Finite Group Databasebeta

← #190516

Subgroups — #190516

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#8701C212yesmaximal, complemented
5#8562C212yesmaximal, complemented
6#8561C212yesmaximal
7#8300C212yesmaximal
8#8155C212yesmaximal
9#8154C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11#2314—24no—
12#2314—24no—
13C2×#388C411yescharacteristic
14C2×#388C411yescharacteristic
15C2×#388C2211yescharacteristic
16C2×#388C2211yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C411yescharacteristic
19C2×#387C2211yescharacteristic
20C2×#387C2211yescharacteristic
21C2×#384C411yescharacteristic
22C2×#382C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×#380C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1917—24no—
28#1916—24no—
29#1817C2212yes—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C2×#117—24no—
41C2×#117—24no—
42C2×#117—22no—
43C2×#117—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97—22no—
71C2×C4×C8—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—24no—
77#388—24no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—24no—
83#387—24no—
84#384—24no—
85#382—24no—
86#380—24no—
87#380—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D414yes—
91C4×C4⋊C4D414yes—
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—24no—
107#336—24no—
108#336—22no—
109#336—22no—
110#324—28no—
111#323—28no—
112C25C22×C411yescharacteristic
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—42no—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
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×C4C2411yescharacteristic
132C22×C8—24no—
133C22×C8—24no—
134#117—216no—
135#117—216no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—81no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4#4114yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—81no—
170C2×C22⋊C4—81no—
171C2×C22⋊C4—81no—
172C2×C22⋊C4—81no—
173C2×C22⋊C4#4114yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C4⋊C8—82no—
192C4⋊C8—82no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197#97—82no—
198C4×C8—82no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95C22⋊C414yes—
212#95C22⋊C414yes—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24#11712yes—
222C24C23×C411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—28no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—48no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
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×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4#11712yes—
284C22×C4#11712yes—
285C22×C4#11712yes—
286C22×C4#13114yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—28no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—28no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—28no—
316C42—24no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—82no—
323C42—82no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—24no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23C22×C22⋊C411yescharacteristic
346C23C2×#11711yescharacteristic
347C23C2×#11711yescharacteristic
348C23C2×#13112yes—
349C23#56812yes—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—416no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#40814yes—
414C2×C4#52512yes—
415C2×C4#52512yes—
416C8—84no—
417C8—84no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#228011yescharacteristic
436C22C2×#40811yescharacteristic
437C22#287911yescharacteristic
438C22#288311yescharacteristic
439C22#288411yescharacteristic
440C22#288811yescharacteristic
441C22#288911yescharacteristic
442C22#292412yes—
443C22#292412yes—
444C4—24no—
445C4—22no—
446C4—22no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C2—22no—
460C2—22no—
461C2—82no—
462C2#1738811yescharacteristic
463C2#1739011yescharacteristic
464C2#1740811yescharacteristic
465C2#1741011yescharacteristic
466C2#2071711yescharacteristic
467C2#2072511yescharacteristic
468C2#2073111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #21912C2semidirect
4. #8701C2semidirect
5. #8562C2semidirect
9. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
#219162
#219122
#87012
#85622
#85612
#83002
#81552
#81542