Finite Group Databasebeta

← #194767

Subgroups — #194767

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#20732C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic, complemented
5#9025C214yesmaximal, complemented
6#9017C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C22×#121—21no—
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2212yes—
13#2535—24no—
14#2535C418yescomplemented
15C2×#397C2212yes—
16C2×#392C2212yes—
17#2349—24no—
18D4×C4⋊C4—24no—
19#2316—21no—
20#2315C2214yes—
21C2×#386—22no—
22#2285C2212yes—
23C2×#382C2211yescharacteristic
24C2×#380—22no—
25C2×#380—21no—
26#1823C2214yes—
27#1802—24no—
28#1786C2214yes—
29C4×#102—24no—
30#1460—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×D4—22no—
34C23×D4D411yescharacteristic
35C23×D4C2311yescharacteristic
36C24×C4—21no—
37C24×C4D412yes—
38C2×#121—28no—
39C2×#121—44no—
40C2×#121—44no—
41#518—44no—
42#518—44no—
43C2×C4×D4—28no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4C2×C414yescomplemented
53C2×C4×D4D414yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—24no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4C2312yes—
64C22×C42C2311yescharacteristic
65#431—24no—
66#431—24no—
67C2×C4⋊C8—22no—
68C2×#102—28no—
69C2×C22⋊C8C2314yes—
70C2×C4×C8—22no—
71#397—24no—
72#397C2×C418yescomplemented
73#392—24no—
74#392C2×C418yescomplemented
75#389—28no—
76#386—216no—
77#385—216no—
78#384D414yescomplemented
79#382—22no—
80#382—22no—
81#382C2×C414yescomplemented
82#380—24no—
83#380—24no—
84#380—24no—
85#380—22no—
86#380—28no—
87#380C2314yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yescomplemented
90C2×#95—24no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95C2312yes—
94#340—28no—
95#323—28no—
96C25—21no—
97C25C2×D412yes—
98C22×D4—28no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—22no—
106C22×D4—44no—
107C22×D4—48no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—84no—
111C22×D4C22⋊C414yes—
112C22×D4D814yes—
113C22×D4SD1614yes—
114C22×D4C22×C412yes—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—28no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—84no—
129C23×C4C22⋊C418yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135#121—432no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—432no—
143C4×D4—44no—
144C4×D4—44no—
145#117—44no—
146#117—44no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—88no—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D418yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—84no—
177C2×C22⋊C4C22×C414yes—
178C2×C22⋊C4C2×D414yes—
179C2×C42—24no—
180C2×C42C22×C412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42#4114yes—
185C4⋊C8—44no—
186#102—416no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C4×C8—44no—
190#95—28no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—24no—
196#95C22×C414yes—
197C24—22no—
198C24—22no—
199C24—22no—
200C24—22no—
201C24—24no—
202C24—22no—
203C24—44no—
204C24—44no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24C2×C22⋊C414yes—
215C24C2×D814yes—
216C24C2×SD1614yes—
217C24C22×D411yescharacteristic
218C2×D4—28no—
219C2×D4—24no—
220C2×D4—28no—
221C2×D4—28no—
222C2×D4—28no—
223C2×D4—28no—
224C2×D4—28no—
225C2×D4—28no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—48no—
229C2×D4—44no—
230C2×D4—416no—
231C2×D4—416no—
232C2×D4—42no—
233C2×D4—88no—
234C2×D4—816no—
235C2×D4—816no—
236C2×D4#102116yescomplemented
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—416no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—84no—
263C22×C4—88no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4—88no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12012yes—
277C22×C4#12312yes—
278C22×C4#12312yes—
279C22×C4C2×D812yes—
280C22×C4C2×SD1612yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4112yes—
287C2×C8—48no—
288C2×C8—44no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—42no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—88no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—416no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—416no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—88no—
306C42—48no—
307C42C4×D414yes—
308C42C4×D414yes—
309C42#12114yes—
310C42#12114yes—
311C23—22no—
312C23—22no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—44no—
321C23—44no—
322C23—48no—
323C23—48no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—42no—
335C23—42no—
336C23—44no—
337C23—42no—
338C23—82no—
339C23—84no—
340C23—84no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23C2×#10218yes—
353C23C22×C22⋊C411yescharacteristic
354C23C2×#12011yescharacteristic
355C23C2×#12311yescharacteristic
356C23C22×D811yescharacteristic
357C23C22×SD1611yescharacteristic
358C23C2×AGL(1,ℤ/8)11yescharacteristic
359C23C2×#13711yescharacteristic
360D4—28no—
361D4—232no—
362D4—48no—
363D4—44no—
364D4—832no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—416no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—88no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—816no—
395C2×C4—1616no—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38614yes—
399C2×C4C2×#10214yes—
400C2×C4#41714yes—
401C2×C4#44714yes—
402C2×C4#44814yes—
403C2×C4#48014yes—
404C2×C4#48114yes—
405C2×C4C2×C4×D412yes—
406C2×C4C2×#12112yes—
407C8—84no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—28no—
417C22—48no—
418C22—44no—
419C22—44no—
420C22—48no—
421C22—48no—
422C22—44no—
423C22—44no—
424C22—44no—
425C22—84no—
426C22—84no—
427C22—84no—
428C22—82no—
429C22—82no—
430C22—84no—
431C22—84no—
432C22—84no—
433C22—84no—
434C22—82no—
435C22—82no—
436C22C2×#38611yescharacteristic
437C22C22×#10211yescharacteristic
438C22C2×#41711yescharacteristic
439C22C2×#44711yescharacteristic
440C22C2×#44811yescharacteristic
441C22#301912yes—
442C22#302512yes—
443C22C2×#48011yescharacteristic
444C22C2×#48111yescharacteristic
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—44no—
450C4—84no—
451C4—88no—
452C4—82no—
453C4#187418yes—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—48no—
459C2—84no—
460C2#1821012yes—
461C2#1821212yes—
462C2C2×#187411yescharacteristic
463C2#2095311yescharacteristic
464C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #20732C2semidirect
4. C2×#1460C2semidirect
5. #9025C2semidirect
6. #9017C2semidirect
7. #8026C2semidirect
14. #2535C4semidirect
52. C2×C4×D4C2×C4semidirect
72. #397C2×C4semidirect
74. #392C2×C4semidirect
78. #384D4semidirect
81. #382C2×C4semidirect
89. C4×C4⋊C4D4semidirect
236. C2×D4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
#341812
#207322
C2×#14602
#90252
#90172
#80262