Finite Group Databasebeta

← #192604

Subgroups — #192604

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34129C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#20797C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8889C212yesmaximal, complemented
7#8888C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8884C212yesmaximal, complemented
10#8038C211yesmaximal, characteristic, complemented
11#7890C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#427—22no—
16C22×C4×C8C2211yescharacteristic
17#2354—22no—
18#2354C2212yes—
19Q8×C4⋊C4C2212yes—
20#2346—22no—
21#2346—21no—
22#2341—21no—
23#2293—21no—
24#2293—22no—
25#2285—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C42×Q8—21no—
29C4×#117—21no—
30C4×#117C2211yescharacteristic
31#1948C2212yes—
32#1947C2212yes—
33#1854—22no—
34#1846—22no—
35#1845C2212yes—
36#1844C2212yes—
37#1825—21no—
38#1803C2212yes—
39#1802C2212yes—
40#1788—21no—
41C4×#107—21no—
42C4×#106—21no—
43C4×#103—21no—
44C4×#102—21no—
45#1476C418yescomplemented
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41D411yescharacteristic
55C23×C8—21no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41C2×C414yes—
59C4×#41Q814yes—
60C2×C4×Q8—22no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D412yes—
72C2×#117D411yescharacteristic
73C2×#117D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yes—
80#427—48no—
81C2×#107—24no—
82C2×#107—22no—
83C2×#106—24no—
84C2×#106—22no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—22no—
89C2×#102—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8C2×C414yes—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#390—24no—
100#390—22no—
101#389—22no—
102#389—24no—
103#388—24no—
104#388—24no—
105#388—24no—
106#384—22no—
107#384C2312yes—
108#384C2312yes—
109#382—22no—
110#382—21no—
111#382D412yes—
112#380—22no—
113#380—21no—
114#380—21no—
115#380—22no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#376—22no—
134#376—21no—
135C43—22no—
136C43—21no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141Q8⋊C8C2×C414yes—
142Q8⋊C8C2×C414yes—
143D4⋊C8C2×C414yes—
144D4⋊C8C2×C414yes—
145C2×#41—22no—
146C2×#41—24no—
147C2×#41—42no—
148C2×#41C4⋊C414yes—
149C22×Q8C2×D411yescharacteristic
150C22×D4—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C22×C8—24no—
159C22×C8—24no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8C4⋊C414yes—
172C4×Q8C22×C412yes—
173C4×Q8C2×Q812yes—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4C4⋊C414yes—
182C4×D4C22×C412yes—
183C4×D4C2×Q812yes—
184#117—22no—
185#117—24no—
186#117—24no—
187#117—24no—
188#117—42no—
189#117—44no—
190#117—44no—
191#117—44no—
192#117—44no—
193#117—84no—
194#117C4⋊C414yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—41no—
230C2×C42C22×C412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42C2411yescharacteristic
238#107—44no—
239#107—48no—
240#106—48no—
241#106—44no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248#103—44no—
249#102—44no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C4×C8—24no—
253C4×C8—24no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257C4×C8C22×C412yes—
258C4×C8C22×C412yes—
259#95—24no—
260#95—24no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—41no—
267#95—41no—
268C24—21no—
269C24#12011yescharacteristic
270#41—28no—
271#41—44no—
272#41—48no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8C2×C4⋊C412yes—
276C2×Q8#13514yes—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—44no—
282C2×D4C2×C4⋊C412yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4C2×C4⋊C412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4#12312yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—216no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—84no—
351C4⋊C4C2×C4⋊C412yes—
352C4⋊C4C2×C4⋊C412yes—
353C4⋊C4#13514yes—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—48no—
360C22⋊C4—82no—
361C22⋊C4—84no—
362C22⋊C4—82no—
363C42—22no—
364C42—22no—
365C42—24no—
366C42—24no—
367C42—44no—
368C42—48no—
369C42—48no—
370C42—42no—
371C42—42no—
372C42—44no—
373C42—42no—
374C42—42no—
375C42—82no—
376C42C2×C4⋊C412yes—
377C42C4×D412yes—
378C42C4×D412yes—
379C42C22⋊Q812yes—
380C42C22⋊Q812yes—
381C42#12312yes—
382C42C23×C411yescharacteristic
383C42C22×Q811yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C23—22no—
387C23—22no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23#38512yes—
396C23C2×#12011yescharacteristic
397Q8—24no—
398Q8—22no—
399Q8#42714yescomplemented
400D4—22no—
401D4—22no—
402D4—48no—
403D4—42no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4#45312yes—
444C2×C4#45612yes—
445C2×C4C22×C4⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×#13512yes—
452C2×C4C2×#13512yes—
453C2×C4#57512yes—
454C2×C4#57512yes—
455C8—24no—
456C8—24no—
457C8—42no—
458C8—42no—
459C8—82no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—82no—
469C22—82no—
470C22C2×#38511yescharacteristic
471C22#300212yes—
472C22#300212yes—
473C22C2×#45311yescharacteristic
474C22C2×#45611yescharacteristic
475C4—22no—
476C4—24no—
477C4—21no—
478C4—21no—
479C4—22no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—84no—
485C4—162no—
486C4#185914yes—
487C4C2×#42711yescharacteristic
488C4#291111yescharacteristic
489C4C4×#13511yescharacteristic
490C4#294311yescharacteristic
491C4#307612yes—
492C4#309112yes—
493C2—42no—
494C2—82no—
495C2#1850312yes—
496C2#1851012yes—
497C2#2084311yescharacteristic
498C2C2×#185911yescharacteristic
499C2#2087211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×#120
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21362C2semidirect
3. #20797C2semidirect
6. #8889C2semidirect
7. #8888C2semidirect
8. #8885C2semidirect
9. #8884C2semidirect
10. #8038C2semidirect
11. #7890C2semidirect
45. #1476C4semidirect
399. Q8#427semidirect

Maximal subgroups

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

subgroupindex
#341292
#213622
#207972
#207372
C2×#14762
#88892
#88882
#88852
#88842
#80382
#78902