Finite Group Databasebeta

← #428411

Subgroups — #428411

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2094C211yesmaximal, characteristic, complemented
3C2×#2033C212yesmaximal
4C2×#1853C212yesmaximal, complemented
5C2×#1847C211yesmaximal, characteristic, complemented
6#15150C211yesmaximal, characteristic, complemented
7#11372C211yesmaximal, characteristic, complemented
8#11325C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#10927C211yesmaximal, characteristic, complemented
11#10922C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21#2504—24no—
22#2504—24no—
23C2×#397—22no—
24C2×#397—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#386—22no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2094—24no—
34#2033—28no—
35#1853—28no—
36#1847—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1445—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—24no—
54C2×#121—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—42no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—24no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—42no—
81C2×#106—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8D418yes—
91C2×#97—24no—
92C2×#97—22no—
93#397—216no—
94#397—48no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390D414yes—
99#386—216no—
100#384—22no—
101#384—22no—
102#382—22no—
103#382—22no—
104#382—22no—
105#382D414yes—
106#380—24no—
107#380—24no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—24no—
117#340—24no—
118#340—24no—
119#336—24no—
120#336—22no—
121C25—21no—
122C25C2×D411yescharacteristic
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—42no—
134C22×D4—44no—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—84no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×D414yes—
168C22×C8C2×D414yes—
169C22×C8#4114yes—
170C22×C8#4114yes—
171#121—48no—
172#121—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4—84no—
216C2×C22⋊C4—84no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225#107—48no—
226#106—48no—
227#106—48no—
228#102—48no—
229#102—48no—
230C22⋊C8—216no—
231#97—28no—
232#95—28no—
233#95—28no—
234#95—24no—
235#95—44no—
236#95—48no—
237#95—44no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—24no—
244C24—22no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—44no—
252C24#12712yes—
253C24AGL(1,ℤ/8)14yes—
254C24C22×D411yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—48no—
260C2×D4—416no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—84no—
274C2×D4—88no—
275C2×D4—84no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—48no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4#12014yes—
333C22×C4#12712yes—
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+412yes—
341C22×C42−1+412yes—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8#12318yes—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—88no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C42—22no—
378C42—22no—
379C42—24no—
380C42—41no—
381C42—41no—
382C42—42no—
383C23—28no—
384C23—28no—
385C23—22no—
386C23—24no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—44no—
392C23—44no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23—81no—
408C23—81no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#49614yes—
415C23C2×#12011yescharacteristic
416C23C2×#12711yescharacteristic
417C23#53511yescharacteristic
418C23#54712yes—
419C23#54712yes—
420C23C2×AGL(1,ℤ/8)12yes—
421C23#57512yes—
422C23#57714yes—
423D4—44no—
424D4—44no—
425D4—816no—
426D4—82no—
427D4—82no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—816no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4C2×#12312yes—
472C2×C4#52511yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53612yes—
476C2×C4#53611yescharacteristic
477C2×C4#53912yes—
478C2×C4#54212yes—
479C2×C4D4212yes—
480C2×C4D4212yes—
481C2×C4#54712yes—
482C2×C4#54712yes—
483C2×C4#54712yes—
484C2×C4#54812yes—
485C2×C4#55312yes—
486C2×C4#56312yes—
487C8—28no—
488C8—44no—
489C22—28no—
490C22—24no—
491C22—42no—
492C22—44no—
493C22—48no—
494C22—44no—
495C22—44no—
496C22—42no—
497C22—82no—
498C22—84no—
499C22—84no—
500C22—82no—
501C22—82no—
502C22—162no—
503C22—164no—
504C22—162no—
505C22#240611yescharacteristic
506C22#243211yescharacteristic
507C22#243311yescharacteristic
508C22#250112yes—
509C22#271812yes—
510C22#300612yes—
511C22#300912yes—
512C22C2×#49611yescharacteristic
513C22#315311yescharacteristic
514C22#315412yes—
515C22#321312yes—
516C22#322212yes—
517C22#334514yes—
518C22#334514yes—
519C22#335014yes—
520C22#335114yes—
521C4—42no—
522C4—44no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—162no—
528C4—162no—
529C4#242212yes—
530C4#244312yes—
531C4#267412yes—
532C4#268012yes—
533C2—44no—
534C2—162no—
535C2#1538411yescharacteristic
536C2#2410112yes—
537C2#2571412yes—
538C2#2573111yescharacteristic
539C2#2573211yescharacteristic
540C2#2793412yes—
541C2#2794712yes—
542C2#3229814yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×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
2. C2×#2094C2semidirect
4. C2×#1853C2semidirect
5. C2×#1847C2semidirect
6. #15150C2semidirect
7. #11372C2semidirect
8. #11325C2semidirect
9. #11121C2semidirect
10. #10927C2semidirect
11. #10922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#20942
C2×#20332
C2×#18532
C2×#18472
#151502
#113722
#113252
#111212
#109272
#109222