Finite Group Databasebeta

← #201202

Subgroups — #201202

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#17742C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9453C218yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2975—21no—
10#2974—24no—
11#2973C2214yes—
12#2972—21no—
13D4×M16—24no—
14#2920C2212yes—
15#2920C2212yes—
16C2×#407—22no—
17#2563—21no—
18#2563—21no—
19#2563—21no—
20#2563—21no—
21#2563C414yes—
22#2563C2214yes—
23C2×#393—22no—
24C2×#390C412yes—
25C2×#382C412yes—
26C2×#380—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1556C2218yescomplemented
29#1549—24no—
30#1546—24no—
31#1466C2214yes—
32#1462—22no—
33#1460—21no—
34#1460—21no—
35#1285C2218yescomplemented
36C23×Q8—21no—
37C23×D4C2×C411yescharacteristic
38C24×C4D412yes—
39C22×M16—24no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2312yes—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4D412yes—
64C22×C42C2×C412yes—
65C22×C42C2311yescharacteristic
66#436—28no—
67#435—216no—
68C8×D4—28no—
69#432—22no—
70#432—22no—
71#432C2314yes—
72#432C2314yes—
73#431—22no—
74#431—22no—
75#431—22no—
76#431D414yes—
77#423—22no—
78#423D414yes—
79#407—24no—
80#407—24no—
81#407—22no—
82#407—24no—
83#407—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C4×M16—22no—
87C4×M16—22no—
88#393—24no—
89#393—24no—
90#393—24no—
91#393—24no—
92#390—22no—
93#390—22no—
94#390C2×C418yes—
95#382—22no—
96#382—22no—
97#382C2×C418yes—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2312yes—
106C2×#95—24no—
107C2×#95—21no—
108#332—216no—
109#323—24no—
110#323—24no—
111C25C22⋊C412yes—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—84no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—42no—
121C22×D4C22⋊C411yescharacteristic
122C22×D4C22⋊C411yescharacteristic
123C22×D4C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—48no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—28no—
141C2×M16—42no—
142C2×M16—88no—
143C22×C8—28no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—24no—
163C4×D4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22×C414yes—
181C2×C4⋊C4C22×C412yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—88no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—22no—
192C2×C42—44no—
193C2×C42C22×C414yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42C2411yescharacteristic
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—42no—
205C4⋊C8C2×D418yes—
206C22⋊C8—28no—
207C22⋊C8—28no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210C22⋊C8—24no—
211C22⋊C8—42no—
212#97—24no—
213#97—24no—
214C4×C8—24no—
215C4×C8—42no—
216#95—22no—
217#95—28no—
218#95—28no—
219#95—22no—
220C24—24no—
221C24—22no—
222C24—22no—
223C24—42no—
224C24—44no—
225C24—44no—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C412yes—
228C24C2×C22⋊C412yes—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—88no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—88no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—42no—
245C2×D4#10114yes—
246C22×C4—22no—
247C22×C4—22no—
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—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12311yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—48no—
294M16—88no—
295M16—84no—
296C2×C8—28no—
297C2×C8—44no—
298C2×C8—88no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—28no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—88no—
315C22⋊C4—88no—
316C42—44no—
317C42—44no—
318C42C4×D412yes—
319C42C4×D412yes—
320C42#12012yes—
321C42#12112yes—
322C42#12112yes—
323C42#12312yes—
324C42C22×D412yes—
325C42C2×#4112yes—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—416no—
342C23—44no—
343C23—44no—
344C23—82no—
345C23—82no—
346C23—88no—
347C23#38611yescharacteristic
348C23#38611yescharacteristic
349C23C2×#10114yes—
350C23C22×C22⋊C411yescharacteristic
351Q8—88no—
352Q8—88no—
353D4—28no—
354D4—48no—
355D4—44no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—88no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4#38612yes—
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38612yes—
395C2×C4C2×#10112yes—
396C2×C4#41312yes—
397C2×C4#45514yes—
398C2×C4C2×C4×D412yes—
399C2×C4C2×#12011yescharacteristic
400C2×C4C2×#12112yes—
401C2×C4C2×#12311yescharacteristic
402C8—48no—
403C8—84no—
404C22—24no—
405C22—24no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—24no—
413C22—44no—
414C22—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—88no—
422C22—88no—
423C22—82no—
424C22#188512yes—
425C22#191012yes—
426C22C2×#38611yescharacteristic
427C22C22×#10111yescharacteristic
428C22C2×#41311yescharacteristic
429C4—22no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—22no—
434C4—42no—
435C4—42no—
436C4—84no—
437C4—84no—
438C4#188512yes—
439C4#190114yes—
440C4C2×#45512yes—
441C2—21no—
442C2—21no—
443C2—22no—
444C2—22no—
445C2—44no—
446C2—82no—
447C2C2×#188511yescharacteristic
448C2C2×#190111yescharacteristic
449C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×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
3. #17742C2semidirect
5. #9453C2semidirect
6. #7950C2semidirect
28. #1556C22semidirect
35. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25632
#177422
#173642
#94532
#79502