Finite Group Databasebeta

← #192621

Subgroups — #192621

Order 512.

1578 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×D4×C4⋊C4C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic, complemented
3#20764C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1490C211yesmaximal, characteristic
6#8894C214yesmaximal, complemented
7#8890C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4—21no—
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16D4×C4⋊C4—22no—
17D4×C4⋊C4—22no—
18D4×C4⋊C4—22no—
19D4×C4⋊C4C2214yes—
20C2×#389—21no—
21C2×#385—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948C2212yescomplemented
27#1947C2212yescomplemented
28#1853—21no—
29#1852—21no—
30#1836—22no—
31#1834C2212yescomplemented
32#1834C2212yescomplemented
33#1824—21no—
34#1802C2212yes—
35#1802C2212yes—
36#1786—21no—
37#1776—22no—
38#1762—22no—
39#1736—22no—
40#1490C418yescomplemented
41C2×D4⋊C8C2214yes—
42#1296C2214yescomplemented
43#1279C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C22×M16—21no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—48no—
55C2×C4×D4C2×C414yes—
56C2×C4×D4D412yes—
57C2×C4×D4Q814yes—
58C2×C4×D4C2312yes—
59C2×#117—21no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—48no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#432—22no—
77#432C2×C414yescomplemented
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#102—22no—
83C2×#102—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—22no—
87C4×M16C2×C414yescomplemented
88C2×#97—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×C4×C8C2312yes—
92#389—24no—
93#389—24no—
94#385—28no—
95#385—28no—
96#384—21no—
97#384—21no—
98#384D412yes—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2312yes—
102#382—21no—
103#380—24no—
104#380—24no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C2×#95—22no—
114C2×#95—21no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121D4⋊C8C2×C4116yescomplemented
122C25—21no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4C4⋊C414yes—
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—28no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—48no—
141C23×C4—48no—
142C23×C4—84no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—416no—
160C4×D4—48no—
161C4×D4—48no—
162C4×D4C4⋊C418yes—
163C4×D4C22×C414yes—
164C4×D4C2×Q814yes—
165#117—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—416no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C4⋊C414yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—44no—
206C2×C42—41no—
207C2×C42C22×C412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215#107—44no—
216#106—44no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8C22×C414yescomplemented
221#102—44no—
222#102—44no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225#97—24no—
226#97—24no—
227#97—42no—
228C4×C8—22no—
229C4×C8C22×C414yescomplemented
230#95—28no—
231#95—22no—
232#95—22no—
233#95—41no—
234#95—41no—
235C24—24no—
236C24—24no—
237C24—22no—
238C24—24no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24#12011yescharacteristic
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—216no—
247C2×D4—216no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—24no—
252C2×D4C2×C4⋊C414yes—
253C2×D4AGL(1,ℤ/8)14yes—
254C2×D4#13714yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—416no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—88no—
291C22×C4—832no—
292C22×C4C2×C4⋊C412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4#12312yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304M16—48no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—416no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4C2×C4⋊C414yes—
335C4⋊C4AGL(1,ℤ/8)14yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—28no—
339C22⋊C4—416no—
340C22⋊C4—82no—
341C42—44no—
342C42—44no—
343C42—82no—
344C42C2×C4⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×Q811yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—216no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—48no—
363C23—48no—
364C23—48no—
365C23—48no—
366C23—48no—
367C23—48no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23#38512yes—
373C23#45412yes—
374C23#45512yes—
375C23C2×#12011yescharacteristic
376D4—216no—
377D4—24no—
378D4—24no—
379D4—24no—
380D4#42818yes—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—416no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—832no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#38512yes—
422C2×C4#38512yes—
423C2×C4#38512yes—
424C2×C4C22×C4⋊C411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)12yes—
431C2×C4C2×AGL(1,ℤ/8)12yes—
432C2×C4C2×#13712yes—
433C2×C4#57512yes—
434C8—44no—
435C8—44no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—28no—
444C22—432no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22#186014yes—
452C22C2×#38511yescharacteristic
453C22#300311yescharacteristic
454C22#300311yescharacteristic
455C22#300411yescharacteristic
456C22#300511yescharacteristic
457C22C2×#45511yescharacteristic
458C22C2×#45411yescharacteristic
459C4—22no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—22no—
464C4—44no—
465C4—44no—
466C4—84no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4C2×#42812yes—
471C4C4×AGL(1,ℤ/8)12yes—
472C4#307712yes—
473C4#309212yes—
474C2—24no—
475C2—48no—
476C2#1852911yescharacteristic
477C2#1853011yescharacteristic
478C2#1853311yescharacteristic
479C2#1853411yescharacteristic
480C2#2084411yescharacteristic
481C2C2×#186011yescharacteristic
482C2#2087311yescharacteristic
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
2. #21365C2semidirect
3. #20764C2semidirect
6. #8894C2semidirect
7. #8890C2semidirect
8. #8040C2semidirect
9. #7904C2semidirect
26. #1948C22semidirect
27. #1947C22semidirect
31. #1834C22semidirect
32. #1834C22semidirect
40. #1490C4semidirect
42. #1296C22semidirect
43. #1279C22semidirect
77. #432C2×C4semidirect
87. C4×M16C2×C4semidirect
121. D4⋊C8C2×C4semidirect
220. C4⋊C8C22×C4semidirect
229. C4×C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
#213652
#207642
#207382
C2×#14902
#88942
#88902
#80402
#79042