Finite Group Databasebeta

← #6458397

Subgroups — #6458397

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27967C214yesmaximal, complemented
4#27966C212yesmaximal, complemented
5#27965C212yesmaximal, complemented
6#27964C214yesmaximal, complemented
7#27963C214yesmaximal, complemented
8#23747C214yesmaximal, complemented
9#21325C212yesmaximal, complemented
10#20765C212yesmaximal, complemented
11#20702C214yesmaximal
12C4×#407C211yesmaximal, characteristic
13C22×#125C2211yescharacteristic
14C22×#117C2212yes—
15C23×C42C2211yescharacteristic
16C2×#407C2212yes—
17C2×C4×M16C2212yes—
18#2851—22no—
19#2851—22no—
20#2851C2212yes—
21#2851C2212yes—
22#2851C2212yes—
23#2851C2212yes—
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#385—22no—
27#2285—22no—
28#2285—22no—
29#2285C2212yes—
30#2285C2212yes—
31#2285C2212yes—
32#2285C2212yes—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#380—22no—
38C2×#380C2212yes—
39C2×C4×C4⋊C4—22no—
40#2101C22116yescomplemented
41#2100C2218yescomplemented
42#2099C2218yescomplemented
43#1992C2214yescomplemented
44#1992C2214yescomplemented
45#1991C2218yescomplemented
46#1836C2214yescomplemented
47#1836C2214yescomplemented
48#1835C2218yescomplemented
49#1825C22116yes—
50#1824C2218yes—
51#1823C2218yes—
52#1749C2212yes—
53#1749C2212yes—
54C4×C22⋊C8C2212yes—
55C4×C22⋊C8C2212yes—
56#1736C2218yes—
57#1736C2218yes—
58#1728C2212yes—
59C24×C4D412yes—
60C24×C4C2311yescharacteristic
61C22×M16C2314yes—
62C2×#125—22no—
63C2×#125—28no—
64C2×#125C2312yes—
65C2×#125C2312yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—216no—
69C2×#117C2314yes—
70C2×#117C2314yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—28no—
82C22×C42—22no—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42D412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#407—28no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C4×M16—28no—
96C2×#97C2314yes—
97C2×C4×C8C2314yes—
98#389—22no—
99#389—22no—
100#389C2314yes—
101#389C2314yes—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#385—24no—
107#385—24no—
108#385—24no—
109#385—24no—
110#384—22no—
111#384—22no—
112#384C2312yes—
113#384C2312yes—
114#384C2312yes—
115#384C2312yes—
116#382—22no—
117#382—24no—
118#382C2312yes—
119#382C2312yes—
120#382C2314yes—
121#382C2314yes—
122#380—22no—
123#380—22no—
124#380—24no—
125#380—24no—
126#380—24no—
127#380—24no—
128#380C2314yes—
129#380C2314yes—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C2×#95—24no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138#340C23116yes—
139#340C23116yes—
140#340C23116yes—
141#340C23116yes—
142#336C2314yes—
143#336C2314yes—
144C25C2×D411yescharacteristic
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4#4118yes—
164C23×C4C2411yescharacteristic
165C23×C4C2411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C22×C8C2418yes—
171#125—28no—
172#125—28no—
173#125—48no—
174#117—216no—
175#117—216no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—48no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42#4118yes—
226C2×C42C2411yescharacteristic
227C2×C42C2411yescharacteristic
228C2×C42C2411yescharacteristic
229C2×C42C2411yescharacteristic
230C22⋊C8—28no—
231C22⋊C8—28no—
232#97—28no—
233C4×C8—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—42no—
246C24#12414yes—
247C24C22×D411yescharacteristic
248C24C22×D411yescharacteristic
249C24C22×D411yescharacteristic
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—232no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
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—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—432no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—48no—
293C22×C4#12318yes—
294C22×C4#12414yes—
295C22×C4#12414yes—
296C22×C4#12414yes—
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4114yes—
305C22×C4C2×#4114yes—
306C22×C4C2×#4114yes—
307C22×C4C2×#4114yes—
308C22×C42−1+414yes—
309C22×C4C2511yescharacteristic
310M16—416no—
311C2×C8—28no—
312C2×C8—28no—
313C2×C8—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—28no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—48no—
342C42—48no—
343C42#12318yes—
344C42C2×#4114yes—
345C42C2×#4114yes—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—28no—
352C23—21no—
353C23—21no—
354C23—28no—
355C23—22no—
356C23—22no—
357C23—44no—
358C23—48no—
359C23—48no—
360C23C2×#12312yes—
361C23C2×#12412yes—
362C23C2×#12412yes—
363C23C2×#12412yes—
364C23#53612yes—
365C23#57514yes—
366C23C23×D411yescharacteristic
367C2×C4—216no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—28no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—88no—
402C2×C4—88no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4C2×#12314yes—
406C2×C4C2×#12314yes—
407C2×C4C2×#12312yes—
408C2×C4C2×#12412yes—
409C2×C4C2×#12412yes—
410C2×C4C2×#12412yes—
411C2×C4#53612yes—
412C2×C4#552116yes—
413C2×C4#57514yes—
414C2×C4C22×#4112yes—
415C2×C4C22×#4112yes—
416C2×C4C2×2−1+412yes—
417C8—48no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—416no—
430C22—42no—
431C22—42no—
432C22#250314yes—
433C22#314214yes—
434C22C22×#12311yescharacteristic
435C22C22×#12411yescharacteristic
436C22C2×#53611yescharacteristic
437C22C2×#57512yes—
438C22C2×#57512yes—
439C4—24no—
440C4—24no—
441C4—24no—
442C4—22no—
443C4—22no—
444C4—42no—
445C4—84no—
446C4—84no—
447C4—84no—
448C4—84no—
449C4#250314yes—
450C4C2×#55214yes—
451C2—24no—
452C2—24no—
453C2—42no—
454C2C2×#250311yescharacteristic
455C2#6051112yes—
456C2C2×#314211yescharacteristic
457C2#6061811yescharacteristic
458C2#6086612yes—
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. #27967C2semidirect
4. #27966C2semidirect
5. #27965C2semidirect
6. #27964C2semidirect
7. #27963C2semidirect
8. #23747C2semidirect
9. #21325C2semidirect
10. #20765C2semidirect
40. #2101C22semidirect
41. #2100C22semidirect
42. #2099C22semidirect
43. #1992C22semidirect
44. #1992C22semidirect
45. #1991C22semidirect
46. #1836C22semidirect
47. #1836C22semidirect
48. #1835C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22852
#279672
#279662
#279652
#279642
#279632
#237472
#213252
#207652
#207022
C4×#4072