Finite Group Databasebeta

← #362898

Subgroups — #362898

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#14347C211yesmaximal, characteristic
3#13778C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11353C212yesmaximal
6#10912C214yesmaximal
7#10846C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9#2329—24no—
10#2329—24no—
11C2×#391C412yescomplemented
12C2×#389C412yescomplemented
13C2×#387C412yescomplemented
14C2×#387C412yescomplemented
15C2×#385C2212yes—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C22×#95C2211yescharacteristic
22C2×#342C2218yes—
23C2×#342C2218yes—
24#1442—24no—
25#1436—24no—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C22⋊#95C2212yes—
29C23×Q8C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—28no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C412yes—
44C22×C22⋊C4C2314yes—
45C22×C22⋊C4C2312yes—
46C22×C22⋊C4C2314yes—
47C22×C22⋊C4C2312yes—
48C22×C42—22no—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51#391—28no—
52#389—28no—
53#387—28no—
54#387—28no—
55#385—24no—
56#385—24no—
57#380—22no—
58#380—22no—
59#380—24no—
60#380—28no—
61#380Q818yescomplemented
62C4×C22⋊C4—24no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4—44no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95C2×C412yes—
75C2×#95C2×C412yes—
76C2×#95C2×C412yes—
77C2×#95C2×C412yes—
78C2×#95C2×C412yes—
79C2×#95C2×C412yes—
80C2×#95C2×C412yes—
81C2×#95D414yes—
82C2×#95D414yes—
83#342—216no—
84#342—48no—
85C25C2411yescharacteristic
86C22×Q8—22no—
87C22×Q8—24no—
88C22×Q8—81no—
89C22×Q8—81no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—81no—
96C23×C4—81no—
97C23×C4—81no—
98C23×C4—81no—
99C23×C4—81no—
100C23×C4—81no—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×Q814yes—
116C23×C4#4114yes—
117C22⋊Q8—216no—
118C22⋊Q8—216no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—81no—
135C2×C4⋊C4—81no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4—82no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4C2×Q818yes—
153C2×C22⋊C4#4118yes—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—44no—
157C2×C42—82no—
158C2×C42—82no—
159C2×C42—82no—
160C2×C42—82no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—28no—
167#95—44no—
168#95—44no—
169#95—82no—
170#95—82no—
171#95—82no—
172#95—82no—
173#95—82no—
174#95—82no—
175#95C22⋊C418yes—
176#95C22⋊C418yes—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24C23×C411yescharacteristic
182C24C22×D411yescharacteristic
183C24C22×D411yescharacteristic
184C24C22×Q812yes—
185C24C2×#4114yes—
186C24C2×#4112yes—
187C24H(5,𝔽2)12yes—
188C242−1+412yes—
189C2×Q8—28no—
190C2×Q8—82no—
191C2×Q8—82no—
192C2×Q8—88no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—88no—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4#11714yes—
261C22×C4#11714yes—
262C22×C4C4×Q818yes—
263C22×C4C22⋊Q814yes—
264C22×C4C22⋊Q814yes—
265C22×C4#12414yes—
266C22×C4C4⋊Q814yes—
267C4⋊C4—28no—
268C4⋊C4—216no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—88no—
274C4⋊C4—88no—
275C22⋊C4—28no—
276C22⋊C4—28no—
277C22⋊C4—216no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C42—84no—
285C42—84no—
286C42—88no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—44no—
307C23—42no—
308C23—42no—
309C23C22×C22⋊C411yescharacteristic
310C23C2×#11712yes—
311C23C2×C4×Q812yes—
312C23#52012yes—
313C23C2×C22⋊Q812yes—
314C23#52512yes—
315C23#52512yes—
316C23C2×#12411yescharacteristic
317C23C2×C4⋊Q811yescharacteristic
318C23#53511yescharacteristic
319C23#53611yescharacteristic
320C23#53712yes—
321C23#55414yes—
322C23#55614yes—
323C23#55714yes—
324C23#55914yes—
325Q8—88no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#39118yes—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—44no—
409C22#228011yescharacteristic
410C22C2×#39111yescharacteristic
411C22#231111yescharacteristic
412C22#235514yes—
413C22#258012yes—
414C22#258112yes—
415C22#258412yes—
416C22#259212yes—
417C22#288311yescharacteristic
418C22#288311yescharacteristic
419C22#288311yescharacteristic
420C22#288411yescharacteristic
421C4—28no—
422C4—44no—
423C4—42no—
424C4—42no—
425C4—44no—
426C4—44no—
427C4—88no—
428C4—82no—
429C4—82no—
430C4—164no—
431C2—22no—
432C2—22no—
433C2—22no—
434C2—22no—
435C2—24no—
436C2#1392511yescharacteristic
437C2#2072411yescharacteristic
438C2#2072511yescharacteristic
439C2#2133311yescharacteristic
440C2#2133411yescharacteristic
441C2#2187611yescharacteristic
442C2#2187711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C2×#391C4semidirect
12. C2×#389C4semidirect
13. C2×#387C4semidirect
14. C2×#387C4semidirect
61. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#143472
#137782
#113812
#113532
#109122
#108462