Finite Group Databasebeta

← #194830

Subgroups — #194830

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#21759C212yesmaximal, complemented
4C22×#323C211yesmaximal, characteristic
5#9021C214yesmaximal, complemented
6#9013C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C23×C22⋊C4—21no—
10C23×C22⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2626—24no—
13#2626C418yescomplemented
14C2×#399C2212yes—
15C2×#396—22no—
16C2×#393C2212yes—
17#2315—24no—
18#2315—22no—
19#2315C2214yes—
20C2×#385—21no—
21C2×#385—24no—
22C2×#380—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2212yes—
25C2×#379—21no—
26#1892—24no—
27#1872—24no—
28#1824C2214yescomplemented
29#1787C2214yescomplemented
30C2×#323—24no—
31C2×#323C2214yes—
32#1294C2218yescomplemented
33C26D411yescharacteristic
34C23×Q8C2311yescharacteristic
35C24×C4—22no—
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8C2×C414yescomplemented
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D414yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—44no—
62C22×C22⋊C4—44no—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4D412yes—
65C22×C22⋊C4D414yes—
66C22×C22⋊C4C2312yes—
67C2×#103—28no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2314yes—
72#399—24no—
73#399C2×C418yescomplemented
74#396—216no—
75#393—24no—
76#393C2×C418yescomplemented
77#385—216no—
78#385—28no—
79#385—216no—
80#380—24no—
81#380—22no—
82#380—22no—
83#380—24no—
84#380—22no—
85#380C2×C414yescomplemented
86#380D414yes—
87#380C2314yes—
88#379—28no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95C2312yes—
93#340—28no—
94#323—28no—
95C25—21no—
96C25—21no—
97C25—22no—
98C25—22no—
99C25—22no—
100C25—82no—
101C25—82no—
102C25C22⋊C414yes—
103C25SD1614yes—
104C25Q1614yes—
105C25C2×D412yes—
106C25C2×D411yescharacteristic
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—82no—
111C22×Q8C22×C412yescomplemented
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—28no—
120C23×C4—48no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4—84no—
124C23×C4—84no—
125C23×C4C22⋊C414yes—
126C23×C4C22×C412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C22×C8—44no—
136C22×C8—84no—
137C22⋊Q8—28no—
138C22⋊Q8—28no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—84no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4—84no—
155C2×C4⋊C4C22×C412yescomplemented
156C2×C4⋊C4C2×D418yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—416no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—84no—
184C2×C22⋊C4—84no—
185C2×C22⋊C4C22⋊C418yes—
186C2×C22⋊C4C22×C414yes—
187C2×C22⋊C4C2×D414yes—
188#103—88no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—44no—
192#95—28no—
193#95—28no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95C22×C414yescomplemented
198C24—22no—
199C24—24no—
200C24—24no—
201C24—24no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—24no—
206C24—28no—
207C24—24no—
208C24—24no—
209C24—28no—
210C24—28no—
211C24—42no—
212C24—42no—
213C24—44no—
214C24—44no—
215C24—44no—
216C24—44no—
217C24—82no—
218C24—82no—
219C24—84no—
220C24—84no—
221C24—88no—
222C24—88no—
223C24—84no—
224C24—84no—
225C24—88no—
226C24—88no—
227C24#103116yescomplemented
228C24C2×C22⋊C414yes—
229C24C2×C22⋊C412yes—
230C24#12011yescharacteristic
231C24#12011yescharacteristic
232C24#12312yes—
233C24C2×SD1614yes—
234C24C2×SD1612yes—
235C24C2×Q1614yes—
236C24C2×Q1612yes—
237C24AGL(1,ℤ/8)12yes—
238C24#13712yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—84no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—416no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—48no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—82no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—832no—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C4×D414yes—
286C22×C4C4×D414yes—
287C22×C4#12012yes—
288C22×C4#12114yes—
289C22×C4#12114yes—
290C22×C4#12312yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4112yes—
295C2×C8—48no—
296C2×C8—44no—
297C2×C8—88no—
298C4⋊C4—24no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—48no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—88no—
305C4⋊C4—88no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—216no—
309C22⋊C4—416no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—88no—
317C22⋊C4—88no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—216no—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—216no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—416no—
341C23—416no—
342C23—48no—
343C23—48no—
344C23—44no—
345C23—44no—
346C23—48no—
347C23—48no—
348C23—44no—
349C23—44no—
350C23—84no—
351C23—84no—
352C23—82no—
353C23—82no—
354C23—88no—
355C23—88no—
356C23—84no—
357C23—84no—
358C23—816no—
359C23—816no—
360C23—816no—
361C23—816no—
362C23#38612yes—
363C23#38612yes—
364C23C2×#10318yes—
365C23C2×#10314yes—
366C23#41714yes—
367C23#45012yes—
368C23#45012yes—
369C23#45112yes—
370C23#45112yes—
371C23#48314yes—
372C23#48414yes—
373C23C22×C22⋊C411yescharacteristic
374C23C2×#12011yescharacteristic
375C23C2×#12311yescharacteristic
376C23C22×SD1611yescharacteristic
377C23C22×Q1611yescharacteristic
378C23C2×AGL(1,ℤ/8)11yescharacteristic
379C23C2×#13711yescharacteristic
380Q8—48no—
381Q8—84no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—82no—
412C2×C4—816no—
413C2×C4—82no—
414C2×C4—832no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4#38614yes—
418C2×C4C2×C4×D412yes—
419C2×C4C2×#12112yes—
420C8—84no—
421C22—24no—
422C22—22no—
423C22—22no—
424C22—24no—
425C22—22no—
426C22—22no—
427C22—24no—
428C22—24no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—28no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—48no—
443C22—432no—
444C22—48no—
445C22—48no—
446C22—44no—
447C22—44no—
448C22—84no—
449C22—84no—
450C22—84no—
451C22—816no—
452C22#187214yes—
453C22#187214yes—
454C22C2×#38611yescharacteristic
455C22C22×#10311yescharacteristic
456C22C2×#41711yescharacteristic
457C22C2×#45011yescharacteristic
458C22C2×#45111yescharacteristic
459C22#301912yes—
460C22#302612yes—
461C22C2×#48311yescharacteristic
462C22C2×#48411yescharacteristic
463C4—22no—
464C4—22no—
465C4—24no—
466C4—44no—
467C4—84no—
468C4—88no—
469C4—82no—
470C4—82no—
471C2—21no—
472C2—21no—
473C2—24no—
474C2—21no—
475C2—21no—
476C2—24no—
477C2—48no—
478C2#1828512yes—
479C2#1828712yes—
480C2C2×#187211yescharacteristic
481C2#2095311yescharacteristic
482C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×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
3. #21759C2semidirect
5. #9021C2semidirect
6. #9013C2semidirect
7. #8023C2semidirect
13. #2626C4semidirect
28. #1824C22semidirect
29. #1787C22semidirect
32. #1294C22semidirect
43. C2×C22⋊Q8C2×C4semidirect
73. #399C2×C4semidirect
76. #393C2×C4semidirect
85. #380C2×C4semidirect
111. C22×Q8C22×C4semidirect
155. C2×C4⋊C4C22×C4semidirect
197. #95C22×C4semidirect
227. C24#103semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23152
#217592
C22×#3232
#90212
#90132
#80232