Finite Group Databasebeta

← #195207

Subgroups — #195207

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2#34205C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic, complemented
8#7907C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C2×#518—21no—
11C23×C4⋊C4C2211yescharacteristic
12C2×#481—24no—
13C2×#481—22no—
14C2×#481C418yescomplemented
15C2×#481C2214yes—
16C22×#106—21no—
17C22×#102—21no—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2364—22no—
21#2361—22no—
22#2344—21no—
23#2342—22no—
24D4×C22⋊C4—21no—
25#2325C2214yes—
26#2308—21no—
27C4×#121—22no—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1824—22no—
32#1824C2214yescomplemented
33C2×#340—22no—
34#1519—24no—
35C2×#327C2214yes—
36#1445C2214yescomplemented
37C4⋊#95—22no—
38C22⋊#95—21no—
39C22⋊#95—21no—
40#1294C2214yescomplemented
41#1280C2214yescomplemented
42C23×D4—21no—
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—21no—
47C23×C8—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121C2×C414yes—
53C2×#121D414yes—
54C2×#121D412yes—
55C2×#121C2312yes—
56#518—44no—
57#518—44no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×#117—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C414yescomplemented
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42D412yes—
82#481—216no—
83#481—28no—
84#481C2×C4116yescomplemented
85C2×#106—28no—
86C2×#106—24no—
87C2×#102—28no—
88C2×#102—24no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8C2×C414yescomplemented
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97#390—22no—
98#389—22no—
99#388—24no—
100#388—24no—
101#387—24no—
102#386—24no—
103#386—24no—
104#386—24no—
105#385—24no—
106#381—22no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380D412yes—
112#380C2312yes—
113#379—22no—
114C4×C4⋊C4—22no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95C2312yes—
124C2×#95C2312yes—
125#340—28no—
126#327—28no—
127C25—21no—
128C25C2×D411yescharacteristic
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C22⋊C414yes—
141C22×D4C2×D412yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—42no—
152C23×C4—84no—
153C23×C4C22×C412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—24no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8—42no—
171#121—24no—
172#121—24no—
173#121—24no—
174#121—24no—
175#121—24no—
176#121—22no—
177#121—22no—
178#121C22⋊C418yes—
179#121C22×C414yes—
180#121C2×D414yes—
181C4×D4—416no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—84no—
185#117—48no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4SD1614yes—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241#106—216no—
242#102—216no—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—42no—
248C22⋊C8C22×C414yes—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—84no—
258#95—84no—
259C24—22no—
260C24—24no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—82no—
266C24—82no—
267C24C4×D412yes—
268C24#12011yescharacteristic
269C24#12011yescharacteristic
270C24#12112yes—
271C24C2×SD1612yes—
272C24#13512yes—
273C24AGL(1,ℤ/8)12yes—
274C24#13712yes—
275C24C22×D411yescharacteristic
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—28no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—44no—
294C2×D4—84no—
295C2×D4C2×C22⋊C414yes—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—88no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—88no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12312yes—
353C22×C4#12312yes—
354C22×C4C2×SD1614yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C2×C8—24no—
362C2×C8—44no—
363C2×C8—48no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—84no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—84no—
379C4⋊C4C2×C22⋊C414yes—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—28no—
385C22⋊C4—48no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—48no—
395C42—44no—
396C42—82no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—48no—
417C23—84no—
418C23—84no—
419C23—164no—
420C23—164no—
421C23#38612yes—
422C23#38612yes—
423C23C4×SD1614yes—
424C23#44014yes—
425C23#45012yes—
426C23#45012yes—
427C23#45212yes—
428C23#45212yes—
429C23#46114yes—
430C23#46314yes—
431C23C2×C4×D411yescharacteristic
432C23C2×#12011yescharacteristic
433C23C2×#12111yescharacteristic
434C23C22×SD1611yescharacteristic
435C23C2×#13511yescharacteristic
436C23C2×AGL(1,ℤ/8)11yescharacteristic
437C23C2×#13711yescharacteristic
438D4—44no—
439D4—416no—
440D4—44no—
441D4—44no—
442D4—44no—
443D4—44no—
444D4—48no—
445D4—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—168no—
493C2×C4—164no—
494C2×C4#38612yes—
495C2×C4#38612yes—
496C2×C4#45814yes—
497C2×C4C22×C22⋊C411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C8—44no—
502C8—82no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—24no—
511C22—21no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—22no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—84no—
527C22—84no—
528C22—168no—
529C22#187714yes—
530C22#187714yes—
531C22C2×#38611yescharacteristic
532C22C2×C4×SD1611yescharacteristic
533C22C2×#44011yescharacteristic
534C22C2×#45011yescharacteristic
535C22C2×#45211yescharacteristic
536C22#302112yes—
537C22C2×#45812yes—
538C22C2×#46111yescharacteristic
539C22C2×#46311yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—22no—
545C4—44no—
546C4—44no—
547C4—84no—
548C4—82no—
549C4—84no—
550C4—162no—
551C4#189014yes—
552C2—21no—
553C2—24no—
554C2—24no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—84no—
559C2#1820312yes—
560C2#1820712yes—
561C2C2×#187711yescharacteristic
562C2#2094511yescharacteristic
563C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1824C2semidirect
5. #13924C2semidirect
6. #9045C2semidirect
7. #8023C2semidirect
8. #7907C2semidirect
14. C2×#481C4semidirect
32. #1824C22semidirect
36. #1445C22semidirect
40. #1294C22semidirect
41. #1280C22semidirect
71. C22×C4⋊C4C2×C4semidirect
84. #481C2×C4semidirect
94. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
#342052
C2×#18242
C2×#15192
#139242
#90452
#80232
#79072