Finite Group Databasebeta

← #379871

Subgroups — #379871

Order 512.

1696 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4#13870C212yesmaximal
5#11222C214yesmaximal
6#11070C214yesmaximal
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11#2484—24no—
12#2484C418yescomplemented
13C2×#400—21no—
14C2×#398—21no—
15#2317—24no—
16#2316—21no—
17#2316—21no—
18#2315C2214yes—
19#2296—24no—
20C2×#387—22no—
21C2×#385C2212yes—
22#2285—22no—
23C2×#381C2212yes—
24C2×#381C2214yes—
25C2×#381C2212yes—
26C2×#381C2211yescharacteristic
27#1820—24no—
28C2×#342C2214yes—
29C2×#342C2218yes—
30#1441—24no—
31#1436—24no—
32C22⋊#95C2214yes—
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—24no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—48no—
44C2×#117C2×C414yes—
45C2×#117D414yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D414yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2312yes—
61C22×C42D412yes—
62C22×C42D414yes—
63#400—28no—
64#398—28no—
65#389—28no—
66#389—28no—
67#387—216no—
68#385—216no—
69#385—24no—
70#385C2×C418yes—
71#384—24no—
72#382—24no—
73#381—22no—
74#381—24no—
75#381—24no—
76#381—24no—
77#381—48no—
78#381C2×C414yes—
79#381C2×C418yes—
80#380—24no—
81#380—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—28no—
90C2×#95—22no—
91C2×#95—28no—
92C2×#95—22no—
93#342—28no—
94#342—48no—
95C25C2411yescharacteristic
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—44no—
111C23×C4C22×C412yes—
112C23×C4C22×C412yes—
113C23×C4C22×C412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D418yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4#4114yes—
123#117—28no—
124#117—28no—
125#117—28no—
126#117—28no—
127#117—48no—
128#117—48no—
129#117—48no—
130#117—48no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22⋊C414yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4C22×C414yes—
169C2×C22⋊C4C22×C414yes—
170C2×C22⋊C4C2×D414yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—48no—
177C2×C42C22⋊C418yes—
178#95—28no—
179#95—28no—
180#95—24no—
181#95—28no—
182#95—28no—
183#95—24no—
184#95—28no—
185#95—416no—
186#95—416no—
187#95—44no—
188#95—44no—
189C24—42no—
190C24—42no—
191C24C23×C411yescharacteristic
192C24C22×D411yescharacteristic
193C24C22×D412yes—
194C24C22×D411yescharacteristic
195C24C2×#4112yes—
196C24H(5,𝔽2)14yes—
197C242−1+414yes—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—48no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C4×D414yes—
259C22×C4C4×D414yes—
260C22×C4#12011yescharacteristic
261C22×C4#12012yes—
262C22×C4#12011yescharacteristic
263C22×C4#12114yes—
264C22×C4#12114yes—
265C22×C4#12312yes—
266C22×C4#12312yes—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—416no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—88no—
276C4⋊C4—88no—
277C22⋊C4—28no—
278C22⋊C4—28no—
279C22⋊C4—28no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C42—28no—
286C42—28no—
287C42—48no—
288C42—48no—
289C23—24no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—48no—
300C23—48no—
301C23C22×C22⋊C411yescharacteristic
302C23C2×C4×D412yes—
303C23#52014yes—
304C23C2×#12011yescharacteristic
305C23C2×#12112yes—
306C23C2×#12311yescharacteristic
307C23#53414yes—
308C23#53512yes—
309C23#53612yes—
310C23#53614yes—
311C23#53714yes—
312C23#53918yes—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—416no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—88no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4#38612yes—
366C2×C4#38612yes—
367C2×C4#38614yes—
368C22—28no—
369C22—24no—
370C22—24no—
371C22—28no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—44no—
381C22—44no—
382C22—48no—
383C22C2×#38611yescharacteristic
384C22#231112yes—
385C22#232214yes—
386C22#261311yescharacteristic
387C22#261411yescharacteristic
388C22#261814yes—
389C22#262312yes—
390C22#302512yes—
391C22#302812yes—
392C4—22no—
393C4—24no—
394C4—22no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4—84no—
399C4—82no—
400C4—88no—
401C4—164no—
402C2—22no—
403C2—24no—
404C2—24no—
405C2—22no—
406C2#1425611yescharacteristic
407C2#2095911yescharacteristic
408C2#2096211yescharacteristic
409C2#2963712yes—
410C2#2965912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2484C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
#340522
C2×#18202
#138702
#112222
#110702
#107262