Finite Group Databasebeta

← #6519067

Subgroups — #6519067

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic, complemented
3#27609C214yesmaximal, complemented
4#25014C214yesmaximal, complemented
5#25012C214yesmaximal, complemented
6#24972C214yesmaximal, complemented
7#24966C214yesmaximal, complemented
8#24012C214yesmaximal, complemented
9#21981C212yesmaximal, complemented
10#21787C211yesmaximal, characteristic, complemented
11#20737C212yesmaximal, complemented
12C2×#537C2212yes—
13C2×#532C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#481C2214yes—
16C2×#483C2214yes—
17C2×#431C2212yes—
18#2857—21no—
19#2856—21no—
20#2620—22no—
21#2538C2214yes—
22#2537C2214yes—
23#2416—24no—
24#2379—22no—
25#2369—21no—
26#2366—21no—
27#2354C2214yes—
28D4×C4⋊C4C2214yes—
29#2324—24no—
30#2311C2214yes—
31#2311C2212yes—
32#2285C2212yes—
33C2×#384C2211yescharacteristic
34#2077C2218yes—
35#2077C2218yes—
36#2035C2218yes—
37#2034C2218yes—
38#1954C2214yes—
39#1954C2214yes—
40#1948C2214yes—
41#1948C2214yes—
42#1875C2214yes—
43#1873C2214yes—
44#1853C2218yes—
45#1853C2218yes—
46#1844C2214yes—
47#1834C2214yes—
48#1825C2214yes—
49#1825C2218yes—
50#1803C2212yes—
51#1803C2212yes—
52#1802C2212yes—
53#1802C2212yes—
54#1788C2214yes—
55#1780C2214yes—
56C4×#103C2214yes—
57C4×#102C2214yes—
58C22×#41C2311yescharacteristic
59C22×#41C2312yes—
60#537—44no—
61#537—44no—
62#532—28no—
63C2×C4⋊Q8D411yescharacteristic
64C2×#127D411yescharacteristic
65C2×#124D412yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2314yes—
68C2×#121—22no—
69C2×#121—24no—
70C2×#121—24no—
71C2×#121D414yes—
72C2×#121C2314yes—
73C4×#41—28no—
74C2×C4×Q8C2312yes—
75C2×C4×D4—22no—
76C2×C4×D4—24no—
77C2×C4×D4C2312yes—
78C2×C4×D4C2314yes—
79C2×#117—22no—
80C2×#117—21no—
81C2×#117C2311yescharacteristic
82C2×#117C2314yes—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#483—48no—
89#481—48no—
90#431—28no—
91C2×#106C2318yes—
92C2×#106C2318yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103C2314yes—
96C2×#103C2314yes—
97C2×#102C2314yes—
98C2×#102C2314yes—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2318yes—
103C2×C4×C8C2314yes—
104#396—28no—
105#394—28no—
106#391—22no—
107#391—24no—
108#390—24no—
109#390—22no—
110#390—24no—
111#390—24no—
112#389—24no—
113#389—24no—
114#388—28no—
115#386—24no—
116#386—24no—
117#386—24no—
118#385—28no—
119#384—21no—
120#384—21no—
121#384C2312yes—
122#384C2312yes—
123#384C2314yes—
124#382—22no—
125#380—22no—
126#380—24no—
127#380C2314yes—
128#380C2318yes—
129#380C2318yes—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2314yes—
133#340C2314yes—
134#340C2314yes—
135#340C2314yes—
136#340C2314yes—
137#340C2318yes—
138#340C2318yes—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—28no—
142C2×#41—24no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41—44no—
146C2×#41—42no—
147C22×Q8C2×D412yes—
148C22×Q8C2411yescharacteristic
149C22×D4—21no—
150C22×D4—24no—
151C22×D4C2×D414yes—
152C22×D4C2×D414yes—
153C22×D4C2×D412yes—
154C22×D4C2411yescharacteristic
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4C2×D412yes—
160C23×C4C2414yes—
161C23×C4C2411yescharacteristic
162C22×C8C2414yes—
163C22×C8C2414yes—
164C4⋊Q8—24no—
165#127—24no—
166#124—28no—
167C22⋊Q8—28no—
168C22⋊Q8—44no—
169C22⋊Q8—48no—
170#121—216no—
171#121—28no—
172#121—44no—
173#121—48no—
174#121—416no—
