Finite Group Databasebeta

← #475510

Subgroups — #475510

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14686C211yesmaximal, characteristic
2#14686C211yesmaximal, characteristic
3#11371C212yesmaximal, complemented
4#11357C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11200C214yesmaximal
8#10712C211yesmaximal, characteristic
9C23×C42C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11C2×#384C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#380—21no—
16C2×#380—21no—
17C2×#380—21no—
18C2×#380—21no—
19C2×#380C2212yes—
20C2×#380C2212yes—
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—
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8C2314yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C42—24no—
52C22×C42—24no—
53C22×C42—42no—
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C22⋊C8—24no—
59C2×C22⋊C8—42no—
60C2×C22⋊C8—42no—
61#384—24no—
62#384—24no—
63#384—42no—
64#384—42no—
65#380—28no—
66#380—44no—
67#380—44no—
68#380—44no—
69#380—44no—
70#380—44no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77#340—44no—
78#340—44no—
79#340—44no—
80#340—44no—
81#336—24no—
82#336—24no—
83C25C2×D411yescharacteristic
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4SD1614yes—
108C23×C4Q1614yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4114yes—
115C23×C4C2411yescharacteristic
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—416no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42#4114yes—
180C2×C42#4114yes—
181C22⋊C8—48no—
182#95—44no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24#12412yes—
201C24C2×SD1612yes—
202C24C2×Q1612yes—
203C24C22×D411yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
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#12312yes—
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#12312yes—
282C22×C4#12412yes—
283C22×C4C2×SD1612yes—
284C22×C4C2×SD1612yes—
285C22×C4C2×Q1612yes—
286C22×C4C2×Q1612yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4#13712yes—
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4112yes—
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+411yescharacteristic
298C22×C42−1+412yes—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C42—28no—
324C42—28no—
325C42—44no—
326C42—44no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23#48714yes—
362C23C2×#12311yescharacteristic
363C23C2×#12311yescharacteristic
364C23C2×#12411yescharacteristic
365C23#53411yescharacteristic
366C23#53511yescharacteristic
367C23#53611yescharacteristic
368C23#53711yescharacteristic
369C23C22×SD1611yescharacteristic
370C23C22×Q1611yescharacteristic
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373C23C2×AGL(1,ℤ/8)11yescharacteristic
374C23C2×#13711yescharacteristic
375C23#57511yescharacteristic
376C23#57511yescharacteristic
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—416no—
405C2×C4—416no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
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#48114yes—
441C2×C4#48314yes—
442C2×C4#48414yes—
443C2×C4#48414yes—
444C2×C4#53812yes—
445C2×C4#53912yes—
446C2×C4#54012yes—
447C2×C4#54112yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4#55114yes—
451C2×C4#55214yes—
452C2×C4#55314yes—
453C2×C4#56214yes—
454C8—84no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22#248311yescharacteristic
471C22#250311yescharacteristic
472C22#250411yescharacteristic
473C22#262411yescharacteristic
474C22#262511yescharacteristic
475C22#263911yescharacteristic
476C22#281611yescharacteristic
477C22C2×#48311yescharacteristic
478C22C2×#48111yescharacteristic
479C22C2×#48411yescharacteristic
480C22C2×#48411yescharacteristic
481C22#309411yescharacteristic
482C22#309411yescharacteristic
483C22#309411yescharacteristic
484C22#309511yescharacteristic
485C22C2×#48711yescharacteristic
486C22#313511yescharacteristic
487C22#313811yescharacteristic
488C22#314111yescharacteristic
489C22#318711yescharacteristic
490C22#318911yescharacteristic
491C22#319211yescharacteristic
492C22#319311yescharacteristic
493C22#320811yescharacteristic
494C22#320911yescharacteristic
495C22#321411yescharacteristic
496C22#321411yescharacteristic
497C22#322711yescharacteristic
498C22#323011yescharacteristic
499C22#323511yescharacteristic
500C22#323711yescharacteristic
501C22#324611yescharacteristic
502C22#324911yescharacteristic
503C22#325311yescharacteristic
504C22#325411yescharacteristic
505C4—24no—
506C4—24no—
507C4—24no—
508C4—24no—
509C4—42no—
510C4—162no—
511C4—162no—
512C4—162no—
513C4—162no—
514C4#274114yes—
515C4#274714yes—
516C2—44no—
517C2#1705811yescharacteristic
518C2#2760111yescharacteristic
519C2#2760311yescharacteristic
520C2#2796111yescharacteristic
521C2#2796811yescharacteristic
522C2#2797211yescharacteristic
523C2#2798111yescharacteristic
524C2#2992111yescharacteristic
525C2#2992311yescharacteristic
526C2#2993811yescharacteristic
527C2#2993911yescharacteristic
528C2#3018711yescharacteristic
529C2#3019711yescharacteristic
530C2#3277311yescharacteristic
531C2#3277411yescharacteristic
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
3. #11371C2semidirect
4. #11357C2semidirect
5. #11267C2semidirect
6. #11267C2semidirect

Maximal subgroups

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

subgroupindex
#146862
#113712
#113572
#112672
#112002
#107122