Finite Group Databasebeta

← #366385

Subgroups — #366385

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1824C211yesmaximal, characteristic
2C2×#1753C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#14129C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#381—22no—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1824C418yescomplemented
24#1753—24no—
25#1736—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1445—22no—
37#1439—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—24no—
47C22×M16—42no—
48C22×M16D414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2×C414yes—
79C2×#97—22no—
80C2×#97—24no—
81C2×C4×C8—22no—
82C2×C4×C8—42no—
83#389—24no—
84#389—42no—
85#381—48no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380C2×C414yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340C2×C418yes—
109#340C2×C418yes—
110#336—22no—
111#336—22no—
112C25C2×D411yescharacteristic
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C22×C412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136C2×M16—28no—
137C2×M16—28no—
138C2×M16—44no—
139C2×M16—48no—
140C2×M16—48no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C22×C412yes—
157C22×C8C22×C412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8#4114yes—
161#117—28no—
162#117—48no—
163#117—44no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42—42no—
217C22⋊C8—28no—
218#97—44no—
219C4×C8—44no—
220#95—24no—
221#95—24no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—44no—
229#95—82no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24C4×D412yes—
237C24#12112yes—
238C24#12412yes—
239C24C22×D411yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—416no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—88no—
302C22×C4#11714yes—
303C22×C4C4×D414yes—
304C22×C4C4×D412yes—
305C22×C4#12112yes—
306C22×C4#12314yes—
307C22×C4#12412yes—
308C22×C4#12614yes—
309C22×C4AGL(1,ℤ/8)14yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)12yes—
319C22×C42−1+412yes—
320M16—416no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8#12314yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—88no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—84no—
362C22⋊C4—84no—
363C42—48no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—42no—
369C42—42no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23#38814yes—
391C23C2×C4×D411yescharacteristic
392C23C2×#12111yescharacteristic
393C23C2×#12411yescharacteristic
394C23#54612yes—
395C23#54712yes—
396C23C2×AGL(1,ℤ/8)12yes—
397C23#57512yes—
398C23#57712yes—
399C23#57812yes—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—816no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#38814yes—
454C2×C4C2×#11711yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C4×#4112yes—
457C2×C4#52012yes—
458C2×C4C2×#12311yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C2×C4#52511yescharacteristic
461C2×C4C2×#12611yescharacteristic
462C2×C4#52912yes—
463C2×C4#53411yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#53912yes—
466C2×C4#55112yes—
467C2×C4#55112yes—
468C2×C4#55112yes—
469C2×C4#55212yes—
470C2×C4#55212yes—
471C2×C4#55212yes—
472C8—24no—
473C8—42no—
474C8—42no—
475C8—42no—
476C22—24no—
477C22—24no—
478C22—24no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22C2×#38811yescharacteristic
486C22#235912yes—
487C22#248611yescharacteristic
488C22#248812yes—
489C22#253411yescharacteristic
490C22C4×AGL(1,ℤ/8)12yes—
491C22#294911yescharacteristic
492C22#295011yescharacteristic
493C22#305512yes—
494C22#306312yes—
495C22#313712yes—
496C22#313811yescharacteristic
497C22#314011yescharacteristic
498C22#330612yes—
499C22#330812yes—
500C22#331512yes—
501C22#331712yes—
502C22#335112yes—
503C22#335212yes—
504C22#335312yes—
505C22#335412yes—
506C4—24no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—84no—
511C4—84no—
512C4—162no—
513C4C4×#12311yescharacteristic
514C4#232211yescharacteristic
515C4#234312yes—
516C4#248311yescharacteristic
517C4#248411yescharacteristic
518C4#249312yes—
519C2—44no—
520C2#1399211yescharacteristic
521C2#2108811yescharacteristic
522C2#2109511yescharacteristic
523C2#2292512yes—
524C2#2292912yes—
525C2#2767012yes—
526C2#2771912yes—
527C2#2772312yes—
528C2#2810112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11316C2semidirect
7. #11292C2semidirect
8. #10880C2semidirect
9. #10821C2semidirect
23. #1824C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18242
C2×#17532
C2×#17362
#141292
#139042
#113162
#112922
#108802
#108212
#107312
#107222