Finite Group Databasebeta

← #197499

Subgroups — #197499

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic, complemented
3#17574C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic, complemented
5#9204C214yesmaximal, complemented
6#9203C214yesmaximal, complemented
7#7981C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C22×#117—21no—
11C22×#117C2211yescharacteristic
12C2×#431C2212yes—
13#2621C412yescomplemented
14#2620—22no—
15#2619C412yescomplemented
16#2618—21no—
17#2618C2212yes—
18#2617—21no—
19#2617C2212yes—
20#2392—22no—
21#2392C414yescomplemented
22#2317—24no—
23#2316—21no—
24#2316C2212yes—
25#2315—21no—
26#2315C2212yes—
27#2297—21no—
28#2296—24no—
29#2285—21no—
30#2285C2211yescharacteristic
31#2285C2212yes—
32#1519C2218yes—
33#1517—24no—
34#1502—24no—
35#1458—22no—
36#1458—22no—
37C2×#324C2212yes—
38C2×#323C2212yes—
39#1289C2218yes—
40C22×#41—21no—
41C24×C4D411yescharacteristic
42#532—22no—
43#532—22no—
44#532—22no—
45#532—22no—
46C2×C4⋊Q8C2×C411yescharacteristic, complemented
47C2×#127C2×C411yescharacteristic, complemented
48C2×#124—21no—
49#525—44no—
50#525—44no—
51C2×#123—22no—
52C2×C22⋊Q8—22no—
53C2×#121—22no—
54C2×#121—21no—
55C2×#121C2312yes—
56C2×#120—22no—
57C2×C4×D4—24no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yescomplemented
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—44no—
70C2×#117—44no—
71C2×#117C2×C412yes—
72C2×#117C2×C412yes—
73C2×#117D412yes—
74C2×#117D412yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81C22×C42—22no—
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84#431—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8C2314yes—
90C2×C4×C8—22no—
91#398—22no—
92#398C2×C414yescomplemented
93#397—22no—
94#397C2×C414yes—
95#396—24no—
96#396—24no—
97#394—22no—
98#394—24no—
99#394—24no—
100#394C2×C414yescomplemented
101#392—22no—
102#392C2×C414yes—
103#391—22no—
104#390—22no—
105#389—28no—
106#389—28no—
107#386—22no—
108#386—22no—
109#385—28no—
110#385—28no—
111#384—22no—
112#384—22no—
113#384—21no—
114#382—22no—
115#382C2×C412yescomplemented
116#382D414yescomplemented
117#380—22no—
118#380—22no—
119#380—22no—
120#380—21no—
121#380C2312yes—
122#380C2314yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yescomplemented
128C4×C4⋊C4D414yescomplemented
129C4×C22⋊C4—24no—
130#327—216no—
131#324—24no—
132#323—24no—
133C25C2×D411yescharacteristic
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—41no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C22×Q8—21no—
143C22×D4—22no—
144C22×D4—24no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4D814yes—
159C23×C4SD1614yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169#127—22no—
170#127—22no—
171#124—42no—
172#124—42no—
173#123—44no—
174#123—44no—
175C22⋊Q8—48no—
176#121—24no—
177#121—24no—
178#121—48no—
179#121—42no—
180#121—42no—
181#120—44no—
182#120—44no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—416no—
186C4×D4—42no—
187C4×D4—42no—
188#117—24no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—48no—
197#117—44no—
198#117—48no—
199#117—44no—
200#117—44no—
201#117—42no—
202#117—42no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22×C412yes—
243C2×C22⋊C4C2×D412yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—48no—
252C2×C42C22⋊C412yes—
253C2×C42C22⋊C412yes—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C4×C8—42no—
267C4×C8—42no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—24no—
274#95C22×C414yes—
275#95C2×D418yes—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—81no—
282C24—81no—
283C24C2×C22⋊C412yes—
284C24C2×D812yes—
285C24C2×SD1612yes—
286C24C22×D411yescharacteristic
287#41—82no—
288#41—82no—
289#41—82no—
290#41—82no—
291#41—82no—
292#41—82no—
293#41—82no—
294#41—82no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—24no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—48no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—82no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—82no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4#10218yes—
353C22×C4#10218yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4#12012yes—
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12312yes—
365C22×C4#12311yescharacteristic
366C22×C4#12311yescharacteristic
367C22×C4C2×D812yes—
368C22×C4C2×D812yes—
369C22×C4C2×SD1612yes—
370C22×C4C2×SD1612yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4112yes—
377C2×C8—44no—
378C2×C8—84no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—48no—
400C22⋊C4—44no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—48no—
415C42—44no—
416C42—48no—
417C42C4×D412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42C4×D412yes—
421C42#12112yes—
422C42#12112yes—
423C42#12112yes—
424C42#12112yes—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23C2×#10214yes—
449C23C22×C22⋊C411yescharacteristic
450C23C2×#12011yescharacteristic
451C23C2×#12311yescharacteristic
452C23C22×D811yescharacteristic
453C23C22×SD1611yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23C2×#13711yescharacteristic
456Q8—82no—
457Q8—82no—
458D4—84no—
459D4—84no—
460D4—816no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—816no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4#38612yes—
506C2×C4#38612yes—
507C2×C4#38611yescharacteristic
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4#38611yescharacteristic
511C2×C4C2×#10214yes—
512C2×C4C2×#10212yes—
513C2×C4C2×#10212yes—
514C2×C4#41712yes—
515C2×C4#41712yes—
516C2×C4#44712yes—
517C2×C4#44712yes—
518C2×C4#44812yes—
519C2×C4#44812yes—
520C2×C4#48012yes—
521C2×C4#48012yes—
522C2×C4#48112yes—
523C2×C4#48112yes—
524C2×C4C2×C4×D412yes—
525C2×C4C2×#12112yes—
526C8—84no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—24no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—84no—
547C22—84no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#38611yescharacteristic
552C22C22×#10211yescharacteristic
553C22C2×#41711yescharacteristic
554C22C2×#44711yescharacteristic
555C22C2×#44811yescharacteristic
556C22C2×#48011yescharacteristic
557C22C2×#48111yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—88no—
566C4—84no—
567C4—82no—
568C4#187412yes—
569C4#187412yes—
570C4#187412yes—
571C4#187412yes—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—44no—
579C2—162no—
580C2C2×#187411yescharacteristic
581C2#2101211yescharacteristic
582C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×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. #17574C2semidirect
4. C2×#1458C2semidirect
5. #9204C2semidirect
6. #9203C2semidirect
7. #7981C2semidirect
13. #2621C4semidirect
15. #2619C4semidirect
21. #2392C4semidirect
46. C2×C4⋊Q8C2×C4semidirect
47. C2×#127C2×C4semidirect
59. C2×C4×D4C2×C4semidirect
92. #398C2×C4semidirect
100. #394C2×C4semidirect
115. #382C2×C4semidirect
116. #382D4semidirect
127. C4×C4⋊C4C2×C4semidirect
128. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#346332
#340522
#175742
C2×#14582
#92042
#92032
#79812