Finite Group Databasebeta

← #400781

Subgroups — #400781

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#442C211yesmaximal, characteristic
2C2×#2092C212yesmaximal, complemented
3C2×#1943C211yesmaximal, characteristic, complemented
4C2×#1862C212yesmaximal, complemented
5#13979C212yesmaximal, complemented
6#11325C212yesmaximal, complemented
7#11123C212yesmaximal, complemented
8#11022C212yesmaximal
9#10798C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×C4×D4C2212yes—
12C2×#442—28no—
13C2×#442—24no—
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#382C2212yes—
26#2092—28no—
27#1943—24no—
28#1862—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1439—24no—
35C4⋊#95C2212yes—
36C22⋊#95—24no—
37C4×#95C2211yescharacteristic
38C23×D4C2312yes—
39C24×C4—22no—
40C22×D8—24no—
41C22×D8—24no—
42C22×D8—22no—
43C22×D8—22no—
44C22×D8Q814yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—42no—
53C2×C4×D4—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—24no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#442—232no—
64C2×#107—22no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—42no—
69C2×#106—22no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102D418yes—
82C2×#97—24no—
83C2×#97D414yes—
84#390—24no—
85#390—24no—
86#390—24no—
87#390—42no—
88#389—24no—
89#389—24no—
90#382—24no—
91#382—42no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95D414yes—
96#340—24no—
97#340—24no—
98#336—24no—
99#336—24no—
100#336D414yes—
101C25—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4C2×D414yes—
114C22×D4C2×Q814yes—
115C22×D4#4114yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—48no—
122C23×C4—44no—
123C23×C4—84no—
124C23×C4C2×D414yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C2×D8—28no—
128C2×D8—216no—
129C2×D8—28no—
130C2×D8—216no—
131C2×D8—28no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D414yes—
144C22×C8C2×D414yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×Q812yes—
148C22×C8#4112yes—
149C4×D4—216no—
150C4×D4—48no—
151C4×D4—48no—
152C4×D4—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—216no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—84no—
180C2×C22⋊C4—84no—
181C2×C22⋊C4—84no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192#107—28no—
193#107—48no—
194#106—28no—
195#106—44no—
196#102—216no—
197#102—216no—
198#102—48no—
199#97—28no—
200#95—44no—
201#95—48no—
202#95—44no—
203#95—84no—
204C24—24no—
205C24—24no—
206C24—24no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—44no—
212C24—44no—
213C24—44no—
214C24—44no—
215C24C22×D411yescharacteristic
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4C22⋊Q818yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—416no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—88no—
272C22×C4—88no—
273C22×C4—816no—
274C22×C4—816no—
275C22×C4#12714yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C42−1+411yescharacteristic
285C22×C42−1+411yescharacteristic
286D8—232no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8#12014yes—
303C2×C8C22⋊Q814yes—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C22⋊C4—416no—
318C22⋊C4—816no—
319C22⋊C4—816no—
320C42—22no—
321C42—22no—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C23—28no—
328C23—48no—
329C23—48no—
330C23—48no—
331C23—48no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—48no—
338C23—48no—
339C23—44no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—84no—
349C23—84no—
350C23#52512yes—
351C23C2×#12711yescharacteristic
352C23D4212yes—
353C23#54711yescharacteristic
354C23D4×Q811yescharacteristic
355C23#57514yes—
356C23#57614yes—
357D4—28no—
358D4—28no—
359D4—44no—
360D4—44no—
361D4—44no—
362D4—44no—
363D4—44no—
364D4—44no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—832no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4C2×#12011yescharacteristic
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×C22⋊Q812yes—
396C2×C4#52511yescharacteristic
397C2×C4#52511yescharacteristic
398C2×C4#52512yes—
399C2×C4#52512yes—
400C2×C4#53511yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#53912yes—
403C2×C4#53912yes—
404C2×C4D4212yes—
405C2×C4#54712yes—
406C2×C4#54712yes—
407C2×C4#54711yescharacteristic
408C2×C4D4×Q811yescharacteristic
409C2×C4#55412yes—
410C2×C4#55912yes—
411C8—24no—
412C8—24no—
413C8—44no—
414C22—416no—
415C22—48no—
416C22—84no—
417C22—88no—
418C22—84no—
419C22—84no—
420C22—84no—
421C22—84no—
422C22—88no—
423C22—88no—
424C22—84no—
425C22#239711yescharacteristic
426C22#243311yescharacteristic
427C22#244912yes—
428C22#245512yes—
429C22#254211yescharacteristic
430C22#311114yes—
431C22#312214yes—
432C22#315312yes—
433C22#315512yes—
434C22#326814yes—
435C22#327314yes—
436C22#334714yes—
437C22#338914yes—
438C4—44no—
439C4—84no—
440C4—84no—
441C4—84no—
442C4—162no—
443C4#240611yescharacteristic
444C4#244912yes—
445C4#245512yes—
446C4#258011yescharacteristic
447C4#263311yescharacteristic
448C4#284811yescharacteristic
449C2—88no—
450C2#1480711yescharacteristic
451C2#2380411yescharacteristic
452C2#2381011yescharacteristic
453C2#2579312yes—
454C2#2639414yes—
455C2#2656514yes—
456C2#2854412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2092C2semidirect
3. C2×#1943C2semidirect
4. C2×#1862C2semidirect
5. #13979C2semidirect
6. #11325C2semidirect
7. #11123C2semidirect
9. #10798C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4422
C2×#20922
C2×#19432
C2×#18622
#139792
#113252
#111232
#110222
#107982