Finite Group Databasebeta

← #194107

Subgroups — #194107

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#21313C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8988C214yesmaximal, complemented
7#8985C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13#3206—24no—
14#3206C418yescomplemented
15C2×#492—21no—
16C2×#487—21no—
17C2×#461C2214yes—
18C2×#448—22no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2361—22no—
23#2361—22no—
24#2354—21no—
25#2354C2212yes—
26D4×C4⋊C4—21no—
27D4×C4⋊C4C2212yes—
28#2326—22no—
29#2311—21no—
30C4×C4⋊Q8—21no—
31C4×#127—21no—
32C2×#384C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1990—22no—
35#1990C2214yes—
36#1988C2212yes—
37#1988C2212yes—
38#1987—21no—
39#1987—21no—
40#1736—21no—
41#1736—21no—
42C2×#340—22no—
43#1492—24no—
44C2×D4⋊C8C2214yes—
45#1296C2214yes—
46#1279C2212yes—
47#1279C2212yes—
48C22×#41—21no—
49C23×Q8—21no—
50C22×SD16—24no—
51C22×M16—21no—
52C23×C8—21no—
53#532—22no—
54#532—22no—
55#532C2×C414yes—
56#532D414yes—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yescomplemented
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×#127D411yescharacteristic
67C2×#124—21no—
68C2×#121—21no—
69C2×#121D412yes—
70#520—44no—
71#520—44no—
72C2×C4×Q8—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—21no—
78C2×C4×D4C2312yes—
79C2×#117—22no—
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D412yes—
88C22×C42C2311yescharacteristic
89#492—28no—
90#487—28no—
91#461—28no—
92#461C2×C4116yescomplemented
93#448—216no—
94#431—22no—
95#431C2×C414yescomplemented
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×#103—24no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8C2312yes—
108#391—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#389—22no—
113#389—22no—
114#388—24no—
115#388—24no—
116#386—24no—
117#385—24no—
118#385—24no—
119#384—21no—
120#384—21no—
121#384D412yes—
122#384C2312yes—
123#380—22no—
124#380—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#376—21no—
134C43—21no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—28no—
140D4⋊C8—28no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8—21no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8—82no—
153C22×Q8—82no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C2×D412yes—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×SD16—216no—
166C2×SD16—84no—
167C2×SD16—84no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—42no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8C22⋊C414yes—
188C4⋊Q8C22×C414yes—
189#127—22no—
190#127C22⋊C414yes—
191#124—24no—
192#124—24no—
193#121—24no—
194#121—24no—
195#121—24no—
196#121C22⋊C418yes—
197C4×Q8—48no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—48no—
203C4×D4—416no—
204C4×D4—84no—
205C4×D4—84no—
206C4×D4C22×C414yes—
207C4×D4C2×D414yes—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—44no—
212#117—44no—
213#117SD1614yescomplemented
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C42—21no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C22×C412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42C2411yescharacteristic
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—42no—
267C4⋊C8C22×C414yes—
268#103—216no—
269C22⋊C8—24no—
270C4×C8—24no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—42no—
274#95—24no—
275#95—24no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280C24—22no—
281C24#12011yescharacteristic
282#41—48no—
283#41—48no—
284#41—48no—
285#41—48no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—44no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—88no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—88no—
298C2×D4—28no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4C2×C22⋊C414yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12312yes—
346C22×C4C2×SD1612yes—
347C22×C4#13512yes—
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353SD16—816no—
354M16—84no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—48no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—82no—
387C4⋊C4C2×C22⋊C414yes—
388C4⋊C4C2×SD1612yes—
389C4⋊C4C2×SD1612yes—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—48no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—24no—
399C42—48no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42C2×C22⋊C412yes—
406C42C4×D412yes—
407C42C4×D412yes—
408C42#12112yes—
409C42#12112yes—
410C42#12312yes—
411C42C23×C411yescharacteristic
412C42C22×D411yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—24no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23#38612yes—
427C23#45012yes—
428C23#45212yes—
429C23C2×#12011yescharacteristic
430Q8—48no—
431Q8—48no—
432Q8—88no—
433Q8—84no—
434D4—28no—
435D4—416no—
436D4—44no—
437D4—48no—
438D4—48no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—88no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—168no—
474C2×C4—164no—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4#38612yes—
478C2×C4C4×SD1614yes—
479C2×C4#44014yes—
480C2×C4#45012yes—
481C2×C4#45212yes—
482C2×C4#45512yes—
483C2×C4#45612yes—
484C2×C4#46114yes—
485C2×C4#46314yes—
486C2×C4C22×C22⋊C411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4C22×SD1611yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4#57612yes—
497C2×C4#57712yes—
498C8—44no—
499C8—44no—
500C8—82no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22#187714yes—
515C22C2×#38611yescharacteristic
516C22C2×#45011yescharacteristic
517C22C2×#45211yescharacteristic
518C22#300811yescharacteristic
519C22#301011yescharacteristic
520C22C2×#45511yescharacteristic
521C22C2×#45611yescharacteristic
522C4—22no—
523C4—21no—
524C4—21no—
525C4—44no—
526C4—42no—
527C4—42no—
528C4—84no—
529C4—84no—
530C4—81no—
531C4—81no—
532C4#187714yes—
533C4#188214yes—
534C4#289712yes—
535C4C2×C4×SD1611yescharacteristic
536C4C2×#44011yescharacteristic
537C4C2×#46111yescharacteristic
538C4C2×#46311yescharacteristic
539C4#309612yes—
540C2—24no—
541C2—84no—
542C2#1863111yescharacteristic
543C2#1863711yescharacteristic
544C2#1865111yescharacteristic
545C2#1865711yescharacteristic
546C2#2091411yescharacteristic
547C2C2×#187711yescharacteristic
548C2C2×#188211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. C2×#1990C2semidirect
4. #21313C2semidirect
6. #8988C2semidirect
7. #8985C2semidirect
8. #8043C2semidirect
9. #7890C2semidirect
14. #3206C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
92. #461C2×C4semidirect
95. #431C2×C4semidirect
213. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#342072
C2×#19902
#213132
C2×#14922
#89882
#89852
#80432
#78902