Finite Group Databasebeta

← #393343

Subgroups — #393343

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×M16C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3C2×#1998C212yesmaximal, complemented
4#20626C211yesmaximal, characteristic
5#10913C214yesmaximal, complemented
6#10858C212yesmaximal, complemented
7#10739C214yesmaximal
8C23×M16C2212yes—
9C22×#125C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423C2212yes—
12C2×#407C2212yes—
13C2×C4×M16—21no—
14C2×C4×M16—21no—
15C2×C4×M16—28no—
16C2×C4×M16C2212yes—
17C2×C4×M16C2212yes—
18C22×#97—21no—
19C22×C4×C8—21no—
20#2851—22no—
21#2851—22no—
22#2851—21no—
23#2851—21no—
24#2851—21no—
25#2851—21no—
26C2×#389C2212yes—
27C2×#387C414yes—
28C2×#385C414yes—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31#1998—28no—
32#1797—22no—
33#1797—22no—
34C2×#343C2218yes—
35C2×#343C2218yes—
36#1728C2212yes—
37C24×C4C2×C412yes—
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C22×M16—24no—
41C22×M16—28no—
42C22×M16—28no—
43C22×M16Q818yes—
44C22×M16C2314yes—
45C22×M16C2314yes—
46C23×C8—22no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—22no—
50C2×#125—24no—
51C2×#125—24no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4C2×C414yes—
57C22×C42—24no—
58C22×C42—24no—
59C22×C42C2×C412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#423—24no—
69#423—24no—
70C2×C4⋊C8—24no—
71#407—28no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C4×M16—28no—
75C4×M16—28no—
76C4×M16—28no—
77C4×M16—24no—
78C4×M16D418yes—
79C2×#97—28no—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—28no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#385—24no—
95#385—24no—
96#385—24no—
97#382—21no—
98#382—21no—
99#382—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104C2×#95C2×C414yes—
105C2×#95C2×C414yes—
106C2×#95C2×C414yes—
107#343—216no—
108#343—216no—
109#336—24no—
110C25C22×C411yescharacteristic
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—28no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C414yes—
124C23×C4C22×C414yes—
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q814yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C2×M16—232no—
134C2×M16—28no—
135C2×M16—28no—
136C2×M16—28no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—232no—
141C2×M16—24no—
142C2×M16—28no—
143C2×M16—28no—
144C2×M16—24no—
145C2×M16C2×D418yes—
146C2×M16C2×Q818yes—
147C2×M16#4118yes—
148C22×C8—28no—
149C22×C8—28no—
150C22×C8—28no—
151C22×C8—24no—
152C22×C8—24no—
153#125—42no—
154#125—42no—
155#125—48no—
156#125—42no—
157#125—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—48no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—48no—
174C2×C42—28no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D414yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42#4114yes—
199C4⋊C8—48no—
200C22⋊C8—44no—
201C22⋊C8—44no—
202#97—28no—
203#97—28no—
204C4×C8—28no—
205C4×C8—28no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212C24—22no—
213C24—22no—
214C24—24no—
215C24—42no—
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C411yescharacteristic
218C24C4×Q814yes—
219C24C23×C411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—432no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C4×D418yes—
269C22×C4C4×Q814yes—
270C22×C4C22⋊Q814yes—
271C22×C4C22⋊Q814yes—
272C22×C4#12412yes—
273C22×C4#12412yes—
274C22×C4C4⋊Q812yes—
275C22×C4C4⋊Q812yes—
276C22×C4C22×D412yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×Q812yes—
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4114yes—
283M16—232no—
284M16—28no—
285M16—28no—
286M16—28no—
287M16—48no—
288M16—48no—
289M16C22⋊Q8116yes—
290C2×C8—232no—
291C2×C8—28no—
292C2×C8—28no—
293C2×C8—28no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C4⋊C4—48no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C42—21no—
309C42—21no—
310C42—24no—
311C42—21no—
312C42—21no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42#12012yes—
318C42#12012yes—
319C42#12118yes—
320C42#12312yes—
321C42#12312yes—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—28no—
330C23—21no—
331C23—21no—
332C23—28no—
333C23—44no—
334C23—48no—
335C23—48no—
336C23#39112yes—
337C23#39112yes—
338C23#41314yes—
339C23C22×C22⋊C411yescharacteristic
340C23C2×C4×D412yes—
341C23C2×C4×Q812yes—
342C23C4×#4112yes—
343C2×C4—216no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4#38614yes—
380C2×C4#38614yes—
381C2×C4#39112yes—
382C2×C4#39112yes—
383C2×C4#41314yes—
384C2×C4C2×#12011yescharacteristic
385C2×C4C2×#12112yes—
386C2×C4C2×C22⋊Q812yes—
387C2×C4C2×C22⋊Q814yes—
388C2×C4C2×#12311yescharacteristic
389C2×C4#52514yes—
390C2×C4#52512yes—
391C2×C4C2×#12411yescharacteristic
392C2×C4#52914yes—
393C2×C4C2×C4⋊Q811yescharacteristic
394C2×C4#53212yes—
395C2×C4#53314yes—
396C8—216no—
397C8—84no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—416no—
410C22—42no—
411C22—42no—
412C22#199814yescomplemented
413C22C2×#38611yescharacteristic
414C22C2×#39111yescharacteristic
415C22#229711yescharacteristic
416C22C4×C22⋊Q814yes—
417C22C2×#41312yes—
418C22C2×#41312yes—
419C4—22no—
420C4—22no—
421C4—24no—
422C4—24no—
423C4—24no—
424C4—42no—
425C4—84no—
426C4—84no—
427C4#284812yes—
428C4#285012yes—
429C4#285512yes—
430C4#285712yes—
431C2—24no—
432C2—24no—
433C2—42no—
434C2#1470211yescharacteristic
435C2#2098912yes—
436C2C2×#199811yescharacteristic
437C2#2133711yescharacteristic
438C2#2142612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1998C2semidirect
5. #10913C2semidirect
6. #10858C2semidirect
412. C22#1998semidirect

Maximal subgroups

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

subgroupindex
C22×C4×M162
C2×#28512
C2×#19982
#206262
#109132
#108582
#107392