Finite Group Databasebeta

← #238464

Subgroups — #238464

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#21759C212yesmaximal
3C2×#1520C212yesmaximal
4#14275C211yesmaximal, characteristic
5#10407C214yesmaximal, complemented
6#10406C214yesmaximal, complemented
7C2×#1280C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C22×C22⋊C8C2212yes—
10#2659—22no—
11#2659—24no—
12#2659C2214yes—
13C2×#398—22no—
14C2×#395—22no—
15C2×#385—21no—
16C2×#380—21no—
17C2×#380—21no—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1892—24no—
22#1872C2214yes—
23#1872C2214yes—
24#1824—22no—
25#1787—22no—
26#1520—28no—
27C2×#328C2218yes—
28#1444C2212yes—
29#1442C2212yes—
30#1441—22no—
31C22⋊#95—21no—
32#1280C2214yes—
33C23×Q8D412yes—
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8D414yes—
45C2×C22⋊Q8C2314yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C2×#103—24no—
60C2×#103—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#398—28no—
66#398—28no—
67#395—28no—
68#395—28no—
69#385—44no—
70#380—24no—
71#380—24no—
72#380—24no—
73#380—24no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380D414yes—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95C2312yes—
88C2×#95C2312yes—
89#340—24no—
90#340—24no—
91#328—216no—
92C25C2×D411yescharacteristic
93C22×Q8—22no—
94C22×Q8—24no—
95C22×Q8—24no—
96C22×Q8—82no—
97C22×Q8—82no—
98C22×Q8C2×D414yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—81no—
111C23×C4—81no—
112C23×C4—81no—
113C23×C4—81no—
114C23×C4—81no—
115C23×C4—81no—
116C23×C4SD1614yes—
117C23×C4Q1614yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4#4114yes—
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C22×C8—24no—
128C22×C8—44no—
129C22×C8—82no—
130C22×C8—82no—
131C22⋊Q8—28no—
132C22⋊Q8—28no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178#103—84no—
179#103—84no—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—44no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—22no—
188#95—42no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95C2×D414yes—
193C24—41no—
194C24—41no—
195C24—41no—
196C24—41no—
197C24#12011yescharacteristic
198C24#12011yescharacteristic
199C24#12112yes—
200C24#12412yes—
201C24C2×SD1612yes—
202C24C2×Q1612yes—
203C24#13712yes—
204C24#13712yes—
205C24C22×D411yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—88no—
214C2×Q8—88no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#12012yes—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4#12112yes—
279C22×C4#12314yes—
280C22×C4#12314yes—
281C22×C4#12412yes—
282C22×C4C2×SD1614yes—
283C22×C4C2×Q1614yes—
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4C2×#4111yescharacteristic
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—88no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—48no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23#39412yes—
337C23#39412yes—
338C23#44812yes—
339C23#44812yes—
340C23#45112yes—
341C23#45112yes—
342C23#46114yes—
343C23#46214yes—
344C23#48714yes—
345C23#49114yes—
346C23C2×#12011yescharacteristic
347C23C2×#12111yescharacteristic
348C23C2×#12411yescharacteristic
349C23C22×SD1611yescharacteristic
350C23C22×Q1611yescharacteristic
351C23C2×#13711yescharacteristic
352C23C2×#13711yescharacteristic
353Q8—48no—
354Q8—84no—
355Q8—88no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—88no—
397C2×C4—88no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4#39414yes—
409C2×C4C2×#12112yes—
410C2×C4C2×#12312yes—
411C8—84no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—21no—
429C22—22no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—22no—
434C22—21no—
435C22—22no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22#204314yes—
441C22#204314yes—
442C22C2×#39411yescharacteristic
443C22C2×#44811yescharacteristic
444C22C2×#45111yescharacteristic
445C22#301911yescharacteristic
446C22#302011yescharacteristic
447C22#302611yescharacteristic
448C22#302611yescharacteristic
449C22C2×#46111yescharacteristic
450C22C2×#46211yescharacteristic
451C22C2×#48711yescharacteristic
452C22C2×#49111yescharacteristic
453C4—22no—
454C4—22no—
455C4—24no—
456C4—44no—
457C4—44no—
458C4—84no—
459C4—84no—
460C4—84no—
461C4—161no—
462C4—161no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—24no—
468C2—24no—
469C2#1828611yescharacteristic
470C2#1828711yescharacteristic
471C2#1829411yescharacteristic
472C2#1829511yescharacteristic
473C2#2349911yescharacteristic
474C2C2×#204311yescharacteristic
475C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10407C2semidirect
6. #10406C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#217592
C2×#15202
#142752
#104072
#104062
C2×#12802