Finite Group Databasebeta

← #6630457

Subgroups — #6630457

Order 512.

2110 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#33154C214yesmaximal, complemented
4#33114C214yesmaximal, complemented
5#33099C212yesmaximal, complemented
6#33098C212yesmaximal, complemented
7#32985C214yesmaximal
8#21583C218yesmaximal, complemented
9#21394C212yesmaximal, complemented
10#21365C212yesmaximal, complemented
11#20715C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C2×#485C2218yes—
16C2×#432C2212yes—
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—
29C4×#117—22no—
30C4×#117—22no—
31C4×#117C2214yes—
32C4×#117C2214yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4—21no—
38C2×#376—21no—
39C2×C43—21no—
40#1988C2214yescomplemented
41#1981C2214yescomplemented
42#1948C2214yescomplemented
43#1947C2214yescomplemented
44#1928C2218yescomplemented
45#1927C2218yescomplemented
46#1921C2218yescomplemented
47#1920C2218yescomplemented
48#1873C2218yes—
49#1872C2214yes—
50#1872C2214yes—
51#1854C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1825C2218yes—
55#1817C2212yes—
56#1797C2214yes—
57#1779C2212yes—
58#1776C2218yescomplemented
59#1763C2218yescomplemented
60#1749C2214yes—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4C2311yescharacteristic
64#533—28no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8C2312yes—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—216no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117C2314yes—
80C2×#117C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—22no—
84C22×C42—22no—
85C22×C42—28no—
86C22×C42—22no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42D412yes—
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D412yes—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#485—232no—
98#432—28no—
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×C4⋊C8C2314yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×C22⋊C8C2318yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2314yes—
107#391—22no—
108#391C2314yes—
109#389—22no—
110#389—22no—
111#389—22no—
112#389C2314yes—
113#387—24no—
114#387—24no—
115#385—24no—
116#385—22no—
117#385—24no—
118#385—22no—
119#384C2312yes—
120#382—21no—
121#381—22no—
122#380—21no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380C2314yes—
127#380C2312yes—
128#379—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145#376—24no—
146#376—24no—
147C2×#95—22no—
148C2×#95—21no—
149C2×#95—21no—
150C43—24no—
151C43—24no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#336C2318yes—
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8C2412yes—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C22×C8C2418yes—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184#125—44no—
185C22⋊Q8—28no—
186C22⋊Q8—216no—
187C22⋊Q8—48no—
188C22⋊Q8—44no—
189C4×Q8—28no—
190C4×Q8—44no—
191#117—216no—
192#117—216no—
193#117—28no—
194#117—28no—
195#117—28no—
196#117—44no—
197#117—44no—
198#117#41116yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4C2412yes—
213C2×C4⋊C4C2414yes—
214C2×C4⋊C4C2414yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—216no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C42—216no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—28no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—44no—
240C2×C42—48no—
241C2×C42—44no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4118yes—
250C2×C42C2412yes—
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253#107—216no—
254#106—216no—
255C4⋊C8—28no—
256#103—216no—
257#103—216no—
258C22⋊C8—28no—
259C22⋊C8—216no—
260#97—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—24no—
270#95—22no—
271C24—22no—
272C24—24no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24C22×D412yes—
277C24C22×D411yescharacteristic
278C2×Q8—28no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—44no—
282C22×C4—232no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—416no—
318C22×C4—416no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4#12314yes—
324C22×C4#12314yes—
325C22×C4#13714yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4114yes—
335C22×C4C2×#4118yes—
336C22×C4C2511yescharacteristic
337C2×C8—28no—
338C2×C8—28no—
339C2×C8—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4C2×#4118yes—
356C4⋊C4C2×#4118yes—
357C22⋊C4—216no—
358C22⋊C4—216no—
359C22⋊C4—48no—
360C22⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4—44no—
363C42—216no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42—216no—
370C42—24no—
371C42—48no—
372C42—48no—
373C42—44no—
374C42—44no—
375C42—48no—
376C42—48no—
377C42#12314yes—
378C42#12314yes—
379C42C2×#4114yes—
380C42C2×#4114yes—
381C23—22no—
382C23—22no—
383C23—28no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—48no—
391C23—48no—
392C23—44no—
393C23—44no—
394C23C2×#12312yes—
395C23#52514yes—
396C23C2×#13712yes—
397C23#57512yes—
398C23C23×D411yescharacteristic
399Q8—48no—
400Q8—84no—
401C2×C4—216no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—216no—
418C2×C4—22no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—416no—
427C2×C4—416no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4C2×#12312yes—
442C2×C4C2×#12312yes—
443C2×C4C2×#12312yes—
444C2×C4C2×#12312yes—
445C2×C4C2×#12312yes—
446C2×C4#52514yes—
447C2×C4C2×#13714yes—
448C2×C4#57514yes—
449C2×C4C22×#4112yes—
450C2×C4C22×#4114yes—
451C8—48no—
452C22—22no—
453C22—22no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—416no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22#309514yes—
465C22C22×#12311yescharacteristic
466C22C2×#52512yes—
467C22C22×#13711yescharacteristic
468C22C2×#57512yes—
469C22C2×#57511yescharacteristic
470C4—22no—
471C4—24no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—24no—
476C4—42no—
477C4—48no—
478C4—48no—
479C4—82no—
480C4—82no—
481C4—84no—
482C4#309314yes—
483C4#344414yes—
484C2—24no—
485C2—44no—
486C2#3662611yescharacteristic
487C2C2×#309311yescharacteristic
488C2C2×#309511yescharacteristic
489C2#6054612yes—
490C2#6056212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33154C2semidirect
4. #33114C2semidirect
5. #33099C2semidirect
6. #33098C2semidirect
8. #21583C2semidirect
9. #21394C2semidirect
10. #21365C2semidirect
40. #1988C22semidirect
41. #1981C22semidirect
42. #1948C22semidirect
43. #1947C22semidirect
44. #1928C22semidirect
45. #1927C22semidirect
46. #1921C22semidirect
47. #1920C22semidirect
58. #1776C22semidirect
59. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×C4×#1172
#331542
#331142
#330992
#330982
#329852
#215832
#213942
#213652
#207152