Finite Group Databasebeta

← #190404

Subgroups — #190404

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3C2×#1817C211yesmaximal, characteristic
4#8703C214yesmaximal, complemented
5#8701C214yesmaximal, complemented
6#7915C214yesmaximal
7C22×#117C412yes—
8C23×C42C2211yescharacteristic
9C2×#407C2212yes—
10C22×C22⋊C8C2212yes—
11#2285—21no—
12#2285—21no—
13#2285—21no—
14#2285—21no—
15#2285—21no—
16#2285—21no—
17#2285—21no—
18#2285—21no—
19C2×#384C411yescharacteristic
20C2×#384C411yescharacteristic
21C2×#382C2212yes—
22C2×#380C412yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C412yes—
25C2×C4×C4⋊C4C2212yes—
26#1817—21no—
27#1817—21no—
28#1817—21no—
29#1817—21no—
30#1817C2212yes—
31#1817C2212yes—
32#1817C2212yes—
33#1817C2212yes—
34#1728—21no—
35#1728—21no—
36C2×#336—21no—
37C2×#336—21no—
38#1280C22116yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C23×C8—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—48no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4C2×C414yes—
56C22×C42—28no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C414yes—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#407—28no—
71C2×C22⋊C8—28no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×C22⋊C8C2314yes—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95C2×C414yes—
105C2×#95C2314yes—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114C25C22×C411yescharacteristic
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C414yes—
137C23×C4C22×C414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4118yes—
143C23×C4C2411yescharacteristic
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—44no—
150C22×C8—44no—
151C22×C8—44no—
152#117—48no—
153#117—48no—
154#117—48no—
155#117—48no—
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—28no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—48no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—48no—
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C22⋊C8—28no—
209C22⋊C8—28no—
210C22⋊C8—44no—
211C22⋊C8—44no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—41no—
221C24—41no—
222C24C2×C22⋊C411yescharacteristic
223C24C2×C22⋊C411yescharacteristic
224C24#11714yes—
225C24C23×C411yescharacteristic
226C22×C4—232no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—416no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—416no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4#10014yes—
283C22×C4#10114yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4#11714yes—
295C22×C4#12314yes—
296C22×C4#12314yes—
297C22×C4#12314yes—
298C22×C4#12314yes—
299C22×C4#13114yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4114yes—
303M16—88no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—88no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—48no—
342C42—48no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—28no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—28no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23#38112yes—
365C23#38112yes—
366C23C2×#10012yes—
367C23C2×#10112yes—
368C23C22×C22⋊C411yescharacteristic
369C23C2×#11712yes—
370C23C2×#13112yes—
371C23#56812yes—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—28no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4#38112yes—
423C2×C4#38112yes—
424C2×C4#40812yes—
425C2×C4#40812yes—
426C2×C4#41012yes—
427C2×C4#41012yes—
428C2×C4C2×#10012yes—
429C2×C4C2×#10012yes—
430C2×C4C2×#10112yes—
431C2×C4C2×#10112yes—
432C2×C4C2×#12312yes—
433C2×C4C2×#12312yes—
434C8—84no—
435C8—84no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—416no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22#182114yes—
457C22C2×#38111yescharacteristic
458C22C2×#40811yescharacteristic
459C22#287911yescharacteristic
460C22C2×#41011yescharacteristic
461C22C2×#41011yescharacteristic
462C22C22×#10011yescharacteristic
463C22C22×#10111yescharacteristic
464C22#292414yes—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—84no—
477C4—84no—
478C4#182014yes—
479C2—24no—
480C2—24no—
481C2—42no—
482C2#1738411yescharacteristic
483C2#1738511yescharacteristic
484C2#1740311yescharacteristic
485C2#1740411yescharacteristic
486C2#2068511yescharacteristic
487C2C2×#182011yescharacteristic
488C2C2×#182111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8703C2semidirect
5. #8701C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#206822
C2×#18172
#87032
#87012
#79152