Finite Group Databasebeta

← #194193

Subgroups — #194193

Order 512.

1632 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
1C22×#461C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4#21833C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#8985C218yesmaximal, complemented
7#8039C211yesmaximal, characteristic, complemented
8#7893C211yesmaximal, characteristic, complemented
9C23×SD16—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C2×#518—21no—
12C22×C4×D4C2211yescharacteristic
13C2×#461—24no—
14C2×#461—22no—
15C2×#461C418yescomplemented
16C2×#461C2214yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103—21no—
21#2349—22no—
22#2349—22no—
23D4×C4⋊C4—22no—
24D4×C4⋊C4C2214yes—
25#2316—21no—
26#2315—22no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×#380—22no—
30#1990—22no—
31#1990C2214yescomplemented
32#1988—22no—
33#1988C2214yescomplemented
34#1823—22no—
35#1823—22no—
36#1478—24no—
37C2×D4⋊C8C2214yes—
38#1296C2214yescomplemented
39#1279C2214yescomplemented
40C23×Q8—21no—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C22×SD16—28no—
44C22×SD16—24no—
45C22×SD16—44no—
46C22×SD16—44no—
47C23×C8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yescomplemented
55C2×C4⋊Q8C2312yes—
56#518—44no—
57#518—44no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D414yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—24no—
76C22×C42C2311yescharacteristic
77#461—216no—
78#461—28no—
79#461C2×C4116yescomplemented
80#431—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2×C414yescomplemented
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×#103—28no—
91C2×#103—24no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#389—24no—
97#389—24no—
98#385—28no—
99#385—28no—
100#384D412yes—
101#384C2312yes—
102#382—21no—
103#382—21no—
104#380—24no—
105#380—24no—
106#380—21no—
107#380—21no—
108#380—24no—
109#380D412yes—
110#380D412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C2×#95—24no—
116#340—28no—
117#340—28no—
118D4⋊C8—28no—
119C25—21no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C22⋊C414yes—
133C22×D4D814yes—
134C22×D4SD1614yes—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—28no—
142C23×C4—41no—
143C23×C4—84no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×SD16—216no—
148C2×SD16—432no—
149C2×SD16—48no—
150C2×SD16—48no—
151C22×C8—24no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8C22×C414yes—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—48no—
175C4×D4—416no—
176C4×D4—48no—
177C4×D4C22⋊C418yes—
178C4×D4C22×C414yes—
179C4×D4C2×D414yes—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—88no—
200C2×C4⋊C4C22⋊C414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—416no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—42no—
227C4⋊C8C22×C414yes—
228#103—216no—
229C22⋊C8—42no—
230C22⋊C8—42no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—42no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239C24—22no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24—82no—
244C24—82no—
245C24#12011yescharacteristic
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—28no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4#102116yes—
269C2×D4C2×C22⋊C414yes—
270C2×D4C2×D814yes—
271C2×D4C2×SD1614yes—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—416no—
285C22×C4—88no—
286C22×C4—88no—
287C22×C4—84no—
288C22×C4—82no—
289C22×C4—88no—
290C22×C4—88no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12312yes—
302C22×C4C2×D812yes—
303C22×C4C2×SD1612yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4#13712yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309SD16—432no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—48no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—44no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4—88no—
334C4⋊C4C2×C22⋊C414yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—28no—
338C22⋊C4—416no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C42—42no—
342C42—42no—
343C42—44no—
344C42C2×C22⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42#12112yes—
348C42#12112yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×D411yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—22no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—216no—
360C23—44no—
361C23—44no—
362C23—84no—
363C23—84no—
364C23—88no—
365C23—88no—
366C23#38612yes—
367C23#44712yes—
368C23#44812yes—
369C23C2×#12011yescharacteristic
370Q8—44no—
371Q8—44no—
372Q8—82no—
373Q8—82no—
374Q8—88no—
375D4—24no—
376D4—24no—
377D4—24no—
378D4—216no—
379D4C2×#10218yes—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—88no—
399C2×C4—84no—
400C2×C4—88no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—1616no—
404C2×C4#38612yes—
405C2×C4#38612yes—
406C2×C4#38612yes—
407C2×C4C2×#10214yes—
408C2×C4#41714yes—
409C2×C4#44712yes—
410C2×C4#44812yes—
411C2×C4#48014yes—
412C2×C4#48114yes—
413C2×C4C22×C22⋊C411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12111yescharacteristic
418C2×C4C2×#12311yescharacteristic
419C2×C4C22×D811yescharacteristic
420C2×C4C22×SD1611yescharacteristic
421C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
422C2×C4C2×#13711yescharacteristic
423C2×C4#57714yes—
424C8—44no—
425C8—82no—
426C8—82no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—24no—
431C22—28no—
432C22—44no—
433C22—44no—
434C22—48no—
435C22—48no—
436C22—816no—
437C22#187414yes—
438C22C2×#38611yescharacteristic
439C22C2×#44711yescharacteristic
440C22C2×#44811yescharacteristic
441C22#300912yes—
442C22#302212yes—
443C4—22no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—44no—
448C4—84no—
449C4—84no—
450C4—84no—
451C4#187414yes—
452C4C22×#10211yescharacteristic
453C4C2×#41711yescharacteristic
454C4#294912yes—
455C4#304212yes—
456C4C2×#48011yescharacteristic
457C4C2×#48111yescharacteristic
458C2—24no—
459C2—48no—
460C2#1890212yes—
461C2#1890412yes—
462C2C2×#187411yescharacteristic
463C2#2093311yescharacteristic
464C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
4. #21833C2semidirect
6. #8985C2semidirect
7. #8039C2semidirect
8. #7893C2semidirect
15. C2×#461C4semidirect
31. #1990C22semidirect
33. #1988C22semidirect
38. #1296C22semidirect
39. #1279C22semidirect
54. C2×C4⋊Q8C2×C4semidirect
79. #461C2×C4semidirect
87. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4612
#341812
#218332
C2×#14782
#89852
#80392
#78932