Finite Group Databasebeta

← #194837

Subgroups — #194837

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3#21758C212yesmaximal
4C22×#324C211yesmaximal, characteristic, complemented
5#9027C214yesmaximal, complemented
6#9019C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C22×#117—21no—
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2618—24no—
14#2618C418yescomplemented
15C2×#398C2212yes—
16C2×#396—22no—
17C2×#394C2212yes—
18#2316—24no—
19#2316—22no—
20#2316C2214yes—
21C2×#389—24no—
22C2×#386—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26#1893—24no—
27#1874—24no—
28#1823C2214yes—
29#1786C2214yes—
30C2×#324—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×D4—21no—
34C23×D4C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×#123—28no—
40C2×#123—44no—
41C2×#123—44no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—24no—
46C2×#121C2×C414yes—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—44no—
50C2×#117—44no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C2×#102—28no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75#398—24no—
76#398C2×C418yescomplemented
77#396—216no—
78#394—24no—
79#394C2×C418yescomplemented
80#389—216no—
81#389—216no—
82#386—28no—
83#380—24no—
84#380—22no—
85#380—22no—
86#380—24no—
87#380—22no—
88#380C2×C414yes—
89#380D414yes—
90#380C2314yes—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95C2312yes—
95#340—28no—
96#324—28no—
97C25—21no—
98C25C2×D411yescharacteristic
99C22×D4—24no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—41no—
104C22×D4—41no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—82no—
111C22×D4C22×C412yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22⋊C414yes—
126C23×C4C22⋊C414yes—
127C23×C4D814yes—
128C23×C4SD1614yes—
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C22×C8—24no—
139C22×C8—44no—
140C22×C8—84no—
141#123—432no—
142#121—28no—
143#121—28no—
144#121—44no—
145#121—44no—
146#117—48no—
147#117—416no—
148#117—48no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C22⋊C418yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D418yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—84no—
191C2×C42—28no—
192C2×C42—48no—
193#102—88no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197#95—28no—
198#95—28no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95C22×C414yes—
203C24—22no—
204C24—42no—
205C24—42no—
206C24—44no—
207C24—44no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24C2×C22⋊C412yes—
215C24#12011yescharacteristic
216C24#12011yescharacteristic
217C24#12312yes—
218C24C2×D812yes—
219C24C2×SD1612yes—
220C24AGL(1,ℤ/8)12yes—
221C24#13712yes—
222C24C22×D411yescharacteristic
223C2×D4—24no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—48no—
228C2×D4—48no—
229C2×D4—42no—
230C2×D4—82no—
231C2×D4—84no—
232C2×D4—82no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—84no—
238C2×D4—84no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—48no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—82no—
283C22×C4—84no—
284C22×C4#102116yescomplemented
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C414yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C4×D414yes—
289C22×C4C4×D414yes—
290C22×C4#12012yes—
291C22×C4#12114yes—
292C22×C4#12114yes—
293C22×C4#12312yes—
294C22×C4C2×D814yes—
295C22×C4C2×SD1614yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4112yes—
300C2×C8—48no—
301C2×C8—44no—
302C2×C8—88no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—216no—
306C4⋊C4—24no—
307C4⋊C4—48no—
308C4⋊C4—416no—
309C4⋊C4—42no—
310C4⋊C4—416no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—88no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—416no—
320C22⋊C4—88no—
321C42—416no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—82no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—82no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23—82no—
353C23#38612yes—
354C23#38612yes—
355C23C2×#10214yes—
356C23#41714yes—
357C23#44712yes—
358C23#44712yes—
359C23#44812yes—
360C23#44812yes—
361C23#48014yes—
362C23#48114yes—
363C23C22×C22⋊C411yescharacteristic
364C23C2×#12011yescharacteristic
365C23C2×#12311yescharacteristic
366C23C22×D811yescharacteristic
367C23C22×SD1611yescharacteristic
368C23C2×AGL(1,ℤ/8)11yescharacteristic
369C23C2×#13711yescharacteristic
370D4—48no—
371D4—84no—
372D4—88no—
373D4—88no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—216no—
378C2×C4—216no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
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—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—816no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4#38614yes—
416C2×C4C2×#10218yes—
417C2×C4C2×C4×D412yes—
418C2×C4C2×#12112yes—
419C8—84no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—22no—
425C22—22no—
426C22—24no—
427C22—24no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—21no—
433C22—22no—
434C22—21no—
435C22—22no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—84no—
441C22—84no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—162no—
446C22—162no—
447C22—162no—
448C22—162no—
449C22#187414yes—
450C22#187414yes—
451C22C2×#38611yescharacteristic
452C22C22×#10211yescharacteristic
453C22C2×#41711yescharacteristic
454C22C2×#44711yescharacteristic
455C22C2×#44811yescharacteristic
456C22#302012yes—
457C22#302612yes—
458C22C2×#48011yescharacteristic
459C22C2×#48111yescharacteristic
460C4—28no—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—44no—
465C4—84no—
466C4—88no—
467C4—82no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—24no—
473C2—24no—
474C2—162no—
475C2#1829212yes—
476C2#1829412yes—
477C2C2×#187411yescharacteristic
478C2#2095411yescharacteristic
479C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C2×#2316C2semidirect
4. C22×#324C2semidirect
5. #9027C2semidirect
6. #9019C2semidirect
7. #8023C2semidirect
14. #2618C4semidirect
76. #398C2×C4semidirect
79. #394C2×C4semidirect
284. C22×C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#23162
#217582
C22×#3242
#90272
#90192
#80232