Finite Group Databasebeta

← #381121

Subgroups — #381121

Order 512.

1295 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2#20377C211yesmaximal, characteristic, complemented
3#13852C211yesmaximal, characteristic, complemented
4#11226C212yesmaximal, complemented
5#11201C212yesmaximal, complemented
6#11075C212yesmaximal, complemented
7#10971C212yesmaximal
8#10933C212yesmaximal, complemented
9#10886C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#104C2214yes—
13C2×#407C2212yes—
14C2×C4×M16C2212yes—
15#2625C414yescomplemented
16#2624C414yescomplemented
17#2623—22no—
18#2622—22no—
19#2395—24no—
20#2297C2211yescharacteristic
21#2297C2211yescharacteristic
22#2296C2211yescharacteristic
23#2296C2211yescharacteristic
24#2280C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2212yes—
27#1750—24no—
28C2×#339C2214yes—
29C2×#339C2214yes—
30C2×#339C2214yes—
31#1728C2212yes—
32#1441—22no—
33#1439—22no—
34C4×#95C2212yes—
35C22×#41C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16C2314yes—
38#525—44no—
39C2×#123—21no—
40C2×C22⋊Q8C2×C412yes—
41C2×#121C2×C412yes—
42C2×#120—21no—
43C2×C4×D4—22no—
44C2×#117—44no—
45C2×#117C2×C414yes—
46C2×#117D414yes—
47C2×#117C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C22×C4⋊C4—22no—
51C22×C4⋊C4D412yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42C2314yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#104—28no—
58C2×#104—44no—
59C2×#104—44no—
60#407—28no—
61C2×C22⋊C8—24no—
62C4×M16—44no—
63C2×#97—22no—
64C2×C4×C8—22no—
65#400—24no—
66#399—24no—
67#398—24no—
68#397—24no—
69#397—24no—
70#397—24no—
71#394—24no—
72#394—24no—
73#391C2×C412yes—
74#391C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#389C2×C414yes—
78#389C2×C414yes—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#381—21no—
84#380C2×C414yes—
85#379—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—44no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—44no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—22no—
96#339—28no—
97#339—44no—
98#339—44no—
99#336—24no—
100C25C2×D411yescharacteristic
101C2×#41—24no—
102C2×#41—24no—
103C2×#41—22no—
104C2×#41—22no—
105C22×Q8C22×C411yescharacteristic
106C22×D4—21no—
107C22×D4—22no—
108C22×D4C22×C411yescharacteristic
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—21no—
115C23×C4C22×C412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16#4118yes—
127C22×C8—24no—
128#123—44no—
129C22⋊Q8—44no—
130#121—44no—
131#120—44no—
132C4×D4—48no—
133#117—24no—
134#117—24no—
135#117—22no—
136#117—22no—
137#117—48no—
138#117—44no—
139#117—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4C22⋊C414yes—
165C2×C22⋊C4C22⋊C414yes—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181#104—416no—
182C22⋊C8—216no—
183#97—44no—
184C4×C8—44no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—48no—
191#95—44no—
192#95—44no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—22no—
197C24—21no—
198C24—44no—
199C24C2×C22⋊C412yes—
200C24C22×D411yescharacteristic
201#41—44no—
202#41—44no—
203#41—44no—
204#41—44no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—84no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×C22⋊C414yes—
252C22×C4C2×C22⋊C414yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4114yes—
257C22×C4H(5,𝔽2)14yes—
258C22×C4H(5,𝔽2)12yes—
259C22×C42−1+412yes—
260M16—28no—
261M16—28no—
262M16—44no—
263M16—44no—
264C2×C8—28no—
265C2×C8—44no—
266C2×C8—44no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—416no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—416no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C22⋊C4—84no—
288C42—24no—
289C42—24no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42—44no—
298C42—44no—
299C42—44no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—24no—
305C23—22no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—41no—
312C23—44no—
313C23—41no—
314C23—82no—
315C23—82no—
316C23—84no—
317C23C22×C22⋊C411yescharacteristic
318C23#53212yes—
319C23#53412yes—
320C23#53511yescharacteristic
321C23#53611yescharacteristic
322Q8—44no—
323D4—44no—
324D4—88no—
325D4—82no—
326D4—82no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4C4×#4112yes—
362C2×C4#51812yes—
363C2×C4#52012yes—
364C2×C4#52512yes—
365C2×C4#53412yes—
366C2×C4#53712yes—
367C2×C4#53814yes—
368C2×C4#53914yes—
369C2×C4#54014yes—
370C2×C4#54214yes—
371C8—48no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—22no—
376C22—22no—
377C22—42no—
378C22—42no—
379C22—44no—
380C22—44no—
381C22—42no—
382C22—84no—
383C22—84no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—81no—
388C22#229711yescharacteristic
389C22#230811yescharacteristic
390C22#231111yescharacteristic
391C22#266112yes—
392C22#266312yes—
393C22#290114yes—
394C4—44no—
395C4—44no—
396C4—42no—
397C4—88no—
398C4—82no—
399C4—82no—
400C4—82no—
401C4—81no—
402C4#233514yes—
403C4#263014yes—
404C2—22no—
405C2—22no—
406C2—82no—
407C2—161no—
408C2#1428711yescharacteristic
409C2#2138412yes—
410C2#2177512yes—
411C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20377C2semidirect
3. #13852C2semidirect
4. #11226C2semidirect
5. #11201C2semidirect
6. #11075C2semidirect
8. #10933C2semidirect
9. #10886C2semidirect
15. #2625C4semidirect
16. #2624C4semidirect

Maximal subgroups

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

subgroupindex
#346352
#203772
#138522
#112262
#112012
#110752
#109712
#109332
#108862