Finite Group Databasebeta

← #410505

Subgroups — #410505

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1916C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic
3#14041C212yesmaximal
4#11328C211yesmaximal, characteristic
5#11315C212yesmaximal
6#11292C212yesmaximal
7#11256C211yesmaximal, characteristic
8#11232C211yesmaximal, characteristic
9#11232C211yesmaximal, characteristic
10#10820C211yesmaximal, characteristic
11#10807C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1916—24no—
20C4×#107—24no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1445—22no—
34#1442—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—24no—
59C2×#107—24no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65C2×C4×C8D414yes—
66#389—24no—
67#389—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—24no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D411yescharacteristic
83C2×#95D411yescharacteristic
84#340—24no—
85#340—22no—
86#340—22no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C2×D412yes—
124C22×C8C2×D412yes—
125C22×C8#4112yes—
126C22×C8#4112yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4#4114yes—
157C2×C4⋊C4#4114yes—
158C2×C4⋊C4#4114yes—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—44no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42C2×D412yes—
177C2×C42#4112yes—
178#107—28no—
179#107—28no—
180C4⋊C8—44no—
181C4×C8—28no—
182#95—28no—
183#95—22no—
184#95—22no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194C24C22×D411yescharacteristic
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4#12112yes—
253C22×C4#12112yes—
254C22×C4#13512yes—
255C22×C4#13512yes—
256C22×C4#13714yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4112yes—
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4112yes—
264C22×C42−1+411yescharacteristic
265C22×C42−1+412yes—
266C22×C42−1+411yescharacteristic
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8#12314yes—
287C4⋊C4—28no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—88no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—88no—
318C23C2×#12111yescharacteristic
319C23#52512yes—
320C23#54711yescharacteristic
321C23#54711yescharacteristic
322C23#54812yes—
323C23C2×#13511yescharacteristic
324C23C2×#13511yescharacteristic
325C23C2×#13712yes—
326C23#57611yescharacteristic
327C23#57612yes—
328C23#57811yescharacteristic
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#47014yes—
391C2×C4C2×#12311yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4#52511yescharacteristic
394C2×C4#52912yes—
395C2×C4#52912yes—
396C2×C4#52912yes—
397C2×C4#53211yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#54112yes—
401C2×C4#54112yes—
402C2×C4#55212yes—
403C2×C4#55212yes—
404C2×C4#55211yescharacteristic
405C2×C4#55212yes—
406C2×C4#55211yescharacteristic
407C2×C4#55612yes—
408C2×C4#55612yes—
409C8—24no—
410C8—42no—
411C8—81no—
412C8—81no—
413C22#239211yescharacteristic
414C22#246912yes—
415C22#247512yes—
416C22#248611yescharacteristic
417C22#253611yescharacteristic
418C22#305011yescharacteristic
419C22C2×#47011yescharacteristic
420C22#305811yescharacteristic
421C22#306111yescharacteristic
422C22#310712yes—
423C22#310812yes—
424C22#312212yes—
425C22#312412yes—
426C22#333611yescharacteristic
427C22#333911yescharacteristic
428C22#334111yescharacteristic
429C22#334211yescharacteristic
430C22#334612yes—
431C22#334612yes—
432C22#336112yes—
433C22#336112yes—
434C22#337412yes—
435C22#337512yes—
436C4—24no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—81no—
441C4—84no—
442C4—81no—
443C4—84no—
444C4—84no—
445C4—162no—
446C4#249911yescharacteristic
447C4#252412yes—
448C4#252812yes—
449C4#260411yescharacteristic
450C4#263411yescharacteristic
451C4#285311yescharacteristic
452C2#1508011yescharacteristic
453C2#2369911yescharacteristic
454C2#2370311yescharacteristic
455C2#2713312yes—
456C2#2714112yes—
457C2#2736112yes—
458C2#2736912yes—
459C2#2766011yescharacteristic
460C2#2766611yescharacteristic
461C2#2818811yescharacteristic
462C2#2819611yescharacteristic
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×#19162
C2×C4×#1072
#140412
#113282
#113152
#112922
#112562
#112322
#108202
#108072