Finite Group Databasebeta

← #6375049

Subgroups — #6375049

Order 512.

2010 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#34012C211yesmaximal, characteristic
4#21989C214yesmaximal, complemented
5#21988C214yesmaximal, complemented
6#21962C214yesmaximal, complemented
7#21939C214yesmaximal
8#21366C214yesmaximal, complemented
9#21364C214yesmaximal, complemented
10#20823C214yesmaximal, complemented
11C23×C42C2211yescharacteristic
12C2×#432C4116yescomplemented
13C2×#432C2212yes—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2212yes—
20#2285C2212yes—
21#2285C2212yes—
22#2285C2212yes—
23#2285C2212yes—
24#2285C2212yes—
25#2285C2212yes—
26#2283—21no—
27#2283—21no—
28#2283—21no—
29#2283—21no—
30#2283—21no—
31#2283—21no—
32#2283—21no—
33#2283—21no—
34#2283C2212yes—
35#2283C2212yes—
36#2283C2212yes—
37#2283C2212yes—
38#2280—21no—
39#2280—21no—
40#2280—21no—
41#2280—21no—
42#2280C2212yes—
43#2280C2212yes—
44C4×#117—21no—
45C4×#117—21no—
46C4×#117—21no—
47C4×#117—21no—
48C4×#117C2212yes—
49C4×#117C2212yes—
50#1949C22116yescomplemented
51#1948C2218yescomplemented
52#1947C2218yescomplemented
53#1854C22116yes—
54#1854C22116yes—
55#1847C2218yescomplemented
56#1836C2218yescomplemented
57#1808C2218yes—
58#1807C2218yes—
59#1776C22116yescomplemented
60C24×C4D411yescharacteristic
61C24×C4D411yescharacteristic
62C24×C4C2311yescharacteristic
63C23×C8C2314yes—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117C2314yes—
69C2×#117C2314yes—
70C22×C42C2×C418yes—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42Q818yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#432C2×C4116yes—
85#432C2×C4116yes—
86#432C2×C4116yes—
87#432C2×C4116yes—
88C2×#107C23116yes—
89C2×#106C23116yes—
90C2×C4⋊C8C2×C4116yes—
91C2×C4⋊C8C2314yes—
92C2×C4⋊C8C2314yes—
93C2×C22⋊C8C2×C4116yes—
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×#97C2×C4116yes—
97C2×#97C2314yes—
98C2×#97C2314yes—
99#384C2314yes—
100#384C2314yes—
101#383—22no—
102#383—22no—
103#383—22no—
104#383—22no—
105#382—22no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382C2314yes—
110#382C2314yes—
111#381—22no—
112#381—22no—
113#381—22no—
114#381—22no—
115#381—22no—
116#381—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380C2314yes—
126#380C2314yes—
127#380C2314yes—
128#380C2314yes—
129#379—22no—
130#379—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2314yes—
141C4×C4⋊C4C2314yes—
142C4×C4⋊C4C2314yes—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155#376—22no—
156#376—22no—
157#376—22no—
158#376—22no—
159#376—22no—
160#376—22no—
161C43—22no—
162C43—22no—
163#340C23116yes—
164#340C23116yes—
165C25C2×D411yescharacteristic
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C4⋊C418yes—
170C23×C4C4⋊C418yes—
171C23×C4C22×C414yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×Q814yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8—48no—
192C22×C8C22×C4116yes—
193C22×C8C2418yes—
194#117—44no—
195#117—48no—
196#117—44no—
197#117—48no—
198#117—48no—
199#117—48no—
200#117—48no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4C2418yes—
206C2×C4⋊C4C2418yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42C4⋊C418yes—
220C2×C42C4⋊C418yes—
221C2×C42C4⋊C418yes—
222C2×C42C4⋊C418yes—
223C2×C42C4⋊C418yes—
224C2×C42C4⋊C418yes—
225C2×C42C22×C414yes—
226C2×C42C22×C414yes—
227C2×C42C22×C414yes—
228C2×C42C22×C414yes—
229C2×C42C22×C414yes—
230C2×C42C22×C414yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q814yes—
244C2×C42C2×Q814yes—
245C2×C42C2×Q814yes—
246C2×C42C2×Q814yes—
