Finite Group Databasebeta

← #6630443

Subgroups — #6630443

Order 512.

2450 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#33148C214yesmaximal, complemented
4#33108C214yesmaximal, complemented
5#33098C214yesmaximal, complemented
6#32982C214yesmaximal
7C4×#483C218yesmaximal, complemented
8#21385C212yesmaximal, complemented
9#21359C212yesmaximal, complemented
10#20714C211yesmaximal, characteristic
11C2×#533C2211yescharacteristic
12C23×C4⋊C4C2212yes—
13C23×C42—21no—
14C23×C42C2211yescharacteristic
15C2×#483C2218yes—
16C2×#431C2212yes—
17#2857C2212yes—
18#2857C2212yes—
19#2855—21no—
20#2855C2212yes—
21#2851—22no—
22#2848—21no—
23#2848C2212yes—
24C4×C22⋊Q8—22no—
25C4×C22⋊Q8C2214yes—
26#2296—21no—
27#2285C2211yescharacteristic
28#2280—21no—
29C2×C4×C4⋊C4—22no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C22×#95—21no—
38#1987C2214yescomplemented
39#1981C2214yescomplemented
40#1948C2218yescomplemented
41#1927C22116yescomplemented
42#1920C22116yescomplemented
43#1872C2218yes—
44#1872C2218yes—
45#1853C2218yes—
46#1853C2218yes—
47#1823C2218yes—
48#1817C2212yes—
49#1797C2214yes—
50#1779C2212yes—
51C4×#106C2218yescomplemented
52C4×#103C2218yescomplemented
53C4×C22⋊C8C2214yes—
54C24×C4—22no—
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4C2311yescharacteristic
58#533—28no—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2314yes—
65C2×C22⋊Q8C2312yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—216no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C42—28no—
78C22×C42—24no—
79C22×C42—22no—
80C22×C42—24no—
81C22×C42—22no—
82C22×C42—48no—
83C22×C42D412yes—
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D412yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#483—232no—
92#431—28no—
93C2×#106C23116yes—
94C2×C4⋊C8C2314yes—
95C2×#103C23116yes—
96C2×C22⋊C8C2318yes—
97C2×C22⋊C8C2314yes—
98C2×C4×C8C2314yes—
99#391—22no—
100#391C2314yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#387—24no—
106#387—24no—
107#385—24no—
108#385—22no—
109#385—24no—
110#385—22no—
111#384C2312yes—
112#382—21no—
113#381—22no—
114#380—21no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380C2314yes—
119#380C2312yes—
120#379—22no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4C2318yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—24no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—22no—
138C2×#95—22no—
139#340C23116yes—
140#340C23116yes—
141#336C2318yes—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8C2412yes—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—216no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—44no—
155C23×C4—48no—
156C23×C4—48no—
157C23×C4SD1618yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C22×C8C2418yes—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173#125—44no—
174C22⋊Q8—28no—
175C22⋊Q8—216no—
176C22⋊Q8—48no—
177C22⋊Q8—44no—
178C4×Q8—28no—
179C4×Q8—44no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—216no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—216no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—216no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4#41116yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2418yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—28no—
212C2×C42—28no—
213C2×C42—28no—
214C2×C42—28no—
215C2×C42—28no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—48no—
221C2×C42—416no—
222C2×C42—416no—
223C2×C42—48no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4118yes—
232C2×C42C2412yes—
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235#106—232no—
236C4⋊C8—28no—
237#103—232no—
238C22⋊C8—28no—
239C22⋊C8—216no—
240C4×C8—28no—
241#95—28no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—24no—
249#95—22no—
250C24—22no—
251C24—24no—
252C24—22no—
253C24—24no—
254C24C2×SD1614yes—
255C24C22×D411yescharacteristic
256C24C22×D412yes—
257C2×Q8—28no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—42no—
261C22×C4—232no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—232no—
277C22×C4—216no—
278C22×C4—216no—
279C22×C4—22no—
280C22×C4—216no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—416no—
294C22×C4—464no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4#12314yes—
302C22×C4#12314yes—
303C22×C4C2×SD1614yes—
304C22×C4C2×SD1618yes—
305C22×C4C2×SD1618yes—
306C22×C4C2×SD1614yes—
307C22×C4#13714yes—
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×D412yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4114yes—
317C22×C4C2×#4118yes—
318C22×C4C2511yescharacteristic
319C2×C8—28no—
320C2×C8—216no—
321C4⋊C4—216no—
322C4⋊C4—216no—
323C4⋊C4—216no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4C2×#41116yes—
337C22⋊C4—44no—
338C22⋊C4—48no—
339C22⋊C4—44no—
340C22⋊C4—48no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—44no—
348C42—416no—
349C42—48no—
350C42—48no—
351C42—44no—
352C42#12314yes—
353C42#12314yes—
354C42C2×#4114yes—
355C42C2×#4114yes—
356C23—28no—
357C23—28no—
358C23—216no—
359C23—28no—
360C23—216no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23C2×#12312yes—
368C23#52514yes—
369C23C22×SD1612yes—
370C23C22×SD1614yes—
371C23C2×#13712yes—
372C23#57514yes—
373C23C23×D411yescharacteristic
374Q8—48no—
375Q8—84no—
376C2×C4—216no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—216no—
387C2×C4—216no—
388C2×C4—28no—
389C2×C4—216no—
390C2×C4—22no—
391C2×C4—216no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—44no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—432no—
399C2×C4—432no—
400C2×C4—432no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4#483116yescomplemented
411C2×C4C2×#12312yes—
412C2×C4C2×#12312yes—
413C2×C4C2×#12312yes—
414C2×C4C2×#12312yes—
415C2×C4C2×#12312yes—
416C2×C4#52514yes—
417C2×C4C22×SD1614yes—
418C2×C4C22×SD1612yes—
419C2×C4C22×SD1612yes—
420C2×C4C2×#13712yes—
421C2×C4C2×#13712yes—
422C2×C4C22×#4112yes—
423C2×C4C22×#4114yes—
424C8—48no—
425C22—232no—
426C22—22no—
427C22—24no—
428C22—24no—
429C22—22no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—24no—
434C22—24no—
435C22—24no—
436C22C2×#48314yes—
437C22#310118yes—
438C22C22×#12311yescharacteristic
439C22C2×#52512yes—
440C22C23×SD1611yescharacteristic
441C22C22×#13711yescharacteristic
442C22C2×#57512yes—
443C4—22no—
444C4—24no—
445C4—24no—
446C4—22no—
447C4—22no—
448C4—42no—
449C4—416no—
450C4—82no—
451C4—82no—
452C4—84no—
453C4C2×#48314yes—
454C4C2×#48314yes—
455C4#344414yes—
456C2—24no—
457C2—28no—
458C2#3662611yescharacteristic
459C2C22×#48311yescharacteristic
460C2#6051111yescharacteristic
461C2C2×#310112yes—
462C2#6056012yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #33108C2semidirect
5. #33098C2semidirect
7. C4×#483C2semidirect
8. #21385C2semidirect
9. #21359C2semidirect
38. #1987C22semidirect
39. #1981C22semidirect
40. #1948C22semidirect
41. #1927C22semidirect
42. #1920C22semidirect
51. C4×#106C22semidirect
52. C4×#103C22semidirect
410. C2×C4#483semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×C4×C4⋊C42
#331482
#331082
#330982
#329822
C4×#4832
#213852
#213592
#207142