Finite Group Databasebeta

← #208446

Subgroups — #208446

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic
2#21914C211yesmaximal, characteristic
3#13750C211yesmaximal, characteristic
4#9910C212yesmaximal
5#9711C212yesmaximal
6#9145C212yesmaximal
7#9144C212yesmaximal
8#8219C212yesmaximal
9C4×#328C212yesmaximal
10#2857C2211yescharacteristic
11#2855C2212yes—
12#2851C2211yescharacteristic
13#2655—22no—
14#2514C412yescomplemented
15#2506C412yescomplemented
16#2493—22no—
17#2473—22no—
18#2453C412yescomplemented
19#2446C412yescomplemented
20#2336—22no—
21C4×C22⋊Q8C2212yes—
22C2×C4×C4⋊C4—22no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1917—22no—
27#1916—22no—
28#1817C2212yes—
29#1797C2212yes—
30#1749C2211yescharacteristic
31C4×C22⋊C8C2211yescharacteristic
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35C2×#328C2212yes—
36C4×#95—22no—
37C4×#95C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—22no—
45C2×#125C2×C411yescharacteristic
46C2×C22⋊Q8C2312yes—
47C2×C4×Q8C2×C412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D412yes—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—22no—
53C22×C42—22no—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2312yes—
59C2×C22⋊C8C2312yes—
60C2×#97—21no—
61C2×C4×C8—21no—
62#401—22no—
63#400—22no—
64#400—22no—
65#400—22no—
66#400—22no—
67#398—22no—
68#398—22no—
69#398—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2×C412yes—
74#389—22no—
75#389—22no—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#384C2×C411yescharacteristic
93#383—22no—
94#382C2×C411yescharacteristic
95#381—22no—
96#380—22no—
97#380C2312yes—
98C4×C4⋊C4—216no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107#376—24no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114#336—22no—
115#336—24no—
116#336—21no—
117#336—21no—
118#328—24no—
119#328—24no—
120#328—24no—
121#328—24no—
122C22×Q8C22×C412yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—81no—
133#125—48no—
134#125—81no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C4×Q8—82no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
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—22no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—24no—
162C2×C42—28no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—28no—
167C2×C42—28no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—81no—
171C2×C42—81no—
172C2×C42—82no—
173C2×C42—82no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42—82no—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D412yes—
185C4⋊C8—82no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190#97—81no—
191C4×C8—81no—
192#95—24no—
193#95—28no—
194#95—28no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
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#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95#4114yes—
217#95#4114yes—
218C24C22×D411yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—42no—
222C2×Q8—42no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4C2×#4112yes—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—216no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4#10418yes—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C42—216no—
298C42—24no—
299C42—24no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—42no—
304C42—41no—
305C42—41no—
306C42—44no—
307C42—82no—
308C42—88no—
309C42—88no—
310C42—84no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23C22×C22⋊C411yescharacteristic
315C23#52511yescharacteristic
316C23#52511yescharacteristic
317C23#57712yes—
318C23#57812yes—
319Q8—82no—
320Q8—82no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4C2×#10414yes—
377C2×C4C2×#10414yes—
378C2×C4C2×#10414yes—
379C2×C4C4×#4112yes—
380C2×C4#53212yes—
381C8—82no—
382C8—82no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22#229711yescharacteristic
392C22#288311yescharacteristic
393C22#288411yescharacteristic
394C22#289812yes—
395C22C22×#10412yes—
396C22#312012yes—
397C22#312311yescharacteristic
398C22#312711yescharacteristic
399C4—24no—
400C4—24no—
401C4—24no—
402C4—22no—
403C4—22no—
404C4—48no—
405C4—48no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4—82no—
413C2—21no—
414C2—22no—
415C2—21no—
416C2#1754812yes—
417C2#1754912yes—
418C2#2139811yescharacteristic
419C2#2141511yescharacteristic
420C2#2142311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2514C4semidirect
15. #2506C4semidirect
18. #2453C4semidirect
19. #2446C4semidirect

Maximal subgroups

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

subgroupindex
#350872
#219142
#137502
#99102
#97112
#91452
#91442
#82192
C4×#3282