Finite Group Databasebeta

← #238534

Subgroups — #238534

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35001C211yesmaximal, characteristic
2#21758C212yesmaximal, complemented
3C22×#327C212yesmaximal, complemented
4#13908C211yesmaximal, characteristic, complemented
5#10403C214yesmaximal, complemented
6#10402C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C2×#534—21no—
9C2×#534—21no—
10C22×#121C2212yes—
11C22×C22⋊C8C2212yes—
12#2856—22no—
13#2656C2214yes—
14#2539—24no—
15#2501—22no—
16#2433—22no—
17C2×#380—21no—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1893—24no—
22#1874C2218yescomplemented
23#1823—22no—
24#1786—22no—
25C2×#327—28no—
26C2×#327C2214yescomplemented
27C2×#327C2214yescomplemented
28#1444C2212yescomplemented
29#1442C2212yescomplemented
30#1436—22no—
31C22⋊#95—21no—
32#1280C2214yescomplemented
33C23×D4—22no—
34C23×D4D412yes—
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38#534—44no—
39#534—44no—
40#534—44no—
41#534—44no—
42C2×#127—22no—
43C2×#123—24no—
44C2×#123—24no—
45C2×#121—22no—
46C2×#121—24no—
47C2×#121—24no—
48C2×#121—22no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121D414yes—
54C2×#121C2314yes—
55C2×#120—24no—
56C2×#120—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42D412yes—
65C2×#102—28no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70#399—28no—
71#392—28no—
72#390—24no—
73#390—24no—
74#390—24no—
75#384—22no—
76#380—22no—
77#380—22no—
78#380—44no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D414yescomplemented
88C2×#95C2312yes—
89C2×#95C2311yescharacteristic
90C2×#95C2311yescharacteristic
91#340—28no—
92#327—28no—
93#327—28no—
94C25—22no—
95C25C2×D411yescharacteristic
96C22×D4—24no—
97C22×D4—24no—
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—22no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—44no—
104C22×D4—44no—
105C22×D4—44no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—84no—
110C22×D4C2×D414yes—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—44no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4#4114yes—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C22×C8—24no—
131C22×C8—44no—
132C22×C8—84no—
133#127—84no—
134#123—416no—
135#123—416no—
136#121—28no—
137#121—28no—
138#121—24no—
139#121—24no—
140#121—24no—
141#121—24no—
142#121—48no—
143#121—48no—
144#121—44no—
145#121—44no—
146#121—44no—
147#121—44no—
148#121—48no—
149#121—48no—
150#120—416no—
151#120—416no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4D814yes—
177C2×C22⋊C4SD1614yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—82no—
181#102—88no—
182C22⋊C8—24no—
183C22⋊C8—24no—
184C22⋊C8—44no—
185#95—24no—
186#95—24no—
187#95—21no—
188#95—21no—
189#95—44no—
190#95—41no—
191#95—41no—
192#95—44no—
193#95—82no—
194#95C2×D412yes—
195#95C2×D412yes—
196C24—24no—
197C24—42no—
198C24—42no—
199C24—44no—
200C24—44no—
201C24—44no—
202C24—84no—
203C24—84no—
204C24#12012yes—
205C24#12112yes—
206C24#12412yes—
207C24C2×D812yes—
208C24C2×SD1612yes—
209C24AGL(1,ℤ/8)12yes—
210C24AGL(1,ℤ/8)12yes—
211C24C22×D411yescharacteristic
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—48no—
215C2×D4—48no—
216C2×D4—48no—
217C2×D4—48no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—48no—
226C2×D4—48no—
227C2×D4—44no—
228C2×D4—88no—
229C2×D4—88no—
230C2×D4—84no—
231C2×D4—84no—
232C2×D4—84no—
233C2×D4—84no—
234C2×D4—88no—
235C2×D4—88no—
236C2×D4—88no—
237C2×D4—88no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—82no—
274C22×C4—84no—
275C22×C4—82no—
276C22×C4—84no—
277C22×C4—88no—
278C22×C4—88no—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12114yes—
282C22×C4#12114yes—
283C22×C4#12112yes—
284C22×C4#12314yes—
285C22×C4#12314yes—
286C22×C4#12412yes—
287C22×C4C2×D814yes—
288C22×C4C2×SD1614yes—
289C22×C4C22×D412yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4111yescharacteristic
293C2×C8—48no—
294C2×C8—44no—
295C2×C8—88no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—88no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—416no—
306C22⋊C4—416no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—48no—
310C22⋊C4—84no—
311C42—44no—
312C42—82no—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—24no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—44no—
321C23—44no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—48no—
329C23—48no—
330C23—88no—
331C23—88no—
332C23—88no—
333C23—168no—
334C23—168no—
335C23#39414yes—
336C23#44714yes—
337C23#45014yes—
338C23#45914yes—
339C23#46014yes—
340C23#48614yes—
341C23#49014yes—
342C23C2×#12011yescharacteristic
343C23C2×#12111yescharacteristic
344C23C2×#12411yescharacteristic
345C23C22×D811yescharacteristic
346C23C22×SD1611yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×AGL(1,ℤ/8)11yescharacteristic
349D4—48no—
350D4—84no—
351D4—88no—
352D4—84no—
353D4—84no—
354D4—84no—
355D4—84no—
356D4—816no—
357D4—168no—
358D4—168no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—816no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—88no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#39412yes—
404C2×C4#39412yes—
405C2×C4C2×#12112yes—
406C2×C4C2×#12312yes—
407C8—84no—
408C22—21no—
409C22—24no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—28no—
414C22—22no—
415C22—22no—
416C22—28no—
417C22—21no—
418C22—24no—
419C22—21no—
420C22—44no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—84no—
425C22—88no—
426C22—88no—
427C22—84no—
428C22—84no—
429C22—1616no—
430C22#204118yes—
431C22C2×#39411yescharacteristic
432C22C2×#44711yescharacteristic
433C22C2×#45011yescharacteristic
434C22#301812yes—
435C22#302612yes—
436C22C2×#45911yescharacteristic
437C22C2×#46011yescharacteristic
438C22C2×#48611yescharacteristic
439C22C2×#49011yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—82no—
448C4—84no—
449C4—162no—
450C2—21no—
451C2—28no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—164no—
456C2#1838012yes—
457C2#1838112yes—
458C2#2349811yescharacteristic
459C2C2×#204111yescharacteristic
460C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35001C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21758C2semidirect
3. C22×#327C2semidirect
4. #13908C2semidirect
5. #10403C2semidirect
6. #10402C2semidirect
7. #7907C2semidirect
22. #1874C22semidirect
26. C2×#327C22semidirect
27. C2×#327C22semidirect
28. #1444C22semidirect
29. #1442C22semidirect
32. #1280C22semidirect
87. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
#350012
#217582
C22×#3272
#139082
#104032
#104022
#79072