Finite Group Databasebeta

← #409383

Subgroups — #409383

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1823C212yesmaximal
2#14686C211yesmaximal, characteristic
3C2×C4⋊#95C211yesmaximal, characteristic
4#11357C212yesmaximal, complemented
5#11315C214yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#10730C212yesmaximal
8#10712C211yesmaximal, characteristic
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#380—21no—
16C2×#380—21no—
17C2×#380C2212yes—
18C22×#95—21no—
19C22×#95—21no—
20#1823—28no—
21C2×#340C2214yes—
22C2×#340C2214yes—
23C2×#340C2214yes—
24C2×#340C2214yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C4⋊#95—21no—
28C4⋊#95—21no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C24×C4—21no—
35C24×C4—21no—
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8C2314yes—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C42—24no—
59C22×C42—24no—
60C22×C42—42no—
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68#384—24no—
69#384—42no—
70#380—24no—
71#380—24no—
72#380—44no—
73#380—44no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—44no—
82C2×#95—44no—
83#340—28no—
84#340—28no—
85#340—28no—
86#340—44no—
87#336—24no—
88#336—42no—
89C25C2×D411yescharacteristic
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4D814yes—
125C23×C4SD1614yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4112yes—
133C23×C4#4114yes—
134C23×C4#4114yes—
135C23×C4C2411yescharacteristic
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C2×C4⋊C4—216no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—416no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4#4114yes—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—416no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42#4114yes—
196C22⋊C8—48no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24#12112yes—
213C24#12312yes—
214C24#12412yes—
215C24C2×D812yes—
216C24C2×SD1612yes—
217C24AGL(1,ℤ/8)12yes—
218C24#13712yes—
219C24C22×D411yescharacteristic
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—42no—
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—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4#12112yes—
309C22×C4#12318yes—
310C22×C4#12312yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4#12412yes—
314C22×C4#12618yes—
315C22×C4C2×D812yes—
316C22×C4C2×D812yes—
317C22×C4C2×SD1612yes—
318C22×C4C2×SD1612yes—
319C22×C4#13712yes—
320C22×C4#13712yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4C2×#4112yes—
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+412yes—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—48no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—88no—
351C42—28no—
352C42—44no—
353C42—44no—
354C42—44no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23#39714yes—
384C23#45914yes—
385C23#46114yes—
386C23#48014yes—
387C23#48114yes—
388C23#48614yes—
389C23#48914yes—
390C23C2×#12111yescharacteristic
391C23C2×#12311yescharacteristic
392C23C2×#12311yescharacteristic
393C23#52511yescharacteristic
394C23C2×#12411yescharacteristic
395C23#53411yescharacteristic
396C23#53611yescharacteristic
397C23C22×D811yescharacteristic
398C23C22×SD1611yescharacteristic
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×#13711yescharacteristic
401C23C2×#13711yescharacteristic
402C23C2×#13711yescharacteristic
403C23#57511yescharacteristic
404C23#57511yescharacteristic
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—216no—
424C2×C4—216no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—416no—
432C2×C4—416no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—816no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—816no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#39714yes—
472C2×C4#48014yes—
473C2×C4#48314yes—
474C2×C4C2×#12312yes—
475C2×C4C2×#12612yes—
476C2×C4#52912yes—
477C2×C4#53212yes—
478C2×C4#53712yes—
479C2×C4#53912yes—
480C2×C4#55114yes—
481C2×C4#55114yes—
482C2×C4#55214yes—
483C2×C4#55214yes—
484C8—84no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22#207118yes—
500C22C2×#39711yescharacteristic
501C22#248311yescharacteristic
502C22#248411yescharacteristic
503C22#250311yescharacteristic
504C22#256111yescharacteristic
505C22#261811yescharacteristic
506C22#285411yescharacteristic
507C22C2×#45911yescharacteristic
508C22C2×#46111yescharacteristic
509C22#303711yescharacteristic
510C22#303711yescharacteristic
511C22C2×#48011yescharacteristic
512C22C2×#48011yescharacteristic
513C22C2×#48311yescharacteristic
514C22C2×#48111yescharacteristic
515C22#309411yescharacteristic
516C22#309511yescharacteristic
517C22#309511yescharacteristic
518C22#309511yescharacteristic
519C22#310011yescharacteristic
520C22#310111yescharacteristic
521C22#311211yescharacteristic
522C22#311311yescharacteristic
523C22C2×#48611yescharacteristic
524C22C2×#48911yescharacteristic
525C22#313911yescharacteristic
526C22#314011yescharacteristic
527C22#318611yescharacteristic
528C22#318711yescharacteristic
529C22#319311yescharacteristic
530C22#319511yescharacteristic
531C22#322611yescharacteristic
532C22#322911yescharacteristic
533C22#323411yescharacteristic
534C22#323611yescharacteristic
535C4—24no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—84no—
541C4—84no—
542C4—162no—
543C4—162no—
544C4#249314yes—
545C4#249814yes—
546C2—28no—
547C2#1501911yescharacteristic
548C2C2×#207111yescharacteristic
549C2#2362911yescharacteristic
550C2#2760011yescharacteristic
551C2#2760111yescharacteristic
552C2#2761011yescharacteristic
553C2#2761311yescharacteristic
554C2#2796011yescharacteristic
555C2#2796911yescharacteristic
556C2#2910311yescharacteristic
557C2#2911111yescharacteristic
558C2#2975411yescharacteristic
559C2#2975611yescharacteristic
560C2#3310611yescharacteristic
561C2#3310811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11357C2semidirect
5. #11315C2semidirect
6. #11267C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18232
#146862
C2×C4⋊#952
#113572
#113152
#112672
#107302
#107122