Finite Group Databasebeta

← #6487664

Subgroups — #6487664

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C212yesmaximal, complemented
2#35166C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#32902C214yesmaximal, complemented
5#29754C214yesmaximal, complemented
6#24811C218yesmaximal, complemented
7#24726C218yesmaximal, complemented
8#21951C211yesmaximal, characteristic, complemented
9#21859C212yesmaximal, complemented
10C2×#537—21no—
11C2×#537—21no—
12C2×#532C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C23×C4⋊C4—21no—
16#3100—216no—
17C2×#480C2214yes—
18C2×#474C2214yes—
19C2×#459C2214yes—
20C2×#459C2218yes—
21C2×#447C2214yes—
22C2×C4×D8C2218yes—
23C2×#431C2212yes—
24#2856C2214yes—
25#2656—22no—
26#2542C2214yescomplemented
27#2541C2214yescomplemented
28#2540—24no—
29D4×C4⋊C4—24no—
30D4×C4⋊C4C2218yes—
31C2×#389C2212yes—
32C2×#385—22no—
33C2×#384C2211yescharacteristic
34C2×#380—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2083C2218yescomplemented
37#2067C22116yescomplemented
38#2041C2218yescomplemented
39#1990C2214yescomplemented
40#1957C2214yescomplemented
41#1939C2218yescomplemented
42#1879C22116yes—
43#1874C2218yes—
44#1802C2218yes—
45#1802C2218yescomplemented
46#1797C2214yes—
47C22×#41—22no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—22no—
51C22×D8C2318yes—
52#537—48no—
53#537—48no—
54#532—28no—
55#532—28no—
56C2×C4⋊Q8C2312yes—
57C2×#127C2312yes—
58C2×#127C2314yes—
59C2×#124—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×#121—24no—
63C2×#121—24no—
64C2×#121—22no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121C2314yes—
68C2×C4×D4—22no—
69C2×C4×D4—24no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—48no—
73C2×C4×D4D418yescomplemented
74C2×C4×D4C2314yes—
75C2×C4×D4C2318yes—
76C2×C4×D4C2314yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—48no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#480—216no—
91#474—216no—
92#459—216no—
93#459—416no—
94#447—216no—
95C4×D8—232no—
96#431—28no—
97C2×#107C2318yes—
98C2×C4⋊C8C2314yes—
99C2×C4⋊C8C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×#102C23116yes—
103C2×C22⋊C8C2314yes—
104C2×C4×C8C2314yes—
105#399—28no—
106#392—28no—
107#390—24no—
108#390—24no—
109#390C2318yes—
110#389—24no—
111#389C2318yes—
112#385—216no—
113#384—22no—
114#384C2314yes—
115#384C2314yes—
116#380—28no—
117C4×C4⋊C4D414yes—
118C4×C4⋊C4C2314yes—
119C2×#95—22no—
120C2×#95D412yes—
121#340C23116yes—
122#336C2318yes—
123C25—21no—
124C2×#41—48no—
125C2×#41—48no—
126C2×#41—48no—
127C22×Q8—22no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4C2×D414yes—
138C22×D4C2414yes—
139C22×D4C2412yes—
140C23×C4—21no—
141C23×C4—28no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—28no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—48no—
149C23×C4—48no—
150C23×C4—84no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C2×D8—216no—
157C2×D8—216no—
158C2×D8—216no—
159C22×C8C2418yes—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162#127—24no—
163#127—24no—
164#127—48no—
165#124—48no—
166C22⋊Q8—416no—
167C22⋊Q8—416no—
168#121—28no—
169#121—28no—
170#121—416no—
171#121—416no—
172#121—48no—
173#121—416no—
174#121—416no—
175C4×D4—216no—
176C4×D4—216no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—48no—
181C4×D4—416no—
182C4×D4—48no—
183C4×D4C2×D4116yes—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—432no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4D818yescomplemented
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2412yes—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—48no—
214C2×C42—44no—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D414yes—
218C2×C42#4114yes—
219C2×C42C2414yes—
220C2×C42C2412yes—
221#107—216no—
222C4⋊C8—28no—
223C4⋊C8—48no—
224#102—216no—
225#102—216no—
226#102—416no—
227C22⋊C8—28no—
228C4×C8—28no—
229#95—28no—
230#95C2×D418yes—
231C24—24no—
232C24—24no—
233C24—24no—
234C24—22no—
