Finite Group Databasebeta

← #194721

Subgroups — #194721

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic, complemented
3#20732C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic, complemented
5#9027C212yesmaximal, complemented
6#9025C212yesmaximal, complemented
7#9019C212yesmaximal, complemented
8#9017C212yesmaximal
9#8014C212yesmaximal, complemented
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13C22×#117C2211yescharacteristic
14C2×#431C2212yes—
15#2392—24no—
16#2392C418yescomplemented
17C2×#398C2211yescharacteristic
18C2×#397C2211yescharacteristic
19C2×#396—21no—
20C2×#394—21no—
21C2×#394C2211yescharacteristic
22C2×#392C2211yescharacteristic
23#2317—24no—
24#2316—21no—
25#2316C2212yes—
26#2315—21no—
27#2315C2212yes—
28#2296—24no—
29#2285—21no—
30#2285C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1823C2214yes—
34#1802—24no—
35#1786C2214yes—
36C4×#102—24no—
37#1458—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×D4—21no—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C2×C4×D4—28no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4—44no—
52C2×C4×D4—44no—
53C2×C4×D4C2×C414yescomplemented
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—24no—
61C2×#117—44no—
62C2×#117—44no—
63C2×#117C2×C414yes—
64C2×#117D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C42—22no—
80C22×C42—21no—
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83#431—24no—
84#431—24no—
85C2×C4⋊C8—22no—
86C2×#102—28no—
87C2×C22⋊C8C2314yes—
88C2×C4×C8—22no—
89#398—22no—
90#398C2×C414yescomplemented
91#397—22no—
92#397C2×C414yes—
93#396—28no—
94#394—22no—
95#394—28no—
96#394C2×C414yescomplemented
97#392—22no—
98#392C2×C414yes—
99#389—28no—
100#389—28no—
101#385—28no—
102#385—28no—
103#384—22no—
104#384D414yescomplemented
105#382—22no—
106#382—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380C2314yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yescomplemented
116C4×C4⋊C4D414yescomplemented
117C4×C22⋊C4—28no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122#340—28no—
123#324—24no—
124#323—24no—
125C25—21no—
126C25C2×D411yescharacteristic
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4C22×C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C22⋊C414yes—
155C23×C4D814yes—
156C23×C4SD1614yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—24no—
164C4×D4—28no—
165C4×D4—28no—
166C4×D4—432no—
167C4×D4—44no—
168C4×D4—44no—
169#117—28no—
170#117—28no—
171#117—28no—
172#117—28no—
173#117—48no—
174#117—44no—
175#117—48no—
176#117—44no—
177#117—44no—
178#117—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D418yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4C22⋊C414yes—
224C2×C22⋊C4C22×C412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—48no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C414yes—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42#4114yes—
241C4⋊C8—44no—
242#102—416no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C4×C8—44no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—21no—
252#95—21no—
253#95—24no—
254#95C22×C412yes—
255#95C22×C412yes—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—24no—
261C24—22no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24C2×C22⋊C412yes—
274C24C2×D812yes—
275C24C2×SD1612yes—
276C24C22×D411yescharacteristic
277C2×D4—28no—
278C2×D4—24no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—416no—
286C2×D4—416no—
287C2×D4—48no—
288C2×D4—42no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—28no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4#102116yescomplemented
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4#12012yes—
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12312yes—
351C22×C4#12312yes—
352C22×C4C2×D812yes—
353C22×C4C2×D812yes—
354C22×C4C2×SD1612yes—
355C22×C4C2×SD1612yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4#13712yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4112yes—
362C2×C8—48no—
363C2×C8—44no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—48no—
392C42—48no—
393C42C4×D414yes—
394C42C4×D414yes—
395C42#12114yes—
396C42#12114yes—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—42no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—82no—
438C23C2×#10214yes—
439C23C22×C22⋊C411yescharacteristic
440C23C2×#12011yescharacteristic
441C23C2×#12311yescharacteristic
442C23C22×D811yescharacteristic
443C23C22×SD1611yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×#13711yescharacteristic
446D4—28no—
447D4—48no—
448D4—44no—
449D4—432no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—216no—
459C2×C4—216no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—816no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4#38614yes—
497C2×C4#38612yes—
498C2×C4#38612yes—
499C2×C4C2×#10214yes—
500C2×C4C2×#10214yes—
501C2×C4#41714yes—
502C2×C4#44712yes—
503C2×C4#44712yes—
504C2×C4#44812yes—
505C2×C4#44812yes—
506C2×C4#48014yes—
507C2×C4#48114yes—
508C2×C4C2×C4×D412yes—
509C2×C4C2×#12112yes—
510C8—84no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—24no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22—82no—
539C22C2×#38611yescharacteristic
540C22C22×#10211yescharacteristic
541C22C2×#41711yescharacteristic
542C22C2×#44711yescharacteristic
543C22C2×#44811yescharacteristic
544C22#301812yes—
545C22#302512yes—
546C22C2×#48011yescharacteristic
547C22C2×#48111yescharacteristic
548C4—24no—
549C4—24no—
550C4—22no—
551C4—22no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—88no—
558C4#187414yes—
559C4#187414yes—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—44no—
565C2—84no—
566C2#1812212yes—
567C2#1812412yes—
568C2C2×#187411yescharacteristic
569C2#2095211yescharacteristic
570C2#2095911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC2×#392C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34052C2semidirect
3. #20732C2semidirect
4. C2×#1458C2semidirect
5. #9027C2semidirect
6. #9025C2semidirect
7. #9019C2semidirect
9. #8014C2semidirect
16. #2392C4semidirect
53. C2×C4×D4C2×C4semidirect
90. #398C2×C4semidirect
96. #394C2×C4semidirect
104. #384D4semidirect
115. C4×C4⋊C4C2×C4semidirect
116. C4×C4⋊C4D4semidirect
341. C22×C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
#340522
#207322
C2×#14582
#90272
#90252
#90192
#90172
#80142