Finite Group Databasebeta

← #196846

Subgroups — #196846

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#9155C214yesmaximal, complemented
7#9147C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#535—21no—
11C2×#532C2211yescharacteristic
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2847C2214yes—
17#2617—22no—
18#2547C412yescomplemented
19#2544—22no—
20#2541C412yescomplemented
21#2537C414yescomplemented
22#2535—22no—
23#2349—24no—
24#2349—24no—
25#2311—21no—
26C2×#389—21no—
27C2×#385—22no—
28#2285C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C22×#95C2212yes—
33#1954—22no—
34#1954—22no—
35#1779C2214yes—
36#1517—24no—
37#1502—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4—22no—
47C24×C4C2312yes—
48#535—44no—
49#535—44no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8C2×C411yescharacteristic, complemented
55C2×#127—22no—
56C2×#127C2×C411yescharacteristic, complemented
57C2×#124—22no—
58C2×#124—21no—
59C2×#121—24no—
60C2×#121—21no—
61C2×#121C2×C418yescomplemented
62C2×#121C2312yes—
63C2×#120—24no—
64C2×C4×D4—28no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—44no—
71C2×C4×D4—44no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C412yescomplemented
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4D411yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#431—22no—
87#431—22no—
88#431—22no—
89#431—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C22⋊C8—28no—
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2312yes—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#397—24no—
98#394—28no—
99#392—24no—
100#391—22no—
101#390—22no—
102#389—28no—
103#386—24no—
104#386—28no—
105#385—216no—
106#384C2×C412yescomplemented
107#382—22no—
108#382—21no—
109#382D414yescomplemented
110#380—21no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380C2×C418yescomplemented
115#380C2312yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yescomplemented
119C4×C4⋊C4D414yescomplemented
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95C2311yescharacteristic
125C2×#95C2312yes—
126C2×#95C2311yescharacteristic
127#327—24no—
128#327—24no—
129#327—28no—
130C25C2×D412yes—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8—21no—
138C22×D4—22no—
139C22×D4—28no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4C22⋊C412yes—
147C22×D4C22⋊C412yes—
148C22×D4D814yes—
149C22×D4SD1614yes—
150C22×D4C22×C412yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—22no—
160C23×C4—48no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—84no—
164C23×C4—84no—
165C23×C4C22×C414yes—
166C23×C4C2×D412yes—
167C23×C4C2×D414yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175#127—22no—
176#127—22no—
177#127—48no—
178#124—48no—
179#124—42no—
180#124—42no—
181#121—22no—
182#121—22no—
183#121—22no—
184#121—22no—
185#121—416no—
186#121—42no—
187#121—42no—
188#121—88no—
189#120—48no—
190#120—48no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—432no—
202C4×D4—42no—
203C4×D4—42no—
204#117—42no—
205#117—42no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C414yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4C22⋊C414yes—
233C2×C22⋊C4C22⋊C414yes—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—88no—
253C4×C8—42no—
254C4×C8—42no—
255#95—28no—
256#95—22no—
257#95—28no—
258#95—24no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95C2×D412yes—
264#95C2×D414yes—
265#95C2×D412yes—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—44no—
272C24—44no—
273C24C2×C22⋊C414yes—
274C24C2×D814yes—
275C24C2×SD1614yes—
276C24C22×D411yescharacteristic
277#41—84no—
278#41—84no—
279#41—84no—
280#41—84no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—28no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—416no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—82no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4#10218yes—
310C2×D4#10218yes—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—416no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—416no—
336C22×C4—44no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4—816no—
350C22×C4—816no—
351C22×C4C2×C22⋊C414yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4#12012yes—
357C22×C4#12012yes—
358C22×C4#12312yes—
359C22×C4#12312yes—
360C22×C4C2×D812yes—
361C22×C4C2×SD1612yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—82no—
384C4⋊C4—88no—
385C22⋊C4—28no—
386C22⋊C4—28no—
387C22⋊C4—416no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—416no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—88no—
395C42—44no—
396C42—48no—
397C42C4×D412yes—
398C42C4×D412yes—
399C42C4×D412yes—
400C42C4×D412yes—
401C42#12112yes—
402C42#12112yes—
403C42#12112yes—
404C42#12112yes—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—28no—
410C23—24no—
411C23—28no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—48no—
416C23—48no—
417C23—44no—
418C23—44no—
419C23—48no—
420C23—44no—
421C23—48no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23C2×#10218yes—
429C23C22×C22⋊C411yescharacteristic
430C23C2×#12011yescharacteristic
431C23C2×#12311yescharacteristic
432C23C22×D811yescharacteristic
433C23C22×SD1611yescharacteristic
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23C2×#13711yescharacteristic
436Q8—84no—
437D4—216no—
438D4—216no—
439D4—82no—
440D4—82no—
441D4—84no—
442D4—82no—
443D4—82no—
444D4—82no—
445D4—82no—
446D4—84no—
447D4—1616no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—832no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—88no—
489C2×C4#38612yes—
490C2×C4#38612yes—
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4C2×#10212yes—
494C2×C4C2×#10212yes—
495C2×C4#41712yes—
496C2×C4#41712yes—
497C2×C4#44714yes—
498C2×C4#44814yes—
499C2×C4#48014yes—
500C2×C4#48114yes—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C8—82no—
506C8—82no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—28no—
516C22—216no—
517C22—48no—
518C22—48no—
519C22—48no—
520C22—48no—
521C22—48no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—84no—
526C22—84no—
527C22—82no—
528C22—82no—
529C22C2×#38611yescharacteristic
530C22#288312yes—
531C22C22×#10211yescharacteristic
532C22C2×#41711yescharacteristic
533C22#290112yes—
534C22C2×#44711yescharacteristic
535C22C2×#44811yescharacteristic
536C22C2×#48011yescharacteristic
537C22C2×#48111yescharacteristic
538C4—22no—
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—44no—
545C4—44no—
546C4—82no—
547C4—84no—
548C4—82no—
549C4—84no—
550C4#187414yes—
551C4#187414yes—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—48no—
557C2—162no—
558C2#1761312yes—
559C2#1763012yes—
560C2C2×#187411yescharacteristic
561C2#2098011yescharacteristic
562C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2349C2semidirect
3. #21786C2semidirect
4. C2×#1517C2semidirect
5. C2×#1502C2semidirect
6. #9155C2semidirect
7. #9147C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
18. #2547C4semidirect
20. #2541C4semidirect
21. #2537C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
56. C2×#127C2×C4semidirect
61. C2×#121C2×C4semidirect
74. C2×C4×D4C2×C4semidirect
106. #384C2×C4semidirect
109. #382D4semidirect
114. #380C2×C4semidirect
118. C4×C4⋊C4C2×C4semidirect
119. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#23492
#217862
C2×#15172
C2×#15022
#91552
#91472
#80262
#80142