Finite Group Databasebeta

← #201201

Subgroups — #201201

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic
3#17746C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9451C218yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×#124C412yes—
9C22×C4×D4C2211yescharacteristic
10#2975—21no—
11#2974—24no—
12#2973C2214yes—
13#2972—21no—
14D4×M16—24no—
15#2920C2212yes—
16#2920C2212yes—
17C2×#407—22no—
18#2547—21no—
19#2547—21no—
20#2547—21no—
21#2547—21no—
22#2547C414yes—
23#2547C2214yes—
24C2×#397—22no—
25C2×#391C412yes—
26C2×#380—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1558C2218yescomplemented
29#1548—24no—
30#1548—24no—
31#1466C2214yes—
32#1462—22no—
33#1460—21no—
34#1460—21no—
35#1285C2218yescomplemented
36C23×Q8C2×C412yes—
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×#124—22no—
49C2×#124—22no—
50C2×#124—48no—
51C2×#124C2×C418yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4—24no—
67C22×C22⋊C4D412yes—
68C22×C42C2×C412yes—
69C22×C42C2311yescharacteristic
70#436—28no—
71#435—216no—
72C8×D4—28no—
73#432—22no—
74#432—22no—
75#432C2314yes—
76#432C2314yes—
77#431—22no—
78#431—22no—
79#431—22no—
80#431D414yes—
81#423—22no—
82#423D414yes—
83#407—24no—
84#407—24no—
85#407—22no—
86#407—24no—
87#407—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C4×M16—22no—
91C4×M16—22no—
92#397—24no—
93#397—24no—
94#397—24no—
95#397—24no—
96#391—22no—
97#391—22no—
98#391C2×C418yes—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2312yes—
107C2×#95—24no—
108C2×#95—21no—
109#333—216no—
110#323—24no—
111#323—24no—
112C25C22⋊C412yes—
113C22×Q8—22no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—44no—
117C22×Q8—44no—
118C22×Q8C22×C414yes—
119C22×D4—24no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—42no—
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22×C412yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—48no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—28no—
143C2×M16—42no—
144C2×M16—88no—
145C22×C8—28no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154#124—48no—
155#124—48no—
156#124—416no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—24no—
168C4×D4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—416no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C2×D412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42C22×C414yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—42no—
210C4⋊C8C2×D418yes—
211C22⋊C8—28no—
212C22⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—42no—
217#97—24no—
218#97—24no—
219C4×C8—24no—
220C4×C8—42no—
221#95—22no—
222#95—28no—
223#95—28no—
224#95—22no—
225C24—24no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24—44no—
230C24—44no—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C412yes—
234C2×Q8—28no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—48no—
250C2×D4—48no—
251C2×D4—42no—
252C2×D4#10014yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—48no—
302M16—88no—
303M16—84no—
304C2×C8—28no—
305C2×C8—44no—
306C2×C8—88no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—416no—
323C22⋊C4—416no—
324C42—44no—
325C42—44no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12312yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—416no—
350C23—44no—
351C23—44no—
352C23—82no—
353C23—82no—
354C23—88no—
355C23#38611yescharacteristic
356C23#38611yescharacteristic
357C23C2×#10014yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—88no—
360Q8—88no—
361D4—28no—
362D4—48no—
363D4—44no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—88no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4#38612yes—
401C2×C4#38611yescharacteristic
402C2×C4#38611yescharacteristic
403C2×C4#38612yes—
404C2×C4C2×#10012yes—
405C2×C4#41312yes—
406C2×C4#45514yes—
407C2×C4C2×C4×D412yes—
408C2×C4C2×#12011yescharacteristic
409C2×C4C2×#12112yes—
410C2×C4C2×#12311yescharacteristic
411C8—48no—
412C8—84no—
413C22—24no—
414C22—24no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—24no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—82no—
430C22—88no—
431C22—88no—
432C22—82no—
433C22#188512yes—
434C22#190912yes—
435C22C2×#38611yescharacteristic
436C22C22×#10011yescharacteristic
437C22C2×#41311yescharacteristic
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—84no—
446C4—84no—
447C4#188512yes—
448C4#190014yes—
449C4C2×#45512yes—
450C2—21no—
451C2—21no—
452C2—22no—
453C2—22no—
454C2—44no—
455C2—82no—
456C2C2×#188511yescharacteristic
457C2C2×#190011yescharacteristic
458C2C2×#190911yescharacteristic
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. #17746C2semidirect
5. #9451C2semidirect
6. #7950C2semidirect
28. #1558C22semidirect
35. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25472
#177462
#173642
#94512
#79502