Finite Group Databasebeta

← #470079

Subgroups — #470079

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C212yesmaximal
2#15150C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4#11323C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11231C212yesmaximal, complemented
8#11199C212yesmaximal
9#11042C211yesmaximal, characteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#106C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C2×#397—21no—
16C2×#397—21no—
17C2×#384C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21#1853—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1445—21no—
31#1445—21no—
32#1445—21no—
33#1445—21no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C22⋊#95—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4D414yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—41no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74#397—44no—
75#397—44no—
76#384—41no—
77#384—41no—
78#380—24no—
79#380—24no—
80#380—41no—
81#380—41no—
82C4×C22⋊C4—41no—
83C4×C22⋊C4—41no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—41no—
104C25C2×D411yescharacteristic
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8#4114yes—
143C22×C8#4114yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4SD1614yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201#106—48no—
202#106—48no—
203C4⋊C8—44no—
204C22⋊C8—216no—
205#95—41no—
206#95—41no—
207#95—42no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—42no—
212#95—44no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24#12112yes—
226C24C2×SD1612yes—
227C24AGL(1,ℤ/8)12yes—
228C24C22×D411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—48no—
270C22×C4—42no—
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—42no—
284C22×C4—42no—
285C22×C4—42no—
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—88no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—88no—
301C22×C4#12112yes—
302C22×C4C2×SD1612yes—
303C22×C4C2×SD1612yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C42−1+412yes—
314C22×C42−1+412yes—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—84no—
351C22⋊C4—88no—
352C42—42no—
353C42—42no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23#46014yes—
381C23C2×#12111yescharacteristic
382C23#53411yescharacteristic
383C23#53511yescharacteristic
384C23#54712yes—
385C23#54712yes—
386C23C22×SD1611yescharacteristic
387C23C2×AGL(1,ℤ/8)11yescharacteristic
388C23C2×#13711yescharacteristic
389C23#57511yescharacteristic
390C23#57712yes—
391C23#57712yes—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—816no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
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#52512yes—
453C2×C4#53411yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53612yes—
456C2×C4#53811yescharacteristic
457C2×C4#53811yescharacteristic
458C2×C4#53912yes—
459C2×C4#53912yes—
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4#55111yescharacteristic
463C2×C4#55212yes—
464C2×C4#55212yes—
465C2×C4#55312yes—
466C2×C4#55911yescharacteristic
467C2×C4#56311yescharacteristic
468C2×C4#56312yes—
469C2×C4#56311yescharacteristic
470C8—44no—
471C8—44no—
472C22—28no—
473C22—24no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22#241312yes—
484C22#261711yescharacteristic
485C22#271711yescharacteristic
486C22#271812yes—
487C22#272411yescharacteristic
488C22C2×#46011yescharacteristic
489C22#303711yescharacteristic
490C22#304212yes—
491C22#318711yescharacteristic
492C22#319311yescharacteristic
493C22#320011yescharacteristic
494C22#320011yescharacteristic
495C22#320611yescharacteristic
496C22#321311yescharacteristic
497C22#322212yes—
498C22#333512yes—
499C22#333512yes—
500C22#334412yes—
501C22#334512yes—
502C22#334512yes—
503C22#335012yes—
504C22#335112yes—
505C22#335612yes—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—84no—
512C4—162no—
513C4—162no—
514C4—162no—
515C4#260011yescharacteristic
516C4#263812yes—
517C4#276411yescharacteristic
518C4#276512yes—
519C4#277111yescharacteristic
520C4#281711yescharacteristic
521C2—44no—
522C2#1667511yescharacteristic
523C2#2457812yes—
524C2#2459012yes—
525C2#2971311yescharacteristic
526C2#2972311yescharacteristic
527C2#3226011yescharacteristic
528C2#3226911yescharacteristic
529C2#3228012yes—
530C2#3229812yes—
531C2#3245911yescharacteristic
532C2#3246811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
4. #11323C2semidirect
5. #11316C2semidirect
6. #11257C2semidirect
7. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
#151502
#139242
#113232
#113162
#112572
#112312
#111992
#110422