235C24—24no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C22×D411yescharacteristic
241#41—832no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—216no—
248C2×D4—216no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—28no—
255C2×D4—22no—
256C2×D4—28no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—48no—
265C2×D4—44no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—84no—
271C2×D4—84no—
272C2×D4C22×D414yes—
273C2×D4H(5,𝔽2)14yes—
274C2×D42−1+414yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—216no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—416no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—432no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—48no—
299C22×C4—416no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—88no—
303C22×C4—832no—
304C22×C4#12114yes—
305C22×C4#12114yes—
306C22×C4C2×D818yes—
307C22×C4C2×D814yes—
308C22×C4AGL(1,ℤ/8)14yes—
309C22×C4C22×D414yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×D414yes—
314C22×C4C22×D412yes—
315C22×C4C2×#4112yes—
316C22×C4C2511yescharacteristic
317D8—432no—
318C2×C8—28no—
319C2×C8—28no—
320C2×C8—28no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—416no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—84no—
334C4⋊C4C2×D8116yes—
335C4⋊C4C22×D414yes—
336C22⋊C4—216no—
337C22⋊C4—416no—
338C22⋊C4—416no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4—416no—
342C42—28no—
343C42—44no—
344C42#12118yes—
345C42C22×D414yes—
346C42C2×#4114yes—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—216no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—48no—
356C23—48no—
357C23—48no—
358C23—48no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—42no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23C2×#12112yes—
373C23C22×D812yes—
374C23C2×AGL(1,ℤ/8)12yes—
375C23C23×D411yescharacteristic
376Q8—88no—
377D4—216no—
378D4—24no—
379D4—24no—
380D4—24no—
381D4—48no—
382D4—48no—
383D4—88no—
384D4—816no—
385D4—88no—
386D4—168no—
387D4—168no—
388D4#53718yescomplemented
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—432no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—832no—
414C2×C4—84no—
415C2×C4—88no—
416C2×C4—816no—
417C2×C4#459116yescomplemented
418C2×C4C2×#12112yes—
419C2×C4C2×#12114yes—
420C2×C4C2×#12114yes—
421C2×C4C22×D818yes—
422C2×C4C22×D814yes—
423C2×C4C2×AGL(1,ℤ/8)14yes—
424C2×C4C23×D411yescharacteristic
425C2×C4C22×#4111yescharacteristic
426C2×C4C2×H(5,𝔽2)12yes—
427C2×C4C2×2−1+412yes—
428C8—48no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—28no—
436C22—432no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—84no—
447C22—84no—
448C22—84no—
449C22—84no—
450C22C2×#45914yes—
451C22C22×#12111yescharacteristic
452C22#348411yescharacteristic
453C22#348611yescharacteristic
454C22C23×D811yescharacteristic
455C22C22×AGL(1,ℤ/8)11yescharacteristic
456C22#358511yescharacteristic
457C22#358611yescharacteristic
458C4—24no—
459C4—28no—
460C4—22no—
461C4—44no—
462C4—44no—
463C4—84no—
464C4—84no—
465C4C2×#45918yes—
466C4C2×#53712yes—
467C4#349212yes—
468C2—24no—
469C2—48no—
470C2—164no—
471C2#3522311yescharacteristic
472C2C22×#45911yescharacteristic
473C2#6042511yescharacteristic
474C2#6155911yescharacteristic
475C2#6158511yescharacteristic
476C2#6181711yescharacteristic
477C2#6182511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3100C2semidirect
2. #35166C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #32902C2semidirect
5. #29754C2semidirect
6. #24811C2semidirect
7. #24726C2semidirect
8. #21951C2semidirect
9. #21859C2semidirect
26. #2542C22semidirect
27. #2541C22semidirect
36. #2083C22semidirect
37. #2067C22semidirect
38. #2041C22semidirect
39. #1990C22semidirect
40. #1957C22semidirect
41. #1939C22semidirect
45. #1802C22semidirect
73. C2×C4×D4D4semidirect
199. C2×C4⋊C4D8semidirect
388. D4#537semidirect
417. C2×C4#459semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#351662
C2×D4×C4⋊C42
#329022
#297542
#248112
#247262
#219512
#218592