Finite Group Databasebeta

← #405048

Subgroups — #405048

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4C2×#1927C212yesmaximal
5#15240C211yesmaximal, characteristic
6#13812C211yesmaximal, characteristic
7#11358C212yesmaximal
8#11357C211yesmaximal, characteristic
9#11322C211yesmaximal, characteristic
10#11077C211yesmaximal, characteristic
11#11077C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—28no—
26#1948—24no—
27#1928—24no—
28#1927—28no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1445—21no—
37#1445—21no—
38#1445—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—22no—
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42—22no—
57C22×C42—21no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—24no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—24no—
70C2×#103—24no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—22no—
78#391—24no—
79#391—22no—
80#391—24no—
81#391—22no—
82#384—22no—
83#384—22no—
84#382—24no—
85#382—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D418yes—
89C4×C4⋊C4D414yes—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—22no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8#4112yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8#4114yes—
149C22×C8#4112yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—28no—
188C2×C42—24no—
189C2×C42—82no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C2×D414yes—
193C2×C42C2×D412yes—
194C2×C42#4114yes—
195C2×C42#4112yes—
196#107—44no—
197#106—48no—
198C4⋊C8—28no—
199C4⋊C8—216no—
200#103—216no—
201#103—28no—
202#103—44no—
203#95—28no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—44no—
214#95—42no—
215#95—42no—
216#95—82no—
217#95—81no—
218C24C22×D411yescharacteristic
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—24no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
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—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4#12114yes—
290C22×C4#12112yes—
291C22×C4#12114yes—
292C22×C4#12112yes—
293C22×C4#13512yes—
294C22×C4#13712yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4112yes—
302C22×C42−1+412yes—
303C22×C42−1+411yescharacteristic
304C22×C42−1+411yescharacteristic
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4#12714yes—
342C42—22no—
343C42—24no—
344C42—22no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—84no—
350C42—88no—
351C23C2×#12111yescharacteristic
352C23C2×#12112yes—
353C23#54712yes—
354C23#54711yescharacteristic
355C23#54811yescharacteristic
356C23C2×#13511yescharacteristic
357C23C2×#13711yescharacteristic
358C23#57512yes—
359C23#57712yes—
360C23#57711yescharacteristic
361C23#57811yescharacteristic
362Q8—42no—
363Q8—44no—
364Q8—42no—
365Q8—81no—
366Q8—82no—
367Q8—81no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4#46414yes—
426C2×C4#52511yescharacteristic
427C2×C4#52512yes—
428C2×C4#52511yescharacteristic
429C2×C4#52512yes—
430C2×C4C2×#12711yescharacteristic
431C2×C4#53212yes—
432C2×C4#53211yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#53612yes—
435C2×C4#53611yescharacteristic
436C2×C4#53612yes—
437C2×C4#55212yes—
438C2×C4#55212yes—
439C2×C4#55212yes—
440C2×C4#55211yescharacteristic
441C2×C4#55212yes—
442C2×C4#55611yescharacteristic
443C2×C4#55612yes—
444C8—44no—
445C8—42no—
446C8—81no—
447C22#241311yescharacteristic
448C22#241312yes—
449C22#241511yescharacteristic
450C22#241512yes—
451C22#285611yescharacteristic
452C22C2×#46411yescharacteristic
453C22#303512yes—
454C22#303712yes—
455C22#304011yescharacteristic
456C22#304211yescharacteristic
457C22#304212yes—
458C22#304612yes—
459C22#304611yescharacteristic
460C22#333712yes—
461C22#334211yescharacteristic
462C22#334412yes—
463C22#334912yes—
464C22#335012yes—
465C22#335112yes—
466C22#335611yescharacteristic
467C22#336511yescharacteristic
468C22#337112yes—
469C22#337411yescharacteristic
470C4—24no—
471C4—44no—
472C4—44no—
473C4—42no—
474C4—44no—
475C4—82no—
476C4—84no—
477C4—82no—
478C4—162no—
479C4—161no—
480C4#239711yescharacteristic
481C4#260411yescharacteristic
482C4#260412yes—
483C4#261111yescharacteristic
484C4#261112yes—
485C4#265811yescharacteristic
486C2#1491011yescharacteristic
487C2#2457911yescharacteristic
488C2#2458412yes—
489C2#2459012yes—
490C2#2459311yescharacteristic
491C2#2468311yescharacteristic
492C2#2468612yes—
493C2#2469112yes—
494C2#2469711yescharacteristic
495C2#3316511yescharacteristic
496C2#3316811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#19482
C2×#19282
C2×#19272
#152402
#138122
#113582
#113572
#113222
#110772