Finite Group Databasebeta

← #470498

Subgroups — #470498

Order 512.

966 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
1#15128C211yesmaximal, characteristic
2#13911C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11315C212yesmaximal
5#11292C212yesmaximal
6#11267C212yesmaximal
7#11227C212yesmaximal
8#11202C211yesmaximal, characteristic
9#10806C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C2×#395—21no—
13C2×#389C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382—21no—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2212yes—
29#1446—21no—
30#1444—21no—
31#1442—21no—
32#1442—21no—
33#1441—21no—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—41no—
61C2×C4⋊C8—42no—
62#395—44no—
63#389—24no—
64#389—42no—
65#384—41no—
66#384—41no—
67#382—44no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—41no—
70C4×C4⋊C4—41no—
71C4×C4⋊C4Q814yescomplemented
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
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—24no—
86#340—24no—
87#340—24no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#336—24no—
93#336—42no—
94#336—41no—
95C22×Q8—44no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—82no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C4⋊Q8—44no—
126C4⋊Q8—48no—
127C4⋊Q8—44no—
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—22no—
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—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4D814yes—
165C2×C4⋊C4C2×Q814yes—
166C2×C4⋊C4#4114yes—
167C2×C4⋊C4#4114yes—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185C4⋊C8—44no—
186#95—24no—
187#95—41no—
188#95—41no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—42no—
193#95—44no—
194#95—42no—
195#95—82no—
196#95—82no—
197#95—82no—
198C24C22×D411yescharacteristic
199C2×Q8—48no—
200C2×Q8—88no—
201C2×Q8—84no—
202C2×Q8—88no—
203C2×Q8—84no—
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—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
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—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22⋊Q812yes—
258C22×C4C22⋊Q812yes—
259C22×C4C2×D812yes—
260C22×C4C2×D812yes—
261C22×C4C2×D812yes—
262C22×C4#13712yes—
263C22×C4#13712yes—
264C22×C4C22×Q811yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4112yes—
267C22×C4C2×#4111yescharacteristic
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C4H(5,𝔽2)12yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+411yescharacteristic
272C22×C42−1+411yescharacteristic
273C22×C42−1+412yes—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—88no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—42no—
315C42—42no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—48no—
321C23C2×C22⋊Q811yescharacteristic
322C23#53411yescharacteristic
323C23#53611yescharacteristic
324C23#54612yes—
325C23#54812yes—
326C23C22×D811yescharacteristic
327C23C2×#13711yescharacteristic
328C23C2×#13711yescharacteristic
329C23#57511yescharacteristic
330C23#57712yes—
331C23#57712yes—
332Q8—816no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
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—82no—
391C2×C4—82no—
392C2×C4#47414yes—
393C2×C4#53312yes—
394C2×C4#53812yes—
395C2×C4#53912yes—
396C2×C4#53911yescharacteristic
397C2×C4#53911yescharacteristic
398C2×C4#54112yes—
399C2×C4#54212yes—
400C2×C4#54411yescharacteristic
401C2×C4#54412yes—
402C2×C4#54411yescharacteristic
403C2×C4#55111yescharacteristic
404C2×C4#55212yes—
405C2×C4#55511yescharacteristic
406C2×C4#55512yes—
407C2×C4#55612yes—
408C2×C4H(3,𝔽4)11yescharacteristic
409C2×C4#56312yes—
410C2×C4#56311yescharacteristic
411C8—82no—
412C8—82no—
413C22#243912yes—
414C22#262711yescharacteristic
415C22#268411yescharacteristic
416C22#269012yes—
417C22#273111yescharacteristic
418C22C2×#47411yescharacteristic
419C22#307811yescharacteristic
420C22#308412yes—
421C22#318611yescharacteristic
422C22#319511yescharacteristic
423C22#320011yescharacteristic
424C22#320011yescharacteristic
425C22#322611yescharacteristic
426C22#323611yescharacteristic
427C22#324412yes—
428C22#329412yes—
429C22#331112yes—
430C22#331612yes—
431C22#332912yes—
432C22#335812yes—
433C22#336912yes—
434C22#337112yes—
435C22#337212yes—
436C4—24no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—84no—
442C4—81no—
443C4—84no—
444C4—84no—
445C4—162no—
446C4#261211yescharacteristic
447C4#266612yes—
448C4#277512yes—
449C4#278511yescharacteristic
450C4#278911yescharacteristic
451C4#284411yescharacteristic
452C2#1668611yescharacteristic
453C2#2602012yes—
454C2#2603512yes—
455C2#2996611yescharacteristic
456C2#2997711yescharacteristic
457C2#3126011yescharacteristic
458C2#3127211yescharacteristic
459C2#3145412yes—
460C2#3148112yes—
461C2#3262811yescharacteristic
462C2#3264511yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
71. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#151282
#139112
#113842
#113152
#112922
#112672
#112272
#112022
#108062