Finite Group Databasebeta

← #160840

Subgroups — #160840

Order 512.

1010 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
1C2×C4×#125C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#7636C214yesmaximal, complemented
5#7610C212yesmaximal
6#7607C212yesmaximal
7#7585C214yesmaximal, complemented
8C22×#125C411yescharacteristic
9C2×#423—21no—
10C2×C4×M16—21no—
11C4×#125—21no—
12C4×#125—21no—
13C4×#125—21no—
14C4×#125—21no—
15C4×#125C412yes—
16C4×#125C412yes—
17C4×#125C2212yes—
18C4×#125C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#382C411yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C43C2211yescharacteristic
29#1839C2212yes—
30#1831—21no—
31#1817C2212yes—
32#1797—21no—
33#1767—21no—
34#1767C2212yes—
35#1767C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yes—
38#1749—21no—
39#1745—21no—
40#1728C2212yes—
41#1271C2218yes—
42#1271C2218yes—
43C22×M16—22no—
44C2×#125—28no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66#423—28no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C4×M16—44no—
72C2×#97—22no—
73C2×#97—22no—
74C2×C4×C8—22no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2312yes—
82#389C2312yes—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#382—21no—
92#382—21no—
93#382C2×C412yes—
94#382C2×C412yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—28no—
143C2×M16—44no—
144C2×M16—44no—
145C22×C8—24no—
146C22×C8—24no—
147#125—28no—
148#125—28no—
149#125—24no—
150#125—24no—
151#125—24no—
152#125M1618yescomplemented
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—28no—
221C4⋊C8—28no—
222C4⋊C8—44no—
223C22⋊C8—44no—
224#97—44no—
225#97—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—
239C24C2×C22⋊C411yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
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—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#10414yes—
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×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13712yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—416no—
293C2×C8—28no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4C2×M1614yes—
317C4⋊C4C2×M1614yes—
318C42—22no—
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—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—44no—
347C42—44no—
348C42—44no—
349C42—44no—
350C42#10114yes—
351C42C2×M1614yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23#41012yes—
357C23#41712yes—
358C23C2×#10412yes—
359C23C22×C22⋊C411yescharacteristic
360C2×C4—28no—
361C2×C4—21no—
362C2×C4—21no—
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—24no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4#40712yes—
415C2×C4#40712yes—
416C2×C4C2×#10112yes—
417C2×C4C2×#10112yes—
418C2×C4C2×#10112yes—
419C2×C4#41712yes—
420C2×C4C2×#10412yes—
421C2×C4C2×#10412yes—
422C2×C4#42112yes—
423C2×C4C22×M1612yes—
424C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C2×C4#57512yes—
427C8—48no—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22#150914yes—
436C22C2×#40711yescharacteristic
437C22C2×#41011yescharacteristic
438C22C22×#10111yescharacteristic
439C22C2×#41711yescharacteristic
440C22#289611yescharacteristic
441C22C22×#10411yescharacteristic
442C22C2×#42111yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—41no—
454C4—44no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4#152814yes—
462C2—24no—
463C2#1736511yescharacteristic
464C2#1744711yescharacteristic
465C2#1745111yescharacteristic
466C2C2×#150911yescharacteristic
467C2#1758011yescharacteristic
468C2C2×#152811yescharacteristic
469C2#1775111yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7636C2semidirect
7. #7585C2semidirect
152. #125M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
#207782
C4×#4232
#76362
#76102
#76072
#75852