Finite Group Databasebeta

← #413551

Subgroups — #413551

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal
2C2×#1939C211yesmaximal, characteristic
3C2×C4×#106C211yesmaximal, characteristic
4#14150C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11231C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10842C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2088—28no—
25#1939—24no—
26C4×#106—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1444—21no—
37#1444—21no—
38#1442—21no—
39#1442—21no—
40#1436—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106D414yes—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8D414yes—
76#389—24no—
77#389—24no—
78#389—42no—
79#389—42no—
80#382—22no—
81#382—22no—
82#382Q814yes—
83#382Q814yes—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D412yes—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#336—24no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C22×C8#4112yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×Q814yes—
187C2×C4⋊C4C2×Q814yes—
188C2×C4⋊C4#4112yes—
189C2×C4⋊C4#4112yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42C2×D412yes—
201C2×C42C2×Q812yes—
202C2×C42C2×Q812yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206#106—28no—
207#106—28no—
208C4⋊C8—48no—
209C4⋊C8—44no—
210C4×C8—28no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—44no—
222#95—82no—
223#95—82no—
224C24C22×D411yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
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#12112yes—
277C22×C4#12112yes—
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#13514yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4H(5,𝔽2)11yescharacteristic
294C22×C42−1+411yescharacteristic
295C22×C42−1+411yescharacteristic
296C22×C42−1+411yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8#12114yes—
318C4⋊C4—28no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—84no—
351C42—84no—
352C42#12314yes—
353C42#12514yes—
354C42#12514yes—
355C42C4⋊Q814yes—
356C23C2×#12111yescharacteristic
357C23C2×C22⋊Q811yescharacteristic
358C23C2×C22⋊Q811yescharacteristic
359C23D4211yescharacteristic
360C23#54711yescharacteristic
361C23#54811yescharacteristic
362C23#54811yescharacteristic
363C23C2×#13512yes—
364C23#57512yes—
365C23#57712yes—
366C23#57712yes—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4#39814yes—
424C2×C4#39814yes—
425C2×C4C2×#12111yescharacteristic
426C2×C4C2×#12311yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4C2×#12511yescharacteristic
429C2×C4C2×#12511yescharacteristic
430C2×C4#52911yescharacteristic
431C2×C4#52911yescharacteristic
432C2×C4C2×C4⋊Q811yescharacteristic
433C2×C4#53211yescharacteristic
434C2×C4#53211yescharacteristic
435C2×C4#53611yescharacteristic
436C2×C4#53711yescharacteristic
437C2×C4#54111yescharacteristic
438C2×C4#54111yescharacteristic
439C2×C4#54211yescharacteristic
440C2×C4#54211yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4D4×Q811yescharacteristic
443C2×C4D4×Q811yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4#55411yescharacteristic
447C2×C4#55411yescharacteristic
448C2×C4#55511yescharacteristic
449C2×C4#55711yescharacteristic
450C2×C4#55711yescharacteristic
451C2×C4#55711yescharacteristic
452C2×C4#55711yescharacteristic
453C8—24no—
454C8—42no—
455C8—82no—
456C22C2×#39811yescharacteristic
457C22#245211yescharacteristic
458C22#245211yescharacteristic
459C22#245311yescharacteristic
460C22#245311yescharacteristic
461C22#253611yescharacteristic
462C22#253711yescharacteristic
463C22#305011yescharacteristic
464C22#305411yescharacteristic
465C22#305711yescharacteristic
466C22#305711yescharacteristic
467C22#307612yes—
468C22#307612yes—
469C22#308412yes—
470C22#308412yes—
471C22#327512yes—
472C22#328212yes—
473C22#333712yes—
474C22#335112yes—
475C22#336012yes—
476C22#336512yes—
477C22#337112yes—
478C22#337612yes—
479C4—24no—
480C4—24no—
481C4—44no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—84no—
488C4—84no—
489C4#250611yescharacteristic
490C4#250611yescharacteristic
491C4#250811yescharacteristic
492C4#250811yescharacteristic
493C4#254611yescharacteristic
494C4#258411yescharacteristic
495C4#262511yescharacteristic
496C4#285411yescharacteristic
497C2#1514911yescharacteristic
498C2#2366611yescharacteristic
499C2#2366811yescharacteristic
500C2#2646912yes—
501C2#2647012yes—
502C2#2651412yes—
503C2#2651712yes—
504C2#2818911yescharacteristic
505C2#2819211yescharacteristic
506C2#2822411yescharacteristic
507C2#2822911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#20882
C2×#19392
C2×C4×#1062
#141502
#112932
#112922
#112312
#108502
#108422