175#121—88no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—48no—
183C4×D4—88no—
184#117—24no—
185#117—44no—
186#117—48no—
187#117—84no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4C2×D418yes—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—44no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4118yes—
225C2×C42C2412yes—
226#106—48no—
227#106—48no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230#103—28no—
231#103—44no—
232#102—28no—
233#102—44no—
234C22⋊C8—28no—
235C22⋊C8—48no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C4×C8—28no—
239#95—24no—
240#95—28no—
241#95—24no—
242#95—28no—
243C24—21no—
244C24C22×D414yes—
245C24C22×D411yescharacteristic
246#41—48no—
247#41—48no—
248#41—88no—
249#41—88no—
250C2×Q8—24no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8H(5,𝔽2)12yes—
257C2×D4—24no—
258C2×D4—28no—
259C2×D4—28no—
260C2×D4—24no—
261C2×D4—28no—
262C2×D4—28no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—48no—
273C2×D4—88no—
274C2×D4H(5,𝔽2)12yes—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4#12318yes—
300C22×C4C22×D414yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C2×#4114yes—
305C22×C4C2511yescharacteristic
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C4⋊C4—28no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C22⋊C4—216no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C22⋊C4—88no—
339C22⋊C4—88no—
340C42—42no—
341C42—44no—
342C42—48no—
343C42—84no—
344C42#12314yes—
345C42#12314yes—
346C42C2×#4114yes—
347C42C2×#4114yes—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—28no—
357C23—28no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23C2×#12312yes—
363C23C23×D411yescharacteristic
364Q8—24no—
365Q8—42no—
366Q8—84no—
367D4—24no—
368D4—48no—
369D4—416no—
370D4—42no—
371D4—84no—
372D4—88no—
373D4—168no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—216no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—48no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—88no—
402C2×C4—164no—
403C2×C4—168no—
404C2×C4C2×#12312yes—
405C2×C4C2×#12314yes—
406C2×C4C2×#12314yes—
407C2×C4#57714yes—
408C2×C4#57714yes—
409C2×C4C22×#4112yes—
410C2×C4C2×H(5,𝔽2)11yescharacteristic
411C2×C4C2×H(5,𝔽2)11yescharacteristic
412C2×C4#58512yes—
413C8—44no—
414C8—44no—
415C22—24no—
416C22—22no—
417C22—42no—
418C22—41no—
419C22—41no—
420C22—48no—
421C22—48no—
422C22—42no—
423C22—44no—
424C22—42no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—82no—
429C22—82no—
430C22C22×#12311yescharacteristic
431C22#348411yescharacteristic
432C22#348511yescharacteristic
433C22C2×#57712yes—
434C22C2×#57712yes—
435C4—24no—
436C4—22no—
437C4—44no—
438C4—42no—
439C4—84no—
440C4—82no—
441C4—84no—
442C4—84no—
443C4#309918yes—
444C4#349412yes—
445C4#349412yes—
446C2—42no—
447C2—84no—
448C2—82no—
449C2#3554011yescharacteristic
450C2C2×#309912yes—
451C2#6166211yescharacteristic
452C2#6166411yescharacteristic
453C2#6179212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34368C2semidirect
3. #27609C2semidirect
4. #25014C2semidirect
5. #25012C2semidirect
6. #24972C2semidirect
7. #24966C2semidirect
8. #24012C2semidirect
9. #21981C2semidirect
10. #21787C2semidirect
11. #20737C2semidirect

Maximal subgroups

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

subgroupindex
#352732
#343682
#276092
#250142
#250122
#249722
#249662
#240122
#219812
#217872
#207372