Finite Group Databasebeta

← #195639

Subgroups — #195639

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#21395C211yesmaximal, characteristic
4#21395C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic, complemented
6#9065C214yesmaximal, complemented
7#9062C214yesmaximal, complemented
8#7904C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8—21no—
13#3141—24no—
14#3141C418yescomplemented
15C2×#489—22no—
16C2×#487C2212yes—
17C2×#486C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#417—22no—
21C2×C4×M16C2211yescharacteristic
22#2365—22no—
23#2365—22no—
24#2365—22no—
25#2365C2214yes—
26C2×#391—21no—
27C2×#389—21no—
28C2×#387—22no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1987C2212yes—
33#1987C2212yes—
34#1982C2212yes—
35#1982C2212yes—
36#1874—21no—
37#1874—21no—
38#1874—21no—
39#1874—21no—
40#1762—22no—
41#1762—22no—
42#1556—24no—
43C2×#332C2214yes—
44#1279C2214yes—
45#1279C2214yes—
46C22×#41—21no—
47C23×Q8—21no—
48C22×M16—21no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532C2×C414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#127C2311yescharacteristic
63C2×#124—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—48no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#489—216no—
83#487—24no—
84#487C2×C418yescomplemented
85#486—24no—
86#486C2×C418yescomplemented
87#432—22no—
88#432—22no—
89#432—22no—
90#432—22no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#417—216no—
94C2×#103—24no—
95C2×#102—24no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C4×M16—22no—
103C4×M16C2×C414yes—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8C2312yes—
108#391—24no—
109#391—24no—
110#389—24no—
111#389—24no—
112#387—28no—
113#387—28no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4C2312yes—
123C4×C4⋊C4C2312yes—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#332—28no—
133C2×#41—24no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—42no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—42no—
159C2×M16—42no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C22⋊C418yes—
174C4⋊Q8C22×C412yes—
175C4⋊Q8C22×C414yes—
176C4⋊Q8C2×D414yes—
177#127—21no—
178#127—21no—
179#127—21no—
180#127C22×C412yes—
181#124—24no—
182#124—42no—
183#121—24no—
184#121—24no—
185#121—42no—
186#121—42no—
187C4×Q8—416no—
188C4×Q8—48no—
189C4×Q8—48no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—42no—
193C4×D4—42no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4C22⋊C418yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—48no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240#103—216no—
241#102—216no—
242#102—44no—
243#102—44no—
244#102—44no—
245#102—44no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248#97—24no—
249#97—42no—
250#97—42no—
251C4×C8—22no—
252C4×C8C22×C414yes—
253#95—28no—
254#95—28no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24—41no—
260C24#12011yescharacteristic
261#41—416no—
262#41—88no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—28no—
266C2×Q8—22no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—416no—
270C2×Q8—416no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4—82no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12312yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—48no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—416no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4C2×C22⋊C418yes—
364C4⋊C4AGL(1,ℤ/8)14yes—
365C4⋊C4#13714yes—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C42—44no—
369C42—44no—
370C42—416no—
371C42—44no—
372C42—44no—
373C42C2×C22⋊C412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42#12112yes—
377C42#12112yes—
378C42#12312yes—
379C42C23×C411yescharacteristic
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—42no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23#38612yes—
395C23#45412yes—
396C23#45412yes—
397C23C2×#12011yescharacteristic
398Q8—416no—
399Q8—44no—
400Q8—48no—
401Q8—48no—
402Q8—82no—
403D4—44no—
404D4—44no—
405D4—44no—
406D4—82no—
407D4—88no—
408D4—82no—
409D4—82no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—416no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—88no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#38612yes—
449C2×C4#38612yes—
450C2×C4#38612yes—
451C2×C4C22×C22⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)12yes—
458C2×C4C2×#13712yes—
459C2×C4#57512yes—
460C2×C4#57512yes—
461C8—44no—
462C8—82no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22#188814yes—
478C22C2×#38611yescharacteristic
479C22#300311yescharacteristic
480C22#300411yescharacteristic
481C22#300511yescharacteristic
482C22#300511yescharacteristic
483C22C2×#45411yescharacteristic
484C22C2×#45411yescharacteristic
485C4—22no—
486C4—22no—
487C4—21no—
488C4—22no—
489C4—21no—
490C4—44no—
491C4—44no—
492C4—48no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4C4×AGL(1,ℤ/8)12yes—
497C4C4×#13712yes—
498C4#303612yes—
499C4#303712yes—
500C2—24no—
501C2—162no—
502C2#1853311yescharacteristic
503C2#1853411yescharacteristic
504C2#1853511yescharacteristic
505C2#1853611yescharacteristic
506C2#2092511yescharacteristic
507C2#2092611yescharacteristic
508C2C2×#188811yescharacteristic
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×#2365C2semidirect
5. C2×#1556C2semidirect
6. #9065C2semidirect
7. #9062C2semidirect
8. #7904C2semidirect
9. #7904C2semidirect
14. #3141C4semidirect
84. #487C2×C4semidirect
86. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
C2×#23652
#213952
C2×#15562
#90652
#90622
#79042