Finite Group Databasebeta

← #415364

Subgroups — #415364

Order 512.

1444 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#13923C211yesmaximal, characteristic
3#13923C212yesmaximal
4#11390C212yesmaximal
5#11390C212yesmaximal
6#11287C212yesmaximal
7#11287C212yesmaximal
8#10854C212yesmaximal
9C4×#342C211yesmaximal, characteristic
10C2×#543C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12#2857—22no—
13#2659—22no—
14#2549—24no—
15#2440—24no—
16#2315C2212yes—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#342C2212yes—
22C2×#342C2214yes—
23C2×#342C2214yes—
24C2×#342C2212yes—
25#1441—22no—
26#1441—24no—
27#1436—22no—
28#1436—24no—
29C22⋊#95C2211yescharacteristic
30C22⋊#95C2211yescharacteristic
31C22⋊#95C2214yes—
32C22⋊#95C2214yes—
33C22⋊#95C2211yescharacteristic
34C22⋊#95C2211yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38#543—24no—
39#543—24no—
40#543—44no—
41C2×C4⋊Q8—22no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—28no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4Q814yescomplemented
53C22×C22⋊C4C2311yescharacteristic
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2314yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42—22no—
60C22×C42D412yes—
61#398—28no—
62#395—28no—
63#391—24no—
64#389—24no—
65#389—24no—
66#385—28no—
67#384—22no—
68#380—24no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—24no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—24no—
77C4×C22⋊C4—24no—
78C2×#95—28no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—28no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D414yes—
88#342—24no—
89#342—44no—
90#342—44no—
91#342Q818yescomplemented
92C25C2411yescharacteristic
93C22×Q8—24no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4—82no—
107C23×C4—82no—
108C23×C4C2×D418yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×Q812yes—
116C23×C4C2×Q812yes—
117C23×C4C2×Q812yes—
118C23×C4#4112yes—
119C23×C4#4114yes—
120C4⋊Q8—84no—
121C22⋊Q8—416no—
122C22⋊Q8—416no—
123C22⋊Q8—48no—
124C22⋊Q8—48no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—84no—
144C2×C4⋊C4—82no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C2×Q818yes—
167C2×C22⋊C4C2×Q814yes—
168C2×C22⋊C4#4114yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—44no—
172C2×C42—82no—
173C2×C42—84no—
174#95—28no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—44no—
179#95—44no—
180#95—416no—
181#95—44no—
182#95—42no—
183#95—82no—
184#95—82no—
185#95—88no—
186C24—42no—
187C24—42no—
188C24C22×D411yescharacteristic
189C24C22×D412yes—
190C24C22×Q811yescharacteristic
191C24C22×Q812yes—
192C24C2×#4112yes—
193C24C2×#4111yescharacteristic
194C24C2×#4112yes—
195C24H(5,𝔽2)12yes—
196C242−1+412yes—
197C2×Q8—48no—
198C2×Q8—84no—
199C2×Q8—84no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—28no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—48no—
235C22×C4—41no—
236C22×C4—44no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—82no—
258C22×C4#12011yescharacteristic
259C22×C4#12012yes—
260C22×C4#12011yescharacteristic
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q814yes—
264C22×C4C22⋊Q814yes—
265C22×C4#12414yes—
266C22×C4#12412yes—
267C22×C4#12414yes—
268C22×C4#12414yes—
269C22×C4#12412yes—
270C22×C4C4⋊Q814yes—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—416no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—28no—
284C22⋊C4—28no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C42—44no—
294C42—82no—
295C42—88no—
296C23—22no—
297C23—24no—
298C23—22no—
299C23—24no—
300C23—41no—
301C23—42no—
302C23—41no—
303C23—41no—
304C23—42no—
305C23—41no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23C2×#12011yescharacteristic
311C23C2×C22⋊Q811yescharacteristic
312C23C2×C22⋊Q812yes—
313C23C2×#12411yescharacteristic
314C23C2×#12411yescharacteristic
315C23C2×#12412yes—
316C23C2×C4⋊Q811yescharacteristic
317C23#53314yes—
318C23#53511yescharacteristic
319C23#53611yescharacteristic
320C23#54214yes—
321C23#54614yes—
322C23D4×Q814yes—
323C23#55412yes—
324C23#55612yes—
325C23#55712yes—
326C23Q8212yes—
327Q8—168no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—416no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—88no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—88no—
373C2×C4—88no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4#39312yes—
385C2×C4#39314yes—
386C2×C4#39312yes—
387C22—22no—
388C22—22no—
389C22—28no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—28no—
400C22—22no—
401C22—44no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22C2×#39311yescharacteristic
408C22#243411yescharacteristic
409C22#243511yescharacteristic
410C22#245114yes—
411C22#249214yes—
412C22#256711yescharacteristic
413C22#256811yescharacteristic
414C22#264011yescharacteristic
415C22#286211yescharacteristic
416C22#302512yes—
417C22#302612yes—
418C4—22no—
419C4—22no—
420C4—24no—
421C4—44no—
422C4—44no—
423C4—44no—
424C4—88no—
425C4—88no—
426C4—82no—
427C4—161no—
428C4—161no—
429C2—24no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—24no—
435C2#1519111yescharacteristic
436C2#2346711yescharacteristic
437C2#2346811yescharacteristic
438C2#2585412yes—
439C2#2590112yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ 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
52. C22×C4⋊C4Q8semidirect
91. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#139232
#113902
#112872
#108542
C4×#3422