Finite Group Databasebeta

← #223942

Subgroups — #223942

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#22648C212yesmaximal
3#9906C212yesmaximal, complemented
4#9135C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8728C211yesmaximal, characteristic, complemented
7#8723C211yesmaximal, characteristic, complemented
8#8587C211yesmaximal, characteristic, complemented
9#8232C211yesmaximal, characteristic, complemented
10#8041C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#552C412yes—
13C2×#435—22no—
14C2×#423C2212yes—
15C22×C4⋊C8C2212yes—
16#2859C2211yescharacteristic
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2565C412yes—
20#2506—22no—
21#2503C412yes—
22#2499C412yes—
23#2496—22no—
24#2486—22no—
25#2344—22no—
26C4×C4⋊Q8C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1940—22no—
31#1851—22no—
32#1850—22no—
33#1842C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C22×M16—22no—
48C23×C8—22no—
49#552—48no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×#126—22no—
52C2×#125—22no—
53C2×#125C2×C412yes—
54C2×#123—22no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D412yes—
60C2×#117—22no—
61C2×#117C2×C412yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#435—48no—
67#423—24no—
68#423—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×#97—22no—
82#400—24no—
83#398—24no—
84#397—24no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#387—24no—
94#387—24no—
95#384—22no—
96#384C2×C412yes—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#382C2×C412yes—
101#382C2×C412yes—
102#381—24no—
103#380—22no—
104#380—22no—
105#380—22no—
106C4×C4⋊C4—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#333—24no—
119#333—42no—
120C22×Q8C22×C412yes—
121C22×D4C22⋊C411yescharacteristic
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C2×M16—44no—
128C2×M16—44no—
129C2×M16—82no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—44no—
135C22×C8—44no—
136C22×C8—82no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#126—48no—
141#125—44no—
142#125—48no—
143#123—48no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146C4×Q8—44no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150#117—48no—
151#117—44no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊C411yescharacteristic
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—44no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8#4114yes—
207C22⋊C8—44no—
208C22⋊C8—44no—
209#97—44no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218C24C2×C22⋊C411yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×D4—22no—
225C2×D4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
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—41no—
247C22×C4—41no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4AGL(1,ℤ/8)14yes—
258C22×C4#13714yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261M16—88no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—88no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—24no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4#10014yes—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—44no—
317C42—44no—
318C42C2×#4112yes—
319C42H(5,𝔽2)11yescharacteristic
320C422−1+411yescharacteristic
321C23—21no—
322C23—21no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23C2×#10012yes—
330C23#41714yes—
331C23C22×C22⋊C411yescharacteristic
332Q8—82no—
333Q8—82no—
334D4—44no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—88no—
375C2×C4C2×#10012yes—
376C2×C4C2×#10012yes—
377C2×C4#41714yes—
378C2×C4C4×#4111yescharacteristic
379C2×C4#52011yescharacteristic
380C2×C4#54612yes—
381C2×C4#54712yes—
382C2×C4C2×AGL(1,ℤ/8)12yes—
383C2×C4C2×#13712yes—
384C2×C4#57712yes—
385C2×C4#57812yes—
386C8—44no—
387C8—82no—
388C8—82no—
389C8—82no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—24no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22#156014yes—
400C22#233911yescharacteristic
401C22C22×#10011yescharacteristic
402C22#288911yescharacteristic
403C22C2×#41712yes—
404C22#289812yes—
405C4—22no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—24no—
410C4—41no—
411C4—42no—
412C4—41no—
413C4—42no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4#331012yes—
419C4#331212yes—
420C4#334312yes—
421C4#334412yes—
422C2—24no—
423C2—81no—
424C2C2×#156011yescharacteristic
425C2#1776312yes—
426C2#1777911yescharacteristic
427C2#2247911yescharacteristic
428C2#2249112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
3. #9906C2semidirect
4. #9135C2semidirect
5. #9127C2semidirect
6. #8728C2semidirect
7. #8723C2semidirect
8. #8587C2semidirect
9. #8232C2semidirect
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
#361812
#226482
#99062
#91352
#91272
#87282
#87232
#85872
#82322
#80412
C2×#12962