Finite Group Databasebeta

← #449429

Subgroups — #449429

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 456 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#11267C212yesmaximal
5#11256C212yesmaximal
6#11219C212yesmaximal
7#10907C212yesmaximal
8#10881C212yesmaximal
9#10875C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×#97C2211yescharacteristic
13C2×#395—21no—
14C2×#389C2212yes—
15C2×#384C2211yescharacteristic
16C2×#382—21no—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
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×#107—24no—
61C2×#107—24no—
62C2×#107—42no—
63C2×#97—41no—
64C2×#97—42no—
65#395—44no—
66#389—24no—
67#389—42no—
68#384—41no—
69#384—41no—
70#382—44no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—41no—
73C4×C4⋊C4—41no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95D411yescharacteristic
85C2×#95D411yescharacteristic
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—42no—
91#340—42no—
92#340—42no—
93#336—42no—
94#336—41no—
95#336Q818yescomplemented
96C22×Q8—44no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—82no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
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×C8C2×Q814yes—
125C22×C8C2×Q814yes—
126C4⋊Q8—44no—
127C4⋊Q8—48no—
128C4⋊Q8—44no—
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—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4#4114yes—
167C2×C4⋊C4#4114yes—
168C2×C4⋊C4#4114yes—
169C2×C42—22no—
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×Q814yes—
184#107—48no—
185#97—44no—
186#95—24no—
187#95—41no—
188#95—41no—
189#95—44no—
190#95—44no—
191#95—42no—
192#95—42no—
193#95—44no—
194#95—44no—
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—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
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×C4—42no—
258C22×C4C4⋊Q812yes—
259C22×C4C4⋊Q812yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263C22×C4#13712yes—
264C22×C4C22×Q812yes—
265C22×C4C2×#4112yes—
266C22×C4C2×#4112yes—
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)11yescharacteristic
269C22×C4H(5,𝔽2)12yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+412yes—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—42no—
296C4⋊C4—42no—
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—84no—
306C4⋊C4—88no—
307C4⋊C4—84no—
308C42—24no—
309C42—24no—
310C42—48no—
311C42—42no—
312C42—42no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C23#52511yescharacteristic
318C23C2×C4⋊Q811yescharacteristic
319C23#53411yescharacteristic
320C23#54612yes—
321C23#54812yes—
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×AGL(1,ℤ/8)11yescharacteristic
324C23C2×#13711yescharacteristic
325C23C2×#13711yescharacteristic
326C23#57612yes—
327C23#57812yes—
328Q8—816no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—82no—
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#50114yes—
389C2×C4#52912yes—
390C2×C4#53312yes—
391C2×C4#53812yes—
392C2×C4#54212yes—
393C2×C4#54212yes—
394C2×C4#54412yes—
395C2×C4#55112yes—
396C2×C4#55111yescharacteristic
397C2×C4#55212yes—
398C2×C4#55412yes—
399C2×C4#55412yes—
400C2×C4#55412yes—
401C2×C4#55511yescharacteristic
402C2×C4#55712yes—
403C2×C4H(3,𝔽4)11yescharacteristic
404C2×C4#56311yescharacteristic
405C8—44no—
406C8—44no—
407C22#247712yes—
408C22#255012yes—
409C22#258311yescharacteristic
410C22#268411yescharacteristic
411C22#273111yescharacteristic
412C22#310911yescharacteristic
413C22#311311yescharacteristic
414C22#312412yes—
415C22C2×#50111yescharacteristic
416C22#316412yes—
417C22#316511yescharacteristic
418C22#319311yescharacteristic
419C22#319311yescharacteristic
420C22#319911yescharacteristic
421C22#320211yescharacteristic
422C22#330812yes—
423C22#331412yes—
424C22#332512yes—
425C22#332812yes—
426C22#336612yes—
427C22#336712yes—
428C22#337412yes—
429C22#337512yes—
430C4—24no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—84no—
436C4—81no—
437C4—84no—
438C4—84no—
439C4—162no—
440C4#252012yes—
441C4#257412yes—
442C4#276312yes—
443C4#277412yes—
444C2#1594411yescharacteristic
445C2#2744112yes—
446C2#2744312yes—
447C2#2879712yes—
448C2#2880412yes—
449C2#2930111yescharacteristic
450C2#2931011yescharacteristic
451C2#3127011yescharacteristic
452C2#3127311yescharacteristic
453C2#3264411yescharacteristic
454C2#3264711yescharacteristic
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
95. #336Q8semidirect

Maximal subgroups

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

subgroupindex
#151282
#139112
#113842
#112672
#112562
#112192
#109072
#108812
#108752