Finite Group Databasebeta

← #192736

Subgroups — #192736

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8896C214yesmaximal, complemented
7#8888C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7901C211yesmaximal, characteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#419—21no—
16Q8×C4⋊C4—22no—
17Q8×C4⋊C4—22no—
18Q8×C4⋊C4—22no—
19Q8×C4⋊C4C2214yes—
20C2×#389—22no—
21C2×#389—21no—
22#2285C2211yescharacteristic
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1948—22no—
28#1948C2212yes—
29#1948C2212yes—
30#1853—21no—
31#1853—21no—
32#1853—22no—
33#1844C2212yes—
34#1834C2212yes—
35#1805—22no—
36#1803C2212yes—
37#1803C2212yes—
38C4×#106—22no—
39#1736—21no—
40#1479C418yescomplemented
41C2×Q8⋊C8C2214yes—
42#1296C2214yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—44no—
53C2×C4×Q8—44no—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8Q814yes—
57C2×C4×Q8C2312yes—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431C2×C414yes—
75#431C2×C414yes—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—24no—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85#419—44no—
86C2×#103—22no—
87C2×#103—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#389—24no—
93#389—28no—
94#389—24no—
95#389—28no—
96#384—24no—
97#384—24no—
98#384—21no—
99#384—21no—
100#384D412yes—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2312yes—
104#382—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C2×#95—22no—
116C2×#95—21no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—22no—
121#340—22no—
122Q8⋊C8C2×C4116yescomplemented
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8C4⋊C414yes—
128C22×Q8SD1614yes—
129C22×Q8SD1614yes—
130C22×Q8C2×D412yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—42no—
140C2×M16—42no—
141C2×M16—42no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—48no—
153C4×Q8—416no—
154C4×Q8—48no—
155C4×Q8C4⋊C418yes—
156C4×Q8C22×C414yes—
157C4×Q8C2×Q814yes—
158#117—42no—
159#117—42no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—84no—
189C2×C4⋊C4C4⋊C414yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—41no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—48no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—41no—
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214#106—44no—
215#106—44no—
216#106—84no—
217C4⋊C8—24no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—42no—
222C4⋊C8C22×C414yes—
223#103—44no—
224#103—44no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C4×C8—24no—
228C4×C8—22no—
229C4×C8C22×C414yes—
230#95—28no—
231#95—22no—
232#95—22no—
233#95—41no—
234#95—41no—
235C24#12011yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—28no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—28no—
244C2×Q8—28no—
245C2×Q8#106116yescomplemented
246C2×Q8C2×C4⋊C414yes—
247C2×Q8C2×SD1614yes—
248C2×Q8C2×SD1614yes—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×C4⋊C412yes—
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4C22⋊Q812yes—
282C22×C4C22⋊Q812yes—
283C22×C4#12312yes—
284C22×C4C2×SD1612yes—
285C22×C4C2×SD1612yes—
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4#13712yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—84no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—416no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—88no—
328C4⋊C4C2×C4⋊C414yes—
329C4⋊C4#13714yes—
330C22⋊C4—82no—
331C42—24no—
332C42—24no—
333C42—28no—
334C42—416no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42—82no—
339C42C2×C4⋊C412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42C22⋊Q812yes—
343C42C22⋊Q812yes—
344C42#12312yes—
345C42C23×C411yescharacteristic
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#38512yes—
354C23#44812yes—
355C23#45012yes—
356C23C2×#12011yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—216no—
360Q8—24no—
361Q8C2×#10618yes—
362C2×C4—216no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—88no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4#38512yes—
401C2×C4C2×#10614yes—
402C2×C4#42814yes—
403C2×C4#44812yes—
404C2×C4#45012yes—
405C2×C4#48114yes—
406C2×C4#48314yes—
407C2×C4C22×C4⋊C411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C2×C4C22×SD1611yescharacteristic
414C2×C4C22×SD1611yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13712yes—
417C2×C4C2×#13711yescharacteristic
418C2×C4#57512yes—
419C8—44no—
420C8—44no—
421C8—82no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22#185314yes—
429C22C2×#38511yescharacteristic
430C22C2×#45011yescharacteristic
431C22C2×#44811yescharacteristic
432C22#300411yescharacteristic
433C22#300511yescharacteristic
434C22#301911yescharacteristic
435C22#302011yescharacteristic
436C4—28no—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—44no—
442C4—44no—
443C4—82no—
444C4—84no—
445C4—161no—
446C4—161no—
447C4#185314yes—
448C4C22×#10611yescharacteristic
449C4C2×#42811yescharacteristic
450C4C4×#13712yes—
451C4#307812yes—
452C4C2×#48311yescharacteristic
453C4C2×#48111yescharacteristic
454C2—24no—
455C2#1876711yescharacteristic
456C2#1876811yescharacteristic
457C2#1877711yescharacteristic
458C2#1877811yescharacteristic
459C2C2×#185311yescharacteristic
460C2#2086411yescharacteristic
461C2#2087411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8896C2semidirect
7. #8888C2semidirect
40. #1479C4semidirect
122. Q8⋊C8C2×C4semidirect
245. C2×Q8#106semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
#219812
#213592
#206472
C2×#14792
#88962
#88882
#80432
#79012