Finite Group Databasebeta

← #194010

Subgroups — #194010

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#21394C211yesmaximal, characteristic
4#21322C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8983C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic
9#7904C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3113—24no—
14#3113C418yescomplemented
15C2×#483—21no—
16C2×#475C2212yes—
17C2×#462C2212yes—
18C2×#451—21no—
19C2×#441—22no—
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23Q8×C4⋊C4—22no—
24Q8×C4⋊C4—22no—
25Q8×C4⋊C4—22no—
26Q8×C4⋊C4C2214yes—
27C2×#389—22no—
28C2×#389—21no—
29C2×#384—21no—
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1992—22no—
34#1990C2212yes—
35#1990C2212yes—
36#1988C2212yes—
37#1981C2212yes—
38#1872—21no—
39#1872—21no—
40#1763—22no—
41#1736—22no—
42#1491—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C23×Q8D411yescharacteristic
47C22×Q16—22no—
48C22×M16—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533C2×C414yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—48no—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#483—28no—
86#475—24no—
87#475C2×C418yescomplemented
88#462—24no—
89#462C2×C418yescomplemented
90#451—28no—
91#441—216no—
92#432—22no—
93#432—22no—
94#432—22no—
95#432C2×C414yes—
96C2×#106—22no—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C4×M16—22no—
106C4×M16—22no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8C2312yes—
111#389—24no—
112#389—28no—
113#389—24no—
114#389—28no—
115#384—24no—
116#384—24no—
117#384—21no—
118#384—21no—
119#384D412yes—
120#384C2312yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4C2312yes—
127C2×#95—22no—
128C2×#95—21no—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133Q8⋊C8—28no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C22⋊C414yes—
141C22×Q8C2×D412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×Q16—28no—
151C2×Q16—44no—
152C2×Q16—44no—
153C2×M16—24no—
154C2×M16—42no—
155C2×M16—42no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C22×C412yes—
166C4⋊Q8C22×C412yes—
167#125—24no—
168#125—42no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—416no—
183C4×Q8—48no—
184C4×Q8—48no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8C22⋊C418yes—
188C4×Q8C22×C414yes—
189C4×Q8C2×D414yes—
190#117—24no—
191#117—42no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—84no—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C22⋊C4—41no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—28no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—48no—
234C2×C42—44no—
235C2×C42—41no—
236C2×C42C22×C412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42C2411yescharacteristic
244#106—28no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8C22×C414yes—
249#103—28no—
250#103—28no—
251#103—44no—
252#103—44no—
253C22⋊C8—24no—
254C22⋊C8—42no—
255#97—24no—
256#97—42no—
257#97—42no—
258C4×C8—22no—
259C4×C8—22no—
260#95—28no—
261#95—22no—
262#95—22no—
263C24#12011yescharacteristic
264C2×Q8—24no—
265C2×Q8—216no—
266C2×Q8—24no—
267C2×Q8—216no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8C2×C22⋊C414yes—
281C2×Q8AGL(1,ℤ/8)14yes—
282C2×Q8#13714yes—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12312yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323Q16—416no—
324M16—48no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—416no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4—88no—
362C4⋊C4C2×C22⋊C414yes—
363C4⋊C4#13714yes—
364C22⋊C4—44no—
365C42—24no—
366C42—24no—
367C42—28no—
368C42—416no—
369C42—44no—
370C42—44no—
371C42—44no—
372C42C2×C22⋊C412yes—
373C42C4×D412yes—
374C42C4×D412yes—
375C42#12112yes—
376C42#12112yes—
377C42#12312yes—
378C42C23×C411yescharacteristic
379C42C22×D411yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23#38612yes—
387C23#45512yes—
388C23#45612yes—
389C23C2×#12011yescharacteristic
390Q8—216no—
391Q8—24no—
392Q8—24no—
393Q8—24no—
394Q8—44no—
395Q8—44no—
396Q8—82no—
397Q8—82no—
398Q8#41718yes—
399C2×C4—216no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4—84no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#38612yes—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4C22×C22⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)12yes—
449C2×C4C2×#13712yes—
450C2×C4C2×#13712yes—
451C2×C4#57512yes—
452C8—44no—
453C8—44no—
454C8—82no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22#188214yes—
462C22C2×#38611yescharacteristic
463C22#300311yescharacteristic
464C22#300411yescharacteristic
465C22#300411yescharacteristic
466C22#300511yescharacteristic
467C22C2×#45511yescharacteristic
468C22C2×#45611yescharacteristic
469C4—28no—
470C4—22no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—84no—
479C4—82no—
480C4—82no—
481C4C2×#41712yes—
482C4C4×#13712yes—
483C4#303712yes—
484C4#309212yes—
485C2—24no—
486C2#1854011yescharacteristic
487C2#1854111yescharacteristic
488C2#1855011yescharacteristic
489C2#1855111yescharacteristic
490C2#2090911yescharacteristic
491C2#2092611yescharacteristic
492C2C2×#188211yescharacteristic
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. #8987C2semidirect
7. #8983C2semidirect
14. #3113C4semidirect
87. #475C2×C4semidirect
89. #462C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×Q8×C4⋊C42
#213942
#213222
C2×#14912
#89872
#89832
#80402
#79042