Finite Group Databasebeta

← #201146

Subgroups — #201146

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3#17667C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9450C218yesmaximal, complemented
6#7950C212yesmaximal
7C22×#125C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—21no—
11#2983C2214yes—
12#2982—21no—
13Q8×M16—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2565—21no—
18#2565—21no—
19#2565—21no—
20#2565—21no—
21#2565C414yes—
22#2565C2214yes—
23C2×#398—22no—
24C2×#391C411yescharacteristic
25C2×#391C411yescharacteristic
26C2×#387—22no—
27C2×#384C411yescharacteristic
28C2×#382C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1534C2218yes—
31#1528—24no—
32#1528—24no—
33#1467C2214yes—
34#1463—22no—
35#1461—21no—
36#1461—21no—
37#1285C2218yes—
38C23×Q8C2×C411yescharacteristic
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—22no—
45C2×#125C2×C412yes—
46C2×#125C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2312yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#446—216no—
71C8×Q8—28no—
72#432—24no—
73#432—24no—
74#432—22no—
75#432D414yes—
76#432C2314yes—
77#432C2314yes—
78#431—24no—
79#431—24no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84#423—24no—
85#423—22no—
86#423—22no—
87#407—22no—
88#407—22no—
89C4×M16—24no—
90C4×M16—22no—
91C4×M16D414yes—
92#398—24no—
93#398—24no—
94#398—24no—
95#398—24no—
96#391—21no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391C2×C414yes—
101#391C2×C414yes—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#384—21no—
107#384—21no—
108#384C2×C414yes—
109#382—21no—
110#382—21no—
111#382C2×C414yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2312yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120#330—216no—
121#324—24no—
122#324—24no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—42no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C22×C412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141#125—24no—
142#125—24no—
143#125—24no—
144#125—24no—
145#125—42no—
146#125—42no—
147#125—42no—
148#125—42no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—24no—
160C4×Q8—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—28no—
219C4⋊C8—28no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—42no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228#97—28no—
229#97—24no—
230#97C2×D418yes—
231C4×C8—28no—
232C4×C8—24no—
233C4×C8—42no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8#10114yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299M16—48no—
300M16—84no—
301C2×C8—28no—
302C2×C8—44no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12012yes—
339C42#12112yes—
340C42#12112yes—
341C42#12312yes—
342C42C22×D412yes—
343C42C2×#4112yes—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23#38611yescharacteristic
349C23#38611yescharacteristic
350C23C22×C22⋊C411yescharacteristic
351Q8—28no—
352Q8—48no—
353Q8—44no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#38612yes—
400C2×C4#38612yes—
401C2×C4C2×#10114yes—
402C2×C4C2×#10112yes—
403C2×C4#41312yes—
404C2×C4C2×C4×D412yes—
405C2×C4C2×#12011yescharacteristic
406C2×C4C2×#12112yes—
407C2×C4C2×#12311yescharacteristic
408C8—48no—
409C8—84no—
410C22—24no—
411C22—24no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22#188412yes—
419C22#191212yes—
420C22C2×#38611yescharacteristic
421C22C22×#10111yescharacteristic
422C22C2×#41311yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4#190114yes—
439C2—21no—
440C2—21no—
441C2—22no—
442C2—22no—
443C2C2×#188411yescharacteristic
444C2C2×#190111yescharacteristic
445C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
5. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25652
#176672
#173652
#94502
#79502