Finite Group Databasebeta

← #194108

Subgroups — #194108

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8982C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498—21no—
15C2×#487—21no—
16C2×#483—22no—
17C2×#475C2214yes—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#419—21no—
21C2×#419—21no—
22C22×C4×C8C2211yescharacteristic
23#2360—22no—
24#2360—22no—
25#2355—21no—
26#2355C2212yes—
27Q8×C4⋊C4—21no—
28Q8×C4⋊C4C2212yes—
29#2330—22no—
30#2317—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C2×#384C2211yescharacteristic
34#2274—21no—
35C4×#117C2211yescharacteristic
36#1990—22no—
37#1990C2214yes—
38#1981—21no—
39#1981—21no—
40#1981C2212yes—
41#1981C2212yes—
42C2×#340—22no—
43#1493—24no—
44C2×Q8⋊C8C2214yes—
45#1296C2214yes—
46#1279C2212yes—
47#1279C2212yes—
48C23×Q8—21no—
49C22×M16—21no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54#533D414yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2312yes—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C22×C42C2311yescharacteristic
81#498—28no—
82#487—28no—
83#483—216no—
84#475—28no—
85#475C2×C4116yescomplemented
86#431—22no—
87#431C2×C414yes—
88C2×#106—24no—
89#423—22no—
90#423—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94#419—44no—
95#419—44no—
96C2×#103—24no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—24no—
113#389—22no—
114#389—24no—
115#387—24no—
116#387—24no—
117#385—24no—
118#384—22no—
119#384—21no—
120#384—21no—
121#384D412yes—
122#384C2312yes—
123#382—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—24no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#340—28no—
140Q8⋊C8—28no—
141C22×Q8—21no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—41no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C2×D412yes—
151C23×C4—22no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—42no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—24no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—42no—
164C22×C8—42no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22×C414yes—
179#125—24no—
180#125—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8C22⋊C418yes—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—84no—
190C4×Q8—84no—
191C4×Q8C22×C414yes—
192C4×Q8C2×D414yes—
193#117—24no—
194#117—24no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117SD1614yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42C22×C412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42C2411yescharacteristic
249#106—216no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—42no—
254C4⋊C8C22×C414yes—
255#103—216no—
256#103—44no—
257#103—44no—
258#103—44no—
259#103—44no—
260C22⋊C8—24no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—42no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273C24#12011yescharacteristic
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8C2×C22⋊C414yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4C2×C22⋊C412yes—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12312yes—
321C22×C4C2×SD1612yes—
322C22×C4#13512yes—
323C22×C4AGL(1,ℤ/8)12yes—
324C22×C4#13712yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328M16—84no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—48no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4C2×C22⋊C414yes—
368C4⋊C4C2×SD1612yes—
369C4⋊C4C2×SD1612yes—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—84no—
382C42—84no—
383C42C2×C22⋊C412yes—
384C42C4×D412yes—
385C42C4×D412yes—
386C42#12112yes—
387C42#12112yes—
388C42#12312yes—
389C42C23×C411yescharacteristic
390C42C22×D411yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#38612yes—
398C23#44812yes—
399C23#44912yes—
400C23C2×#12011yescharacteristic
401Q8—28no—
402Q8—44no—
403Q8—48no—
404Q8—44no—
405Q8—44no—
406Q8—88no—
407Q8—82no—
408Q8—82no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—88no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—164no—
443C2×C4#38612yes—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4C4×SD1614yes—
447C2×C4#44014yes—
448C2×C4#44812yes—
449C2×C4#44912yes—
450C2×C4#45512yes—
451C2×C4#45612yes—
452C2×C4#46014yes—
453C2×C4#46414yes—
454C2×C4C22×C22⋊C411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C2×C4C22×SD1611yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57712yes—
465C2×C4#57812yes—
466C8—44no—
467C8—44no—
468C8—82no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22#188014yes—
476C22C2×#38611yescharacteristic
477C22C2×#44811yescharacteristic
478C22C2×#44911yescharacteristic
479C22#300911yescharacteristic
480C22#301111yescharacteristic
481C22C2×#45511yescharacteristic
482C22C2×#45611yescharacteristic
483C4—22no—
484C4—21no—
485C4—21no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—84no—
491C4—81no—
492C4—84no—
493C4—81no—
494C4#188014yes—
495C4#188214yes—
496C4#289812yes—
497C4C2×C4×SD1611yescharacteristic
498C4C2×#44011yescharacteristic
499C4C2×#46011yescharacteristic
500C4C2×#46411yescharacteristic
501C4#309712yes—
502C2—24no—
503C2#1863211yescharacteristic
504C2#1863411yescharacteristic
505C2#1865211yescharacteristic
506C2#1865411yescharacteristic
507C2#2091311yescharacteristic
508C2C2×#188011yescharacteristic
509C2C2×#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. #8983C2semidirect
7. #8982C2semidirect
13. #3229C4semidirect
85. #475C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#342142
C2×#19902
#212592
C2×#14932
#89832
#89822
#80432
#78902