Finite Group Databasebeta

← #223949

Subgroups — #223949

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36182C211yesmaximal, characteristic
2#22648C212yesmaximal, complemented
3#9906C212yesmaximal, complemented
4#9141C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#8728C211yesmaximal, characteristic, complemented
7#8725C211yesmaximal, characteristic, complemented
8#8585C211yesmaximal, characteristic, complemented
9#8231C211yesmaximal, characteristic, complemented
10#8041C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#553C412yes—
13C2×#435—22no—
14C2×#423C2212yes—
15C22×C4⋊C8C2212yes—
16#2859C2211yescharacteristic
17#2856C2211yescharacteristic
18#2856C2211yescharacteristic
19#2566C412yes—
20#2512—22no—
21#2504C412yes—
22#2500C412yes—
23#2422—22no—
24#2416—22no—
25#2361—22no—
26C4×#127C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1940—22no—
31#1851—22no—
32#1850—22no—
33#1842C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2212yes—
37C2×#331C2212yes—
38C2×#331C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C22×M16—22no—
48C23×C8—22no—
49#553—48no—
50C2×#127C2311yescharacteristic
51C2×#126—22no—
52C2×#125C2×C412yes—
53C2×#123—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121C2×C412yes—
58C2×#121C2×C412yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D412yes—
64C2×#117C2×C412yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C412yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#435—48no—
70#423—24no—
71#423—42no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—22no—
85#397—24no—
86#396—24no—
87#394—24no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#389—22no—
95#389—22no—
96#388—24no—
97#386—24no—
98#385—24no—
99#384—22no—
100#384C2×C412yes—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382C2×C412yes—
105#382C2×C412yes—
106#380—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4D412yes—
110C4×C22⋊C4C2×C411yescharacteristic
111C4×C22⋊C4C2×C411yescharacteristic
112C4×C22⋊C4D412yes—
113C4×C22⋊C4D412yes—
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#331—24no—
119#331—42no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C2×M16—44no—
132C2×M16—44no—
133C2×M16—82no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—44no—
139C22×C8—44no—
140C22×C8—82no—
141#127—22no—
142#127—21no—
143#127—21no—
144#126—48no—
145#125—44no—
146#123—48no—
147#121—44no—
148#121—48no—
149#121—44no—
150#121—44no—
151#121—44no—
152#121—82no—
153#121—82no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—44no—
158C4×D4—48no—
159C4×D4—82no—
160C4×D4—82no—
161#117—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4C22⋊C411yescharacteristic
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C411yescharacteristic
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C4⋊C8—44no—
214C4⋊C8—82no—
215C4⋊C8#4114yes—
216C22⋊C8—44no—
217C22⋊C8—44no—
218#97—44no—
219#95—24no—
220#95—24no—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224C24—22no—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C411yescharacteristic
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—48no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—84no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4AGL(1,ℤ/8)14yes—
276C22×C4#13714yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D412yes—
279M16—88no—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—88no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—84no—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—48no—
316C22⋊C4—84no—
317C22⋊C4—84no—
318C22⋊C4#10014yes—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—44no—
334C42C2×#4112yes—
335C42H(5,𝔽2)11yescharacteristic
336C422−1+411yescharacteristic
337C23—21no—
338C23—21no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—44no—
349C23—44no—
350C23C2×#10012yes—
351C23#41714yes—
352C23C22×C22⋊C411yescharacteristic
353D4—44no—
354D4—82no—
355D4—82no—
356D4—88no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—164no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
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—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—164no—
399C2×C4—164no—
400C2×C4C2×#10012yes—
401C2×C4C2×#10012yes—
402C2×C4#41714yes—
403C2×C4C4×#4111yescharacteristic
404C2×C4#52011yescharacteristic
405C2×C4#54612yes—
406C2×C4#54712yes—
407C2×C4C2×AGL(1,ℤ/8)12yes—
408C2×C4C2×#13712yes—
409C2×C4#57612yes—
410C2×C4#57712yes—
411C8—44no—
412C8—82no—
413C8—82no—
414C8—82no—
415C22—24no—
416C22—22no—
417C22—22no—
418C22—24no—
419C22—41no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—84no—
426C22—82no—
427C22—82no—
428C22—84no—
429C22#156014yes—
430C22#233911yescharacteristic
431C22C22×#10011yescharacteristic
432C22#288911yescharacteristic
433C22C2×#41712yes—
434C22#289712yes—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—24no—
440C4—41no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4#331812yes—
448C4#332012yes—
449C4#334512yes—
450C4#334612yes—
451C2—24no—
452C2—81no—
453C2—162no—
454C2C2×#156011yescharacteristic
455C2#1776512yes—
456C2#1777011yescharacteristic
457C2#2247911yescharacteristic
458C2#2249312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36182C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22648C2semidirect
3. #9906C2semidirect
4. #9141C2semidirect
5. #9133C2semidirect
6. #8728C2semidirect
7. #8725C2semidirect
8. #8585C2semidirect
9. #8231C2semidirect
10. #8041C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361822
#226482
#99062
#91412
#91332
#87282
#87252
#85852
#82312
#80412
C2×#12962