Finite Group Databasebeta

← #206749

Subgroups — #206749

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#9867C214yesmaximal, complemented
5#9866C214yesmaximal, complemented
6#7904C212yesmaximal
7#7904C212yesmaximal
8C22×#125C411yescharacteristic
9C2×#432C2212yes—
10C2×#423—21no—
11C2×C4×M16C2212yes—
12C4×#125—21no—
13C4×#125—21no—
14C4×#125—21no—
15C4×#125—21no—
16C4×#125C412yes—
17C4×#125C412yes—
18C4×#125C2212yes—
19C4×#125C2212yes—
20C2×#389C411yescharacteristic
21C2×#389C411yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#382C411yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1767—21no—
31#1767—21no—
32#1767C2212yes—
33C4×C4⋊C8C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36#1749—21no—
37C4×C22⋊C8—21no—
38#1745—21no—
39#1728—21no—
40C4×#97—21no—
41#1279C2218yes—
42#1279C2218yes—
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#432—24no—
67#432Q818yes—
68#423—28no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2314yes—
72C2×C22⋊C8—22no—
73C4×M16—28no—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97—22no—
77C2×C4×C8—22no—
78C2×C4×C8C2314yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#382—21no—
96#382—21no—
97#382C2×C412yes—
98#382C2×C412yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2312yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43C2311yescharacteristic
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—28no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—24no—
148C22×C8—24no—
149#125—28no—
150#125—28no—
151#125—24no—
152#125—24no—
153#125—24no—
154#125M1618yescomplemented
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
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×Q814yes—
219C2×C42#4114yes—
220C2×C42C2411yescharacteristic
221C4⋊C8—28no—
222C4⋊C8—28no—
223C4⋊C8—24no—
224C4⋊C8C2×Q818yes—
225C22⋊C8—28no—
226#97—28no—
227#97—44no—
228#97—44no—
229C4×C8—28no—
230C4×C8—44no—
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—
241#95—22no—
242#95—22no—
243C24C2×C22⋊C411yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
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—41no—
266C22×C4—41no—
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—42no—
276C22×C4—42no—
277C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C4×Q814yes—
290C22×C4C22⋊Q814yes—
291C22×C4#12412yes—
292C22×C4C4⋊Q812yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296M16—416no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4C2×M1614yes—
320C4⋊C4C2×M1614yes—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42—44no—
348C42C4×Q814yes—
349C42C22⋊Q814yes—
350C42#12412yes—
351C42C4⋊Q812yes—
352C42C2×M1614yes—
353C42C22×Q812yes—
354C42C2×#4112yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23#39112yes—
360C23C2×#10412yes—
361C23#42112yes—
362C23C22×C22⋊C411yescharacteristic
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—28no—
383C2×C4—21no—
384C2×C4—21no—
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—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—44no—
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—42no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4#39112yes—
416C2×C4#39112yes—
417C2×C4#39112yes—
418C2×C4#40712yes—
419C2×C4#40712yes—
420C2×C4C2×#10412yes—
421C2×C4C2×#10412yes—
422C2×C4C2×C4×Q812yes—
423C2×C4C2×C22⋊Q812yes—
424C2×C4C2×#12411yescharacteristic
425C2×C4C2×C4⋊Q811yescharacteristic
426C2×C4C22×M1612yes—
427C2×C4#56812yes—
428C2×C4#57512yes—
429C2×C4#57512yes—
430C8—48no—
431C8—48no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22#198614yes—
439C22C2×#39111yescharacteristic
440C22C2×#40711yescharacteristic
441C22#287911yescharacteristic
442C22#289611yescharacteristic
443C22#289611yescharacteristic
444C22C22×#10411yescharacteristic
445C22C2×#42111yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4—44no—
459C4—41no—
460C4—41no—
461C4—82no—
462C4—82no—
463C4Q8×M1614yes—
464C4#308014yes—
465C2—24no—
466C2#1744711yescharacteristic
467C2#1745211yescharacteristic
468C2#1749711yescharacteristic
469C2#1750211yescharacteristic
470C2#2130311yescharacteristic
471C2C2×#198611yescharacteristic
472C2#2131411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9866C2semidirect
154. #125M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C4×#4322
C4×#4232
#98672
#98662
#79042