Finite Group Databasebeta

← #192628

Subgroups — #192628

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic, complemented
3#20764C211yesmaximal, characteristic, complemented
4#20738C211yesmaximal, characteristic
5C2×#1490C211yesmaximal, characteristic
6#8897C214yesmaximal, complemented
7#8893C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic, complemented
9#7904C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117—21no—
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2354—22no—
16#2354—22no—
17#2354—22no—
18#2354C2214yes—
19C2×#390—21no—
20C2×#388—22no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380—21no—
24C2×C4×C4⋊C4—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—
46C22×M16—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4D412yes—
53C2×C4×D4Q814yes—
54C2×C4×D4C2312yes—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—21no—
58C2×#117—48no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432—22no—
73#432—22no—
74#432C2×C414yes—
75C2×#107—22no—
76C2×#106—22no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#102—22no—
80C2×#102—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—22no—
84C4×M16C2×C414yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8C2312yes—
89#390—24no—
90#390—24no—
91#388—28no—
92#388—28no—
93#384—21no—
94#384—21no—
95#384D412yes—
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#384C2312yes—
99#382—21no—
100#380—24no—
101#380—24no—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C2×#95—22no—
113C2×#95—21no—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120D4⋊C8C2×C4116yescomplemented
121C25—21no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C4⋊C414yes—
127C22×D4C2×D412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4C4⋊C418yes—
155C4×D4C22×C414yes—
156C4×D4C2×Q814yes—
157#117—416no—
158#117—48no—
159#117—48no—
160#117—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C4⋊C414yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—48no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—84no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—48no—
199C2×C42—44no—
200C2×C42—41no—
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#107—44no—
210#106—44no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8C22×C414yes—
215#102—44no—
216#102—44no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219#97—24no—
220#97—24no—
221#97—42no—
222C4×C8—22no—
223C4×C8C22×C414yes—
224#95—28no—
225#95—22no—
226#95—22no—
227#95—41no—
228#95—41no—
229C24—22no—
230C24—24no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24#12011yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—48no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—48no—
248C2×D4C2×C4⋊C414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—416no—
267C22×C4—416no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C2×C4⋊C412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4#12312yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—48no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4C2×C4⋊C414yes—
324C4⋊C4AGL(1,ℤ/8)14yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—28no—
328C22⋊C4—416no—
329C22⋊C4—88no—
330C22⋊C4—82no—
331C42—44no—
332C42—416no—
333C42—44no—
334C42—82no—
335C42C2×C4⋊C412yes—
336C42C4×D412yes—
337C42C4×D412yes—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42#12312yes—
341C42C23×C411yescharacteristic
342C42C22×Q811yescharacteristic
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—24no—
350C23—48no—
351C23—48no—
352C23—48no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23#38512yes—
364C23#45412yes—
365C23#45512yes—
366C23C2×#12011yescharacteristic
367D4—24no—
368D4—24no—
369D4—48no—
370D4—42no—
371D4—42no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—416no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—88no—
404C2×C4—84no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4#38512yes—
412C2×C4#38512yes—
413C2×C4#38512yes—
414C2×C4C22×C4⋊C411yescharacteristic
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)12yes—
421C2×C4#57512yes—
422C2×C4#57512yes—
423C2×C4#57512yes—
424C8—44no—
425C8—44no—
426C8—82no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—816no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22#186014yes—
442C22C2×#38511yescharacteristic
443C22#300311yescharacteristic
444C22#300511yescharacteristic
445C22#300511yescharacteristic
446C22#300511yescharacteristic
447C22C2×#45511yescharacteristic
448C22C2×#45411yescharacteristic
449C4—22no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—44no—
455C4—44no—
456C4—84no—
457C4—82no—
458C4—161no—
459C4—161no—
460C4#291112yes—
461C4C4×AGL(1,ℤ/8)12yes—
462C4#307712yes—
463C4#309312yes—
464C2—24no—
465C2—84no—
466C2#1853311yescharacteristic
467C2#1853611yescharacteristic
468C2#1855011yescharacteristic
469C2#1855211yescharacteristic
470C2#2084511yescharacteristic
471C2C2×#186011yescharacteristic
472C2#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. #8897C2semidirect
7. #8893C2semidirect
8. #8040C2semidirect
9. #7904C2semidirect
26. #1948C22semidirect
27. #1947C22semidirect
31. #1834C22semidirect
32. #1834C22semidirect
40. #1490C4semidirect
42. #1296C22semidirect
43. #1279C22semidirect
120. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#213652
#207642
#207382
C2×#14902
#88972
#88932
#80402
#79042