Finite Group Databasebeta

← #201211

Subgroups — #201211

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3#17741C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9458C218yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C22×#127C2211yescharacteristic
8C22×#124C412yes—
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—21no—
12#2983C2214yescomplemented
13#2982—21no—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2548—21no—
19#2548—21no—
20#2548—21no—
21#2548—21no—
22#2548C414yes—
23#2548C2214yes—
24C2×#394—22no—
25C2×#390—22no—
26C2×#390C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1555C2218yes—
29#1545—24no—
30#1545—24no—
31#1467C2214yescomplemented
32#1463—22no—
33#1461—21no—
34#1461—21no—
35#1285C2218yescomplemented
36C23×Q8C2×C411yescharacteristic
37C23×D4—21no—
38C23×D4—21no—
39C23×D4C2×C412yes—
40C2×#127—22no—
41C2×#127—22no—
42C2×#127—21no—
43C2×#127—21no—
44C2×#127—22no—
45C2×#127—22no—
46C2×#127C2×C412yes—
47C2×#127C2312yes—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—48no—
51C2×#124C2×C418yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—24no—
67C22×C42C2×C412yes—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70#446—28no—
71#446—216no—
72C8×Q8—28no—
73#432—24no—
74#432—24no—
75#432—22no—
76#432—22no—
77#432C2314yes—
78#432C2314yes—
79#431—24no—
80#431—24no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431D414yes—
85#423—24no—
86#423—22no—
87#423D414yes—
88#407—22no—
89#407—22no—
90C4×M16—24no—
91C4×M16—22no—
92C4×M16—22no—
93#394—24no—
94#394—24no—
95#394—24no—
96#394—24no—
97#390—22no—
98#390—22no—
99#390—24no—
100#390—24no—
101#390—24no—
102#390—24no—
103#390C2×C418yes—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2312yes—
108C2×#95—21no—
109#331—216no—
110#324—24no—
111#324—24no—
112C25—22no—
113C22×Q8—24no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—42no—
117C22×Q8C22⋊C411yescharacteristic
118C22×Q8C22⋊C411yescharacteristic
119C22×Q8C22×C412yes—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—48no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—84no—
137C22×D4C22×C414yes—
138C23×C4—22no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—42no—
146#127—24no—
147#127—24no—
148#127—24no—
149#127—24no—
150#127—42no—
151#127—42no—
152#127—42no—
153#127—42no—
154#124—48no—
155#124—48no—
156#124—416no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—24no—
168C4×Q8—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—416no—
188C2×C22⋊C4—88no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—44no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22×C414yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C4⋊C8C2×D418yes—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218#97—28no—
219#97—24no—
220#97—24no—
221C4×C8—28no—
222C4×C8—24no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226C24—24no—
227C24—44no—
228C24—48no—
229C24—44no—
230C24—48no—
231C24—42no—
232C24—42no—
233C24C2×C22⋊C411yescharacteristic
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8#10014yes—
245C2×D4—28no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—48no—
249C2×D4—48no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—84no—
260C2×D4—84no—
261C2×D4—84no—
262C2×D4—88no—
263C2×D4—88no—
264C2×D4—88no—
265C2×D4—84no—
266C2×D4—84no—
267C2×D4—88no—
268C2×D4—84no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—48no—
303M16—84no—
304C2×C8—28no—
305C2×C8—44no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C22⋊C4—416no—
316C22⋊C4—416no—
317C22⋊C4—88no—
318C22⋊C4—88no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—44no—
328C42—44no—
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12012yes—
332C42#12112yes—
333C42#12112yes—
334C42#12312yes—
335C42C22×D412yes—
336C42C2×#4112yes—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—22no—
341C23—48no—
342C23—48no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23—816no—
351C23—816no—
352C23—84no—
353C23—84no—
354C23#38611yescharacteristic
355C23#38611yescharacteristic
356C23C22×C22⋊C411yescharacteristic
357Q8—28no—
358Q8—48no—
359Q8—44no—
360D4—88no—
361D4—84no—
362D4—84no—
363D4—88no—
364D4—88no—
365D4—84no—
366D4—88no—
367D4—84no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—82no—
397C2×C4—88no—
398C2×C4#38611yescharacteristic
399C2×C4#38612yes—
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4C2×#10014yes—
403C2×C4C2×#10012yes—
404C2×C4#41312yes—
405C2×C4#45414yes—
406C2×C4C2×C4×D412yes—
407C2×C4C2×#12011yescharacteristic
408C2×C4C2×#12112yes—
409C2×C4C2×#12311yescharacteristic
410C8—48no—
411C8—84no—
412C22—24no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—88no—
421C22—88no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—168no—
426C22—168no—
427C22#188812yes—
428C22#191012yes—
429C22C2×#38611yescharacteristic
430C22C22×#10011yescharacteristic
431C22C2×#41311yescharacteristic
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—84no—
442C4#188812yes—
443C4#190014yes—
444C4C2×#45412yes—
445C2—21no—
446C2—21no—
447C2—22no—
448C2—22no—
449C2—164no—
450C2C2×#188811yescharacteristic
451C2C2×#190011yescharacteristic
452C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. #17365C2semidirect
5. #9458C2semidirect
6. #7950C2semidirect
12. #2983C22semidirect
31. #1467C22semidirect
35. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25482
#177412
#173652
#94582
#79502