Finite Group Databasebeta

← #195567

Subgroups — #195567

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34141C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1547C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41—21no—
13#3227—24no—
14#3227C418yescomplemented
15C2×#498C2212yes—
16C2×#486C2212yes—
17C2×#481—22no—
18C2×#476—22no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—22no—
23#2367—22no—
24#2366C2212yes—
25#2365C2212yes—
26#2364—22no—
27#2361—22no—
28#2297—21no—
29#2293—21no—
30#2290—22no—
31C42×D4—21no—
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1988C2214yes—
35#1987—22no—
36#1981—21no—
37#1981—21no—
38#1981C2214yes—
39#1736—21no—
40#1736—21no—
41#1547—24no—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8—21no—
48C22×M16—21no—
49C23×C8—21no—
50#532—22no—
51#532—22no—
52#532C2×C414yes—
53#532D414yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yescomplemented
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124—21no—
65C2×#121—21no—
66C2×#121D412yes—
67C4×#41—44no—
68C4×#41—44no—
69C2×C4×Q8—22no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4D412yes—
75C2×#117—22no—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#498—24no—
85#498C2×C418yescomplemented
86#486—24no—
87#486C2×C418yescomplemented
88#481—216no—
89#476—216no—
90#431—22no—
91#431C2×C414yescomplemented
92C2×#106—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—48no—
105C2×#102—24no—
106C2×C22⋊C8—21no—
107C2×C4×C8C2312yes—
108#391—22no—
109#390—22no—
110#390—22no—
111#389—22no—
112#388—24no—
113#388—24no—
114#387—24no—
115#386—24no—
116#386—24no—
117#385—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C2×#95—21no—
133C43—21no—
134#340—24no—
135#340—24no—
136#332—24no—
137#331—24no—
138C2×#41—24no—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8—21no—
145C22×Q8—24no—
146C22×Q8—22no—
147C22×Q8—44no—
148C22×Q8—44no—
149C22×Q8—41no—
150C22×Q8—42no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C2×D412yes—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—44no—
169C22×C8—42no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8C22×C414yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C2×D412yes—
184#127—21no—
185#127—21no—
186#127C22×C412yes—
187#127C2×D412yes—
188#124—24no—
189#124—24no—
190#121—24no—
191#121—24no—
192#121—24no—
193#121C22⋊C418yes—
194C4×Q8—44no—
195C4×Q8—44no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—416no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4C22⋊C418yes—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—48no—
209#117SD1614yes—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—24no—
235C2×C42—21no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—48no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254#106—216no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262#103—48no—
263#103—48no—
264#102—216no—
265C22⋊C8—24no—
266C4×C8—22no—
267C4×C8C22×C414yes—
268#95—24no—
269#95—24no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275C24—22no—
276C24#12011yescharacteristic
277#41—48no—
278#41—48no—
279#41—48no—
280#41—48no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—416no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×D4—28no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4C2×C22⋊C414yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C4×D412yes—
334C22×C4C4×D412yes—
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12312yes—
341C22×C4C2×SD1612yes—
342C22×C4#13512yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—84no—
349C2×C8—24no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—44no—
355C2×C8—42no—
356C2×C8—84no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4C2×C22⋊C414yes—
375C4⋊C4C2×SD1614yes—
376C4⋊C4AGL(1,ℤ/8)14yes—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—42no—
383C22⋊C4—84no—
384C42—24no—
385C42—48no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—82no—
394C42C2×C22⋊C412yes—
395C42C4×D412yes—
396C42C4×D412yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C23—22no—
405C23—22no—
406C23—24no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23#38612yes—
412C23#45012yes—
413C23#45212yes—
414C23C2×#12011yescharacteristic
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419Q8—48no—
420Q8—88no—
421Q8—82no—
422Q8—82no—
423D4—44no—
424D4—416no—
425D4—42no—
426D4—42no—
427D4—44no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—82no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—416no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—816no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4#38612yes—
463C2×C4#38612yes—
464C2×C4#38612yes—
465C2×C4C4×SD1614yes—
466C2×C4#44014yes—
467C2×C4#45012yes—
468C2×C4#45212yes—
469C2×C4#46114yes—
470C2×C4#46314yes—
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C22×SD1611yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)12yes—
481C2×C4C2×#13711yescharacteristic
482C2×C4#57512yes—
483C8—44no—
484C8—82no—
485C8—82no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—84no—
494C22—84no—
495C22#187714yes—
496C22C2×#38611yescharacteristic
497C22C2×#45011yescharacteristic
498C22C2×#45211yescharacteristic
499C22#300612yes—
500C22#302212yes—
501C4—22no—
502C4—21no—
503C4—21no—
504C4—22no—
505C4—44no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4#187714yes—
512C4C2×C4×SD1611yescharacteristic
513C4C2×#44011yescharacteristic
514C4C4×AGL(1,ℤ/8)12yes—
515C4C2×#46111yescharacteristic
516C4C2×#46311yescharacteristic
517C4#303811yescharacteristic
518C4#304011yescharacteristic
519C2—24no—
520C2—84no—
521C2#1879912yes—
522C2#1880312yes—
523C2C2×#187711yescharacteristic
524C2#2092811yescharacteristic
525C2#2095511yescharacteristic
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. #21313C2semidirect
4. C2×#1981C2semidirect
6. #9059C2semidirect
7. #9057C2semidirect
8. #7901C2semidirect
9. #7891C2semidirect
14. #3227C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
85. #498C2×C4semidirect
87. #486C2×C4semidirect
91. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#341412
#213132
C2×#19812
C2×#15472
#90592
#90572
#79012
#78912