
10. #10879C2semidirect
11. #10878C2semidirect
15. #2645C4semidirect
16. #2644C4semidirect

Maximal subgroups

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

subgroupindex
#346392
#203652
#137592
#112262
#110752
#109332
#109242
#109232
#108862
#108792
#108782