Finite Group Databasebeta

← #242622

Subgroups — #242622

Order 512.

934 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#21964C211yesmaximal, characteristic, complemented
2#21952C211yesmaximal, characteristic, complemented
3#21803C211yesmaximal, characteristic
4#21803C211yesmaximal, characteristic
5C2×#1574C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic
7#10599C212yesmaximal, complemented
8#10595C212yesmaximal, complemented
9#10591C212yesmaximal, complemented
10#10587C212yesmaximal, complemented
11#8038C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#1982—21no—
20#1982—21no—
21#1982C2212yes—
22#1982C2212yes—
23#1957—21no—
24#1957—21no—
25#1957C2212yes—
26#1957C2212yes—
27#1948—21no—
28#1948C2212yescomplemented
29#1947—21no—
30#1947C2212yescomplemented
31#1939—21no—
32#1939—21no—
33#1939C2212yescomplemented
34#1939C2212yescomplemented
35#1809—22no—
36#1798—22no—
37#1786—22no—
38#1786—22no—
39#1574—24no—
40C2×#335C2212yescomplemented
41C2×#334C2212yescomplemented
42#1545—24no—
43C2×#331C2212yes—
44C2×#331C2212yes—
45#1296C2211yescharacteristic, complemented
46#1296C2211yescharacteristic, complemented
47#1296C2211yescharacteristic, complemented
48#1296C2211yescharacteristic, complemented
49C23×D4—21no—
50C23×D4—21no—
51C23×D4—21no—
52C22×M16—21no—
53C2×#127—22no—
54C2×#127—21no—
55C2×#127—21no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—21no—
59C2×#127Q814yes—
60C2×#127C2312yes—
61C2×#117—21no—
62C2×#117—21no—
63C22×C42C2311yescharacteristic
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70C2×#107—22no—
71C2×#106—22no—
72#423—22no—
73#423D414yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D412yes—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#382—21no—
103#382—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380D412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2312yes—
112#340—22no—
113#340—22no—
114#336—22no—
115#336—22no—
116#335—24no—
117#334—24no—
118#331—24no—
119#331—24no—
120C25—41no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—81no—
134C22×D4—81no—
135C22×D4—81no—
136C22×D4—81no—
137C22×D4—81no—
138C22×D4—81no—
139C22×D4—81no—
140C22×D4—81no—
141C22×D4—82no—
142C22×D4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4#4111yescharacteristic
146C2×M16—24no—
147C2×M16—81no—
148C2×M16—81no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156#127—24no—
157#127—24no—
158#127—24no—
159#127—22no—
160#127—22no—
161#127—22no—
162#127—42no—
163#127—42no—
164#127—42no—
165#127C2×Q814yes—
166#117—81no—
167#117—81no—
168#117—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193#107—82no—
194#106—82no—
195C4⋊C8—24no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—82no—
208C4⋊C8—82no—
209C4⋊C8C2×D414yes—
210#102—82no—
211#102—82no—
212#102—82no—
213#102—82no—
214#102—82no—
215#102—82no—
216#102—82no—
217#102—82no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—42no—
224C4×C8—42no—
225#95—22no—
226#95—22no—
227#95—41no—
228#95—41no—
229C24—42no—
230C24—44no—
231C24—82no—
232C24—82no—
233C24—82no—
234C24—82no—
235C24—82no—
236C24—82no—
237C24#12111yescharacteristic
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—84no—
258C2×D4—84no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—88no—
266C2×D4—84no—
267C2×D4—84no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—88no—
271C2×D4—82no—
272C2×D4—82no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4C22⋊Q812yes—
293C22×C4C22⋊Q812yes—
294C22×C4C22⋊Q812yes—
295C22×C4#12612yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299M16—84no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—42no—
342C42—42no—
343C42—82no—
344C42#12112yes—
345C42#12112yes—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12612yes—
350C42C22×D411yescharacteristic
351C42C22×Q811yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—44no—
361C23—88no—
362C23—88no—
363C23—88no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—164no—
371C23—164no—
372C23—164no—
373C23—164no—
374C23#39612yes—
375C23C2×#12111yescharacteristic
376D4—44no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—84no—
381D4—84no—
382D4—88no—
383D4—88no—
384D4—88no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4#39612yes—
418C2×C4#39612yes—
419C2×C4#39612yes—
420C2×C4#47112yes—
421C2×C4#47212yes—
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×C22⋊Q811yescharacteristic
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×#12611yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57611yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C8—44no—
437C8—42no—
438C8—42no—
439C8—81no—
440C8—81no—
441C8—81no—
442C8—81no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—84no—
451C22—84no—
452C22—84no—
453C22—1616no—
454C22—162no—
455C22—162no—
456C22—162no—
457C22—162no—
458C22C2×#39611yescharacteristic
459C22#304111yescharacteristic
460C22#304311yescharacteristic
461C22#306211yescharacteristic
462C22#306211yescharacteristic
463C22C2×#47111yescharacteristic
464C22C2×#47211yescharacteristic
465C4—21no—
466C4—21no—
467C4—22no—
468C4—22no—
469C4—82no—
470C4—82no—
471C4—161no—
472C4—161no—
473C4#206314yes—
474C4#304111yescharacteristic
475C4#304311yescharacteristic
476C4#306211yescharacteristic
477C4#306211yescharacteristic
478C4#308111yescharacteristic
479C4#308111yescharacteristic
480C4#308111yescharacteristic
481C4#308111yescharacteristic
482C2—42no—
483C2—164no—
484C2#1955211yescharacteristic
485C2#1955411yescharacteristic
486C2#1956411yescharacteristic
487C2#1956611yescharacteristic
488C2C2×#206311yescharacteristic
489C2#2361411yescharacteristic
490C2#2361411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21964C2semidirect
2. #21952C2semidirect
5. C2×#1574C2semidirect
7. #10599C2semidirect
8. #10595C2semidirect
9. #10591C2semidirect
10. #10587C2semidirect
11. #8038C2semidirect
28. #1948C22semidirect
30. #1947C22semidirect
33. #1939C22semidirect
34. #1939C22semidirect
40. C2×#335C22semidirect
41. C2×#334C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#219642
#219522
#218032
C2×#15742
C2×#15452
#105992
#105952
#105912
#105872
#80382