Finite Group Databasebeta

← #195206

Subgroups — #195206

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2#34205C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic, complemented
8#7907C211yesmaximal, characteristic, complemented
9C22×#121C2211yescharacteristic
10C2×#518—21no—
11C23×C4⋊C4C2211yescharacteristic
12C2×#480—24no—
13C2×#480—22no—
14C2×#480C418yescomplemented
15C2×#480C2214yes—
16C22×#107—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×#381—21no—
29C2×#381—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1823—22no—
34#1823C2214yescomplemented
35C2×#340—22no—
36#1519—24no—
37C2×#327C2214yes—
38#1445C2214yescomplemented
39C4⋊#95—22no—
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#480—216no—
83#480—28no—
84#480C2×C4116yescomplemented
85C2×#107—28no—
86C2×#107—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#381—44no—
108#381—44no—
109#380—21no—
110#380—21no—
111#380—22no—
112#380—22no—
113#380D412yes—
114#380C2312yes—
115#379—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C2×#95—21no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—28no—
128#327—28no—
129C25—21no—
130C25C2×D411yescharacteristic
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—44no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C22⋊C414yes—
143C22×D4C2×D412yes—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22×C412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—24no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—42no—
176#121—24no—
177#121—24no—
178#121—24no—
179#121—24no—
180#121—24no—
181#121—22no—
182#121—22no—
183#121C22⋊C418yes—
184#121C22×C414yes—
185#121C2×D414yes—
186C4×D4—416no—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—84no—
190#117—48no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4D814yes—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246#107—216no—
247#102—216no—
248C22⋊C8—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—42no—
253C22⋊C8C22×C414yes—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—48no—
263#95—48no—
264C24—22no—
265C24—24no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—82no—
271C24—82no—
272C24C4×D412yes—
273C24#12011yescharacteristic
274C24#12011yescharacteristic
275C24#12112yes—
276C24C2×D812yes—
277C24#13512yes—
278C24AGL(1,ℤ/8)12yes—
279C24AGL(1,ℤ/8)12yes—
280C24C22×D411yescharacteristic
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—84no—
300C2×D4C2×C22⋊C414yes—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12312yes—
364C22×C4#12312yes—
365C22×C4C2×D814yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C2×C8—24no—
373C2×C8—44no—
374C2×C8—48no—
375C2×C8—44no—
376C2×C8—42no—
377C2×C8—84no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—84no—
390C4⋊C4C2×C22⋊C414yes—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—28no—
396C22⋊C4—48no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C22⋊C4—84no—
405C42—48no—
406C42—44no—
407C42—82no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—48no—
428C23—84no—
429C23—84no—
430C23—164no—
431C23—164no—
432C23#38612yes—
433C23#38612yes—
434C23C4×D814yes—
435C23#44214yes—
436C23#44712yes—
437C23#44712yes—
438C23#44912yes—
439C23#44912yes—
440C23#45914yes—
441C23#46314yes—
442C23C2×C4×D411yescharacteristic
443C23C2×#12011yescharacteristic
444C23C2×#12111yescharacteristic
445C23C22×D811yescharacteristic
446C23C2×#13511yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×AGL(1,ℤ/8)11yescharacteristic
449D4—44no—
450D4—416no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—48no—
456D4—82no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—82no—
493C2×C4—88no—
494C2×C4—88no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—168no—
506C2×C4—164no—
507C2×C4#38612yes—
508C2×C4#38612yes—
509C2×C4#45814yes—
510C2×C4C22×C22⋊C411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C8—44no—
515C8—82no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—24no—
523C22—24no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—22no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—84no—
540C22—84no—
541C22—168no—
542C22#187914yes—
543C22#187914yes—
544C22C2×#38611yescharacteristic
545C22C2×C4×D811yescharacteristic
546C22C2×#44211yescharacteristic
547C22C2×#44711yescharacteristic
548C22C2×#44911yescharacteristic
549C22#302212yes—
550C22C2×#45812yes—
551C22C2×#45911yescharacteristic
552C22C2×#46311yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—84no—
561C4—82no—
562C4—84no—
563C4—162no—
564C4#189014yes—
565C2—21no—
566C2—24no—
567C2—24no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—84no—
572C2#1820212yes—
573C2#1820612yes—
574C2C2×#187911yescharacteristic
575C2#2094411yescharacteristic
576C2C2×#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×#1823C2semidirect
5. #13904C2semidirect
6. #9043C2semidirect
7. #8023C2semidirect
8. #7907C2semidirect
14. C2×#480C4semidirect
34. #1823C22semidirect
38. #1445C22semidirect
40. #1294C22semidirect
41. #1280C22semidirect
71. C22×C4⋊C4C2×C4semidirect
84. #480C2×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×#4802
#342052
C2×#18232
C2×#15192
#139042
#90432
#80232
#79072