Finite Group Databasebeta

← #239617

Subgroups — #239617

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic
7#10419C214yesmaximal, complemented
8#10418C214yesmaximal, complemented
9#7904C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2546—22no—
17#2546—22no—
18#2546—22no—
19#2546C2214yes—
20C2×#398—22no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1993—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1948C2212yes—
30#1947C2212yes—
31#1853—21no—
32#1852—21no—
33#1823—21no—
34#1803C2212yes—
35#1803C2212yes—
36#1787—21no—
37#1776—22no—
38#1763—22no—
39#1736—22no—
40#1556—24no—
41C2×#332C2214yes—
42#1491—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—48no—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2312yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#432—22no—
82#432D414yes—
83C2×#107—22no—
84C2×#106—22no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C4×M16—22no—
92C4×M16—22no—
93C2×#97—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8C2312yes—
97#398—28no—
98#398—28no—
99#391—24no—
100#391—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#382—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#332—28no—
126Q8⋊C8—28no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C2×D412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—416no—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—41no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—44no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—41no—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—44no—
227#106—44no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8C2×D414yes—
232#103—44no—
233#103—44no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236#97—24no—
237#97—42no—
238#97—42no—
239C4×C8—22no—
240C4×C8—22no—
241#95—28no—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—88no—
266C2×Q8—88no—
267C2×Q8AGL(1,ℤ/8)14yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308M16—48no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—416no—
327C4⋊C4—416no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4#13714yes—
343C4⋊C4#13714yes—
344C22⋊C4—82no—
345C42—44no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42—44no—
350C42—82no—
351C42#12112yes—
352C42#12112yes—
353C42#12112yes—
354C42#12312yes—
355C42#12312yes—
356C42#12412yes—
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C42C2×#4111yescharacteristic
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#39412yes—
366C23#45412yes—
367C23#45612yes—
368C23C2×#12011yescharacteristic
369Q8—24no—
370Q8—24no—
371Q8—48no—
372Q8—42no—
373Q8—42no—
374Q8—88no—
375Q8—84no—
376Q8—84no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—88no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4#39412yes—
418C2×C4#39412yes—
419C2×C4#39412yes—
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12311yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C2×C4C2×#12411yescharacteristic
426C2×C4C2×AGL(1,ℤ/8)12yes—
427C2×C4C2×#13712yes—
428C2×C4C2×#13712yes—
429C2×C4#57512yes—
430C8—44no—
431C8—44no—
432C8—82no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22#203814yes—
440C22C2×#39411yescharacteristic
441C22#300311yescharacteristic
442C22#300411yescharacteristic
443C22#300411yescharacteristic
444C22#300511yescharacteristic
445C22C2×#45411yescharacteristic
446C22C2×#45611yescharacteristic
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—22no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4—84no—
456C4—82no—
457C4—161no—
458C4—161no—
459C4#303912yes—
460C4#305512yes—
461C4#309412yes—
462C4#313912yes—
463C2—24no—
464C2#1853411yescharacteristic
465C2#1853511yescharacteristic
466C2#1854011yescharacteristic
467C2#1854111yescharacteristic
468C2#2348011yescharacteristic
469C2C2×#203811yescharacteristic
470C2#2350711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10419C2semidirect
8. #10418C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#213652
#213242
#207402
C2×#15562
C2×#14912
#104192
#104182
#79042