247C2×C42C2×Q814yes—
248C2×C42C2×Q814yes—
249C2×C42C2411yescharacteristic
250C2×C42C2411yescharacteristic
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253#107—416no—
254#106—416no—
255C4⋊C8C22×C4116yes—
256C4⋊C8C22×C4116yes—
257C22⋊C8C22×C4116yes—
258C22⋊C8C22×C4116yes—
259#97C22×C4116yes—
260#97C22×C4116yes—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24C2×C4⋊C414yes—
277C24C22×D411yescharacteristic
278C24C22×D411yescharacteristic
279C24C22×D411yescharacteristic
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—416no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—416no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C2×C4⋊C414yes—
305C22×C4C2×C4⋊C414yes—
306C22×C4C2×C4⋊C414yes—
307C22×C4C2×C4⋊C414yes—
308C22×C4C2×C4⋊C414yes—
309C22×C4C2×C4⋊C414yes—
310C22×C4C2×C4⋊C414yes—
311C22×C4C2×C4⋊C414yes—
312C22×C4C2×C4⋊C414yes—
313C22×C4C2×C4⋊C414yes—
314C22×C4C2×C4⋊C414yes—
315C22×C4C2×C4⋊C414yes—
316C22×C4C2×C4⋊C414yes—
317C22×C4C2×C4⋊C414yes—
318C22×C4C2×C4⋊C414yes—
319C22×C4C23×C412yes—
320C22×C4C23×C412yes—
321C22×C4C23×C412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q812yes—
334C22×C4C22×Q812yes—
335C22×C4C22×Q812yes—
336C22×C4C2511yescharacteristic
337C2×C8—216no—
338C2×C8—416no—
339C2×C8C23×C4116yes—
340C4⋊C4—28no—
341C4⋊C4—216no—
342C4⋊C4—28no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C42—48no—
354C42—48no—
355C42—48no—
356C42—48no—
357C42—48no—
358C42—48no—
359C42C2×C4⋊C414yes—
360C42C2×C4⋊C414yes—
361C42C2×C4⋊C414yes—
362C42C2×C4⋊C414yes—
363C42C2×C4⋊C414yes—
364C42C2×C4⋊C414yes—
365C42C2×C4⋊C414yes—
366C42C2×C4⋊C414yes—
367C42C2×C4⋊C414yes—
368C42C2×C4⋊C414yes—
369C42C2×C4⋊C414yes—
370C42C2×C4⋊C414yes—
371C42C23×C412yes—
372C42C23×C412yes—
373C42C23×C412yes—
374C42C23×C412yes—
375C42C22×Q812yes—
376C42C22×Q812yes—
377C42C22×Q812yes—
378C42C22×Q812yes—
379C23—22no—
380C23—22no—
381C23—28no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—44no—
389C23—44no—
390C23—416no—
391C23C22×C4⋊C412yes—
392C23C22×C4⋊C412yes—
393C23C22×C4⋊C412yes—
394C23#57514yes—
395C23C23×D411yescharacteristic
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4C22×C4⋊C412yes—
434C2×C4C22×C4⋊C412yes—
435C2×C4C22×C4⋊C412yes—
436C2×C4C22×C4⋊C412yes—
437C2×C4C22×C4⋊C412yes—
438C2×C4C22×C4⋊C412yes—
439C2×C4C22×C4⋊C412yes—
440C2×C4C22×C4⋊C412yes—
441C2×C4C22×C4⋊C412yes—
442C2×C4C22×C4⋊C412yes—
443C2×C4C22×C4⋊C412yes—
444C2×C4#57514yes—
445C2×C4C24×C411yescharacteristic
446C2×C4C23×Q811yescharacteristic
447C2×C4#58514yes—
448C8—216no—
449C22—22no—
450C22—28no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—416no—
459C22—44no—
460C22—44no—
461C22#291114yes—
462C22C23×C4⋊C411yescharacteristic
463C22#344912yes—
464C22C2×#57512yes—
465C22C2×#57512yes—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—88no—
477C4—88no—
478C4#291114yes—
479C4#342812yes—
480C4#345812yes—
481C2—24no—
482C2—42no—
483C2—42no—
484C2#3405211yescharacteristic
485C2C2×#291111yescharacteristic
486C2C2×#291111yescharacteristic
487C2#6096114yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×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
4. #21989C2semidirect
5. #21988C2semidirect
6. #21962C2semidirect
8. #21366C2semidirect
9. #21364C2semidirect
10. #20823C2semidirect
12. C2×#432C4semidirect
50. #1949C22semidirect
51. #1948C22semidirect
52. #1947C22semidirect
55. #1847C22semidirect
56. #1836C22semidirect
59. #1776C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#340122
#219892
#219882
#219622
#219392
#213662
#213642
#208232