Finite Group Databasebeta

← #194148

Subgroups — #194148

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41—21no—
12C2×C4×#41C2211yescharacteristic
13#3079—24no—
14#3079C418yescomplemented
15C2×#477—21no—
16C2×#475C2212yes—
17C2×#476—21no—
18C2×#474C2212yes—
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#417—21no—
24C22×#103—22no—
25#2351—21no—
26Q8×C4⋊C4C2212yes—
27#2349—22no—
28#2349—21no—
29D4×C4⋊C4—22no—
30D4×C4⋊C4C2212yes—
31#2296—22no—
32#2296—21no—
33#2285—21no—
34#2285C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990C2214yes—
38#1987—22no—
39#1981—21no—
40#1981—21no—
41#1981C2214yes—
42#1824—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yescomplemented
59C2×C4⋊Q8C2312yes—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—44no—
63C4×#41—44no—
64C4×#41C2×C414yes—
65C4×#41D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—22no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D412yes—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#477—28no—
88#476—28no—
89#475—24no—
90#475C2×C418yescomplemented
91#474—24no—
92#474C2×C418yescomplemented
93#431—22no—
94#431—22no—
95#428—28no—
96C2×#107—22no—
97C2×#106—22no—
98#423—22no—
99#423C2×C414yescomplemented
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#417—28no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—48no—
112C2×#102—22no—
113C2×C22⋊C8—21no—
114C2×C4×C8C2312yes—
115#389—24no—
116#389—22no—
117#389—22no—
118#389—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384—22no—
123#384C2312yes—
124#382—22no—
125#382—21no—
126#380—22no—
127#380—21no—
128#380—22no—
129#380D412yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D412yes—
140C4×C4⋊C4C2312yes—
141C2×#95—22no—
142C2×#95—21no—
143#340—28no—
144Q8⋊C8—24no—
145D4⋊C8—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41C22⋊C414yes—
150C2×#41SD1614yes—
151C2×#41Q1614yes—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8C2×D411yescharacteristic
159C22×D4—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—41no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—44no—
177C22×C8—42no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8C22×C414yes—
188C4×Q8—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8C22⋊C414yes—
194C4×Q8C22×C412yes—
195C4×Q8C2×D412yes—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—416no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4C22⋊C414yes—
207C4×D4C22×C412yes—
208C4×D4C2×D412yes—
209#117—22no—
210#117—24no—
211#117—24no—
212#117—48no—
213#117—42no—
214#117—42no—
215#117C22⋊C414yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—48no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42C22×C412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42C2411yescharacteristic
267#107—28no—
268#106—28no—
269C4⋊C8—24no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8C22×C414yes—
276#103—28no—
277#103—48no—
278#103—48no—
279#102—28no—
280C22⋊C8—42no—
281C4×C8—22no—
282C4×C8—22no—
283#95—24no—
284#95—24no—
285#95—21no—
286#95—22no—
287#95—21no—
288#95—22no—
289#95—22no—
290C24—21no—
291C24#12011yescharacteristic
292#41—28no—
293#41—28no—
294#41—28no—
295#41#103116yescomplemented
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—24no—
300C2×Q8—416no—
301C2×Q8—42no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×Q8C2×C22⋊C412yes—
309C2×Q8C2×SD1612yes—
310C2×Q8C2×Q1612yes—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C2×C22⋊C412yes—
318C2×D4C2×SD1612yes—
319C2×D4C2×Q1612yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12312yes—
356C22×C4C2×SD1612yes—
357C22×C4C2×Q1612yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4#13712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—48no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—44no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—84no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—84no—
395C4⋊C4C2×C22⋊C412yes—
396C4⋊C4C2×C22⋊C412yes—
397C4⋊C4AGL(1,ℤ/8)14yes—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—24no—
402C22⋊C4—48no—
403C22⋊C4—44no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—48no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—44no—
413C42—42no—
414C42C2×C22⋊C412yes—
415C42C4×D412yes—
416C42C4×D412yes—
417C42#12112yes—
418C42#12112yes—
419C42#12312yes—
420C42C23×C411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23#38612yes—
432C23#45012yes—
433C23#45112yes—
434C23C2×#12011yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—22no—
438Q8—48no—
439Q8—88no—
440Q8—82no—
441Q8—82no—
442Q8C2×#10314yescomplemented
443D4—216no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4C2×#10314yescomplemented
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—216no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—84no—
478C2×C4—88no—
479C2×C4—816no—
480C2×C4—88no—
481C2×C4—84no—
482C2×C4#38612yes—
483C2×C4#38612yes—
484C2×C4#38612yes—
485C2×C4C2×#10314yes—
486C2×C4#41714yes—
487C2×C4#45012yes—
488C2×C4#45112yes—
489C2×C4#48314yes—
490C2×C4#48414yes—
491C2×C4C22×C22⋊C411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12311yescharacteristic
497C2×C4C22×SD1611yescharacteristic
498C2×C4C22×Q1611yescharacteristic
499C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4C2×#13711yescharacteristic
502C2×C4#57512yes—
503C8—44no—
504C8—82no—
505C8—82no—
506C22—24no—
507C22—24no—
508C22—22no—
509C22—22no—
510C22—24no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—42no—
515C22#187214yes—
516C22C2×#38611yescharacteristic
517C22C2×#45011yescharacteristic
518C22C2×#45111yescharacteristic
519C22#300612yes—
520C22#302212yes—
521C4—24no—
522C4—22no—
523C4—21no—
524C4—22no—
525C4—21no—
526C4—44no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—84no—
531C4—84no—
532C4#187214yes—
533C4C22×#10311yescharacteristic
534C4C2×#41711yescharacteristic
535C4C4×AGL(1,ℤ/8)12yes—
536C4#303811yescharacteristic
537C4#304011yescharacteristic
538C4C2×#48311yescharacteristic
539C4C2×#48411yescharacteristic
540C2—24no—
541C2—44no—
542C2#1879912yes—
543C2#1880112yes—
544C2C2×#187211yescharacteristic
545C2#2092811yescharacteristic
546C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21834C2semidirect
4. C2×#1981C2semidirect
6. #8983C2semidirect
7. #8981C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
14. #3079C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
90. #475C2×C4semidirect
92. #474C2×C4semidirect
99. #423C2×C4semidirect
295. #41#103semidirect
442. Q8C2×#103semidirect
447. D4C2×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C4⋊C4×#412
#218342
C2×#19812
C2×#14772
#89832
#89812
#80452
#79012