Finite Group Databasebeta

← #194006

Subgroups — #194006

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#21395C211yesmaximal, characteristic
4#21200C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic, complemented
6#8979C214yesmaximal, complemented
7#8975C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3112—24no—
14#3112C418yescomplemented
15C2×#480—21no—
16C2×#477C2212yes—
17C2×#460C2212yes—
18C2×#448—21no—
19C2×#440—22no—
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#417—22no—
23C2×C4×M16C2211yescharacteristic
24Q8×C4⋊C4—22no—
25Q8×C4⋊C4—22no—
26Q8×C4⋊C4—22no—
27Q8×C4⋊C4C2214yes—
28C2×#389—22no—
29C2×#389—21no—
30C2×#384—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1987C2212yes—
35#1982C2212yes—
36#1959—22no—
37#1957C2212yes—
38#1956C2212yes—
39#1874—21no—
40#1874—21no—
41#1762—22no—
42#1491—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48C22×SD16—22no—
49C22×M16—21no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532C2×C414yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×#127C2311yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—48no—
66C2×C4×Q8C2×C414yes—
67C2×C4×Q8D414yes—
68C2×C4×Q8D412yes—
69C2×C4×Q8C2312yes—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#480—28no—
86#477—24no—
87#477C2×C418yescomplemented
88#460—24no—
89#460C2×C418yescomplemented
90#448—28no—
91#440—216no—
92#432—22no—
93#432—22no—
94#432—22no—
95#432C2×C414yes—
96C2×#107—22no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#417—48no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—22no—
105C2×#102—22no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8C2312yes—
114#389—24no—
115#389—28no—
116#389—24no—
117#389—28no—
118#384—24no—
119#384—24no—
120#384—21no—
121#384—21no—
122#384D412yes—
123#384C2312yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C2×#95—22no—
131C2×#95—21no—
132#340—22no—
133#340—22no—
134Q8⋊C8—28no—
135C2×#41—24no—
136C2×#41—44no—
137C2×#41—44no—
138C2×#41—42no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8C22⋊C414yes—
145C22×Q8C2×D412yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—41no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×SD16—28no—
160C2×SD16—44no—
161C2×SD16—44no—
162C2×M16—24no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22×C412yes—
172#127—21no—
173#127—21no—
174#127—21no—
175#127C22×C412yes—
176#124—24no—
177#124—42no—
178#121—24no—
179#121—24no—
180#121—42no—
181#121—42no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—416no—
190C4×Q8—48no—
191C4×Q8—48no—
192C4×Q8C22⋊C418yes—
193C4×Q8C22×C414yes—
194C4×Q8C2×D414yes—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—42no—
198C4×D4—42no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—84no—
223C2×C4⋊C4C22⋊C414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—48no—
238C2×C42—44no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—28no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8C22×C414yes—
252#103—28no—
253#103—48no—
254#102—28no—
255#102—44no—
256#102—48no—
257#102—44no—
258C22⋊C8—24no—
259C22⋊C8—42no—
260#97—24no—
261#97—42no—
262#97—42no—
263C4×C8—22no—
264C4×C8—22no—
265#95—28no—
266#95—22no—
267#95—22no—
268C24—41no—
269C24#12011yescharacteristic
270#41—416no—
271#41—88no—
272C2×Q8—24no—
273C2×Q8—216no—
274C2×Q8—24no—
275C2×Q8—216no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8C2×C22⋊C414yes—
285C2×Q8AGL(1,ℤ/8)14yes—
286C2×Q8#13714yes—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4—82no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—82no—
331C22×C4—82no—
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×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344SD16—416no—
345M16—48no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—416no—
370C4⋊C4—44no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—82no—
376C4⋊C4—88no—
377C4⋊C4C2×C22⋊C414yes—
378C4⋊C4#13714yes—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—24no—
382C42—24no—
383C42—28no—
384C42—416no—
385C42—44no—
386C42—44no—
387C42C2×C22⋊C412yes—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12112yes—
391C42#12112yes—
392C42#12312yes—
393C42C23×C411yescharacteristic
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—42no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23#38612yes—
409C23#45312yes—
410C23#45412yes—
411C23C2×#12011yescharacteristic
412Q8—216no—
413Q8—24no—
414Q8—24no—
415Q8—24no—
416Q8—44no—
417Q8—82no—
418Q8#41718yescomplemented
419D4—44no—
420D4—44no—
421D4—44no—
422D4—82no—
423D4—88no—
424D4—82no—
425D4—82no—
426C2×C4—216no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4#38612yes—
465C2×C4#38612yes—
466C2×C4#38612yes—
467C2×C4C22×C22⋊C411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C2×C4C2×#13712yes—
475C2×C4C2×#13712yes—
476C2×C4#57512yes—
477C8—44no—
478C8—44no—
479C8—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22#188314yes—
494C22C2×#38611yescharacteristic
495C22#300311yescharacteristic
496C22#300411yescharacteristic
497C22#300411yescharacteristic
498C22#300511yescharacteristic
499C22C2×#45311yescharacteristic
500C22C2×#45411yescharacteristic
501C4—28no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—22no—
506C4—21no—
507C4—44no—
508C4—44no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4C2×#41712yes—
513C4C4×#13712yes—
514C4#303712yes—
515C4#309212yes—
516C2—24no—
517C2—162no—
518C2#1853411yescharacteristic
519C2#1853511yescharacteristic
520C2#1854511yescharacteristic
521C2#1854611yescharacteristic
522C2#2090911yescharacteristic
523C2#2092611yescharacteristic
524C2C2×#188311yescharacteristic
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
2. C2×Q8×C4⋊C4C2semidirect
5. C2×#1491C2semidirect
6. #8979C2semidirect
7. #8975C2semidirect
8. #8040C2semidirect
9. #7904C2semidirect
14. #3112C4semidirect
87. #477C2×C4semidirect
89. #460C2×C4semidirect
418. Q8#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C2×Q8×C4⋊C42
#213952
#212002
C2×#14912
#89792
#89752
#80402
#79042