Finite Group Databasebeta

← #239780

Subgroups — #239780

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic, complemented
3#21804C211yesmaximal, characteristic, complemented
4#20645C211yesmaximal, characteristic
5C22×#333C211yesmaximal, characteristic, complemented
6C2×#1477C211yesmaximal, characteristic
7#10414C214yesmaximal, complemented
8#10413C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#416—21no—
16#2857—22no—
17#2628—22no—
18#2568—21no—
19#2567C2212yes—
20#2563C2212yes—
21#2542—21no—
22#2487—22no—
23#2409—22no—
24#2317—21no—
25#2311—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#1981—22no—
29#1956C2214yes—
30#1947—22no—
31#1947C2214yes—
32#1808—22no—
33#1805—21no—
34#1804—21no—
35#1803C2212yes—
36#1802C2212yes—
37#1787—22no—
38#1735—21no—
39C2×#333—24no—
40C2×#333C2214yes—
41#1477—24no—
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2214yes—
45C22×#41D411yescharacteristic
46C23×Q8D412yes—
47C23×C8—21no—
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2312yes—
57C2×C22⋊Q8—24no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78C2×#107—24no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D414yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88#416—44no—
89C2×#103—22no—
90C2×#103—28no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×C4×C8C2312yes—
94#400—24no—
95#399—24no—
96#395—24no—
97#393—24no—
98#391—24no—
99#391—22no—
100#390—22no—
101#389—24no—
102#389—24no—
103#386—24no—
104#385—24no—
105#384—22no—
106#384D412yes—
107#384C2312yes—
108#382—22no—
109#382—21no—
110#382—21no—
111#380—24no—
112#380—21no—
113#380—24no—
114#380—21no—
115#380—21no—
116#380D412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2312yes—
121#340—22no—
122#340—22no—
123#340—24no—
124#333—28no—
125Q8⋊C8—24no—
126D4⋊C8—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41Q1614yes—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—84no—
137C22×Q8C2×D411yescharacteristic
138C22×Q8C2×D414yes—
139C22×D4—21no—
140C22×D4C2×D411yescharacteristic
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—82no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C22⋊Q8—88no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183#117—22no—
184#117—22no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—41no—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231#107—84no—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8C2×D414yes—
239#103—44no—
240#103—88no—
241#102—44no—
242C22⋊C8—42no—
243C4×C8—22no—
244C4×C8—22no—
245#95—28no—
246#95—28no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251C24—21no—
252C24#12011yescharacteristic
253#41—28no—
254#41—28no—
255#41—44no—
256#41—44no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—24no—
260C2×Q8—21no—
261C2×Q8—24no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—44no—
267C2×Q8—88no—
268C2×Q8—88no—
269C2×Q8C2×Q1612yes—
270C2×D4—24no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4C2×Q1612yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4#12412yes—
308C22×C4C2×Q1612yes—
309C22×C4#13512yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—24no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—84no—
323C2×C8—84no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—88no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—88no—
353C22⋊C4—82no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—82no—
361C42#12112yes—
362C42#12112yes—
363C42#12112yes—
364C42#12312yes—
365C42#12312yes—
366C42#12412yes—
367C42C22×D411yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—22no—
372C23—22no—
373C23—22no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23#39412yes—
379C23#44912yes—
380C23#45112yes—
381C23C2×#12011yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—48no—
385Q8—41no—
386Q8—41no—
387Q8—84no—
388Q8—88no—
389D4—22no—
390D4—22no—
391D4—48no—
392D4—41no—
393D4—41no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—48no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4—88no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#39412yes—
424C2×C4#39412yes—
425C2×C4#39412yes—
426C2×C4#44912yes—
427C2×C4#45112yes—
428C2×C4#46714yes—
429C2×C4#46814yes—
430C2×C4#48414yes—
431C2×C4#48514yes—
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4C22×Q1611yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57712yes—
443C2×C4#57812yes—
444C8—44no—
445C8—82no—
446C8—82no—
447C22—24no—
448C22—24no—
449C22—22no—
450C22—22no—
451C22—42no—
452C22—44no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22#203614yes—
457C22C2×#39411yescharacteristic
458C22C2×#45111yescharacteristic
459C22C2×#44911yescharacteristic
460C22#301312yes—
461C22#301912yes—
462C4—21no—
463C4—22no—
464C4—21no—
465C4—22no—
466C4—42no—
467C4—84no—
468C4—81no—
469C4—84no—
470C4—81no—
471C4—162no—
472C4#203614yes—
473C4#304211yescharacteristic
474C4#304411yescharacteristic
475C4C2×#46711yescharacteristic
476C4C2×#46811yescharacteristic
477C4C2×#48511yescharacteristic
478C4C2×#48411yescharacteristic
479C4#313712yes—
480C2—24no—
481C2—82no—
482C2#1887312yes—
483C2#1887612yes—
484C2C2×#203611yescharacteristic
485C2#2349311yescharacteristic
486C2#2351211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
2. #21961C2semidirect
3. #21804C2semidirect
5. C22×#333C2semidirect
7. #10414C2semidirect
8. #10413C2semidirect
9. #7893C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#219612
#218042
#206452
C22×#3332
C2×#14772
#104142
#104132
#78932