Finite Group Databasebeta

← #197725

Subgroups — #197725

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×D4×M16C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic, complemented
6#9223C214yesmaximal, complemented
7#9215C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7873C211yesmaximal, characteristic, complemented
10C23×M16—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#476—24no—
14C2×#476—21no—
15C2×#476—21no—
16C2×#476C418yescomplemented
17C2×#476C2212yes—
18C2×#476C2212yes—
19D4×M16—24no—
20D4×M16—24no—
21C2×#436C2212yes—
22C2×#435—22no—
23C2×C8×D4C2212yes—
24C2×#431C2211yescharacteristic
25C22×#106—21no—
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102—21no—
29C2×#407—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#1993—22no—
33#1990C2214yescomplemented
34#1823—21no—
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38C4×#102—21no—
39C4×#102—21no—
40#1736—21no—
41#1736—21no—
42#1548—24no—
43C2×#333C2214yescomplemented
44#1296C2214yescomplemented
45#1277C2214yescomplemented
46C23×Q8—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C22×M16—28no—
50C22×M16—21no—
51C22×M16—22no—
52C22×M16—44no—
53C22×M16—44no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56C23×C8—21no—
57C23×C8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8C2×C414yes—
65C2×C4⋊Q8C2312yes—
66C2×C4×D4—21no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C22⋊C4—21no—
81C22×C42C2311yescharacteristic
82#476—28no—
83#476—28no—
84#476—24no—
85#476—24no—
86#476C2×C418yes—
87#476C2×C418yes—
88#436—24no—
89#436—24no—
90#435—216no—
91C8×D4—24no—
92C8×D4—24no—
93#431—22no—
94#431—22no—
95C2×#106—28no—
96C2×#106—22no—
97C2×#106—22no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2×C414yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#102—28no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112#407—28no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116C4×M16—22no—
117C4×M16—22no—
118C2×#97D412yes—
119C2×C4×C8—21no—
120C2×C4×C8C2312yes—
121#384C2312yes—
122#382—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#340—24no—
128#340—24no—
129#340—22no—
130#340—22no—
131#333—28no—
132C25—21no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—41no—
136C22×Q8—42no—
137C22×Q8—84no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—42no—
142C22×D4C22⋊C414yes—
143C22×D4C2×D411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—28no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—44no—
168C2×M16—432no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—44no—
172C2×M16SD1614yes—
173C22×C8—28no—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—42no—
190C4⋊Q8—42no—
191C4⋊Q8—42no—
192C4⋊Q8C22×C414yes—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—24no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4C22⋊C414yes—
204C4×D4C22⋊C414yes—
205C4×D4C22×C412yes—
206C4×D4C22×C412yes—
207C4×D4C2×D412yes—
208C4×D4C2×D412yes—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C22⋊C414yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—24no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C42—41no—
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42#4111yescharacteristic
241C2×C42#4112yes—
242C2×C42C2411yescharacteristic
243#106—28no—
244#106—28no—
245C4⋊C8—24no—
246C4⋊C8—24no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8C22×C414yes—
252C4⋊C8C2×D414yes—
253#102—28no—
254#102—28no—
255#102—44no—
256#102—44no—
257C22⋊C8—28no—
258C22⋊C8—28no—
259C22⋊C8—42no—
260#97—24no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—42no—
264#95—41no—
265#95—41no—
266C24—21no—
267C24—21no—
268C24—24no—
269C24—44no—
270C24—42no—
271C24—42no—
272C24—82no—
273C24#12111yescharacteristic
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—88no—
280C2×Q8—88no—
281C2×D4—28no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—44no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4C2×C22⋊C412yes—
292C2×D4C2×C22⋊C412yes—
293C2×D4#13714yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—84no—
328C22×C4C2×C22⋊C412yes—
329C22×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4#12012yes—
332C22×C4#12112yes—
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12312yes—
337C22×C4C2×SD1612yes—
338C22×C4#13512yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344M16—28no—
345M16—416no—
346M16—48no—
347M16—44no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—416no—
356C2×C8—48no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8C2×SD1614yes—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—82no—
378C4⋊C4C2×C22⋊C412yes—
379C4⋊C4C2×C22⋊C412yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—82no—
387C42—82no—
388C42C2×C22⋊C412yes—
389C42C4×D412yes—
390C42C4×D412yes—
391C42#12012yes—
392C42#12112yes—
393C42#12312yes—
394C42C23×C411yescharacteristic
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C23—22no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—48no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—48no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—84no—
413C23—88no—
414C23—88no—
415C23#38612yes—
416C23#46112yes—
417C23#46312yes—
418C23C2×#12111yescharacteristic
419Q8—84no—
420Q8—88no—
421Q8—84no—
422D4—22no—
423D4—22no—
424D4—22no—
425D4—24no—
426D4—48no—
427D4—42no—
428D4#41914yes—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—48no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—88no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#38612yes—
461C2×C4#38612yes—
462C2×C4#38612yes—
463C2×C4C4×SD1614yes—
464C2×C4#44014yes—
465C2×C4#45014yes—
466C2×C4#45214yes—
467C2×C4#46112yes—
468C2×C4#46312yes—
469C2×C4C22×C22⋊C411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×#12011yescharacteristic
473C2×C4C2×#12111yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4C22×SD1611yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×#13712yes—
479C2×C4C2×#13711yescharacteristic
480C2×C4#57512yes—
481C8—24no—
482C8—44no—
483C8—44no—
484C8—42no—
485C8—82no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—22no—
490C22—24no—
491C22—42no—
492C22—42no—
493C22—44no—
494C22—44no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—816no—
499C22—82no—
500C22—82no—
501C22#187714yes—
502C22C2×#38611yescharacteristic
503C22C2×#46111yescharacteristic
504C22C2×#46311yescharacteristic
505C22#305612yes—
506C22#306812yes—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—84no—
514C4—162no—
515C4#187714yes—
516C4C2×#41911yescharacteristic
517C4#289611yescharacteristic
518C4C2×C4×SD1611yescharacteristic
519C4C2×#44011yescharacteristic
520C4C2×#45011yescharacteristic
521C4C2×#45211yescharacteristic
522C4#305612yes—
523C2—24no—
524C2—44no—
525C2—82no—
526C2#1916512yes—
527C2#1916812yes—
528C2C2×#187711yescharacteristic
529C2#2097211yescharacteristic
530C2#2100511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21324C2semidirect
5. C2×#1548C2semidirect
6. #9223C2semidirect
7. #9215C2semidirect
8. #8045C2semidirect
9. #7873C2semidirect
16. C2×#476C4semidirect
33. #1990C22semidirect
43. C2×#333C22semidirect
44. #1296C22semidirect
45. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×D4×M162
#213242
#207322
C2×#15482
#92232
#92152
#80452
#78732