Finite Group Databasebeta

← #208184

Subgroups — #208184

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#9900C212yesmaximal, complemented
5#9697C212yesmaximal, complemented
6#9131C212yesmaximal
7#9127C212yesmaximal
8#8237C212yesmaximal, complemented
9#8235C212yesmaximal, complemented
10C2×#432—21no—
11C22×C4⋊C8C2212yes—
12C22×C4×C8—21no—
13#2864—21no—
14#2863—21no—
15#2863C412yes—
16#2862C2211yescharacteristic
17#2859C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2855—21no—
21#2855C412yes—
22#2851—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8C412yes—
28#2295—21no—
29#2295C412yes—
30#2283—21no—
31C2×C43C2211yescharacteristic
32#1842C2212yes—
33#1831—21no—
34#1797—21no—
35#1797C2212yes—
36#1767—21no—
37#1767C2212yes—
38C4×C4⋊C8—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2212yes—
42C2×#336—21no—
43C2×#333C2214yes—
44C2×#332C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47C23×C8—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8C2×C412yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yes—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42C2311yescharacteristic
62#432—44no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2314yes—
68C2×C4⋊C8C2314yes—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—44no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#383—22no—
107#382—21no—
108#382—21no—
109#382—21no—
110#381—22no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380C2×C412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C22⋊C4—22no—
122#376—22no—
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#333—28no—
136#332—28no—
137C22×Q8—21no—
138C22×Q8C22×C412yes—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—28no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—44no—
150C22×C8—44no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153#125—44no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—44no—
157C4×Q8—44no—
158C4×Q8—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42SD1614yes—
199C2×C42Q1614yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—28no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222C4⋊C8#4118yes—
223C22⋊C8—44no—
224#97—44no—
225C4×C8—44no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
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—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4#10314yes—
271C22×C4#10314yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×SD1612yes—
284C22×C4C2×Q1612yes—
285C22×C4AGL(1,ℤ/8)12yes—
286C22×C4#13712yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—28no—
291C2×C8—416no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—44no—
341C42—44no—
342C42#10314yes—
343C42#10314yes—
344C42C2×SD1612yes—
345C42C2×SD1612yes—
346C42C2×Q1612yes—
347C42C2×Q1612yes—
348C42#13514yes—
349C42C2×#4114yes—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23#41312yes—
355C23C2×#10312yes—
356C23#41712yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—84no—
359Q8—82no—
360Q8—82no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4#41312yes—
409C2×C4C2×#10312yes—
410C2×C4C2×#10312yes—
411C2×C4C2×#10312yes—
412C2×C4C2×#10312yes—
413C2×C4C2×#10312yes—
414C2×C4#41612yes—
415C2×C4#41612yes—
416C2×C4#41712yes—
417C2×C4C4×#4112yes—
418C2×C4#52512yes—
419C2×C4#53212yes—
420C2×C4C22×SD1611yescharacteristic
421C2×C4C22×Q1611yescharacteristic
422C2×C4C2×#13512yes—
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57512yes—
426C8—48no—
427C8—84no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22#155014yes—
435C22#229711yescharacteristic
436C22C2×#41311yescharacteristic
437C22C2×#41311yescharacteristic
438C22C22×#10311yescharacteristic
439C22C2×#41611yescharacteristic
440C22C2×#41711yescharacteristic
441C22#289611yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4#311412yes—
459C4#311712yes—
460C4#316712yes—
461C4#317012yes—
462C2—24no—
463C2C2×#155011yescharacteristic
464C2#1770811yescharacteristic
465C2#1771311yescharacteristic
466C2#1771911yescharacteristic
467C2#2139111yescharacteristic
468C2#2140011yescharacteristic
469C2#2142611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9900C2semidirect
5. #9697C2semidirect
8. #8237C2semidirect
9. #8235C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#208102
C2×C4×C4⋊C82
#99002
#96972
#91312
#91272
#82372
#82352