Finite Group Databasebeta

← #194136

Subgroups — #194136

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21385C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8975C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7901C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3078—24no—
14#3078C418yescomplemented
15C2×#479—22no—
16C2×#477C2212yes—
17C2×#475C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C2×#419—21no—
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#1990C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1956C2212yes—
37#1872—22no—
38#1872—21no—
39#1872—21no—
40#1803—22no—
41C4×#103—22no—
42#1479—24no—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45#1279C2214yes—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—44no—
66C2×C4×Q8—44no—
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#479—216no—
86#477—24no—
87#477C2×C418yescomplemented
88#475—24no—
89#475C2×C418yescomplemented
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#428—28no—
95C2×#107—22no—
96C2×#106—22no—
97#423—22no—
98#423C2×C414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103#419—28no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2312yes—
113#389—24no—
114#389—28no—
115#389—24no—
116#389—28no—
117#384—24no—
118#384—24no—
119#384—21no—
120#384—21no—
121#384D412yes—
122#384C2312yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C2×#95—22no—
130C2×#95—21no—
131#340—24no—
132#340—22no—
133#340—22no—
134Q8⋊C8—28no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8C22⋊C414yes—
142C22×Q8SD1614yes—
143C22×Q8Q1614yes—
144C22×Q8C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
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—48no—
183C4×Q8—416no—
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—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—84no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—41no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—28no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—48no—
235C2×C42—41no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246#107—28no—
247#106—28no—
248C4⋊C8—24no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—42no—
253C4⋊C8C22×C414yes—
254#103—28no—
255#103—28no—
256#103—44no—
257#103—44no—
258#103—84no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—42no—
264#95—28no—
265#95—22no—
266#95—22no—
267C24#12011yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—28no—
276C2×Q8—28no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8#103116yescomplemented
286C2×Q8C2×C22⋊C414yes—
287C2×Q8C2×SD1614yes—
288C2×Q8C2×Q1614yes—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4C2×C22⋊C412yes—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12312yes—
324C22×C4C2×SD1612yes—
325C22×C4C2×Q1612yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4#13712yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—48no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—416no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—88no—
370C4⋊C4C2×C22⋊C414yes—
371C4⋊C4#13714yes—
372C22⋊C4—44no—
373C42—24no—
374C42—24no—
375C42—28no—
376C42—416no—
377C42—42no—
378C42—42no—
379C42—44no—
380C42—44no—
381C42C2×C22⋊C412yes—
382C42C4×D412yes—
383C42C4×D412yes—
384C42#12112yes—
385C42#12112yes—
386C42#12312yes—
387C42C23×C411yescharacteristic
388C42C22×D411yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23#38612yes—
396C23#45012yes—
397C23#45112yes—
398C23C2×#12011yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—24no—
402Q8—216no—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407Q8C2×#10318yes—
408C2×C4—216no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—84no—
439C2×C4—88no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4C2×#10314yes—
448C2×C4#41714yes—
449C2×C4#45012yes—
450C2×C4#45112yes—
451C2×C4#48314yes—
452C2×C4#48414yes—
453C2×C4C22×C22⋊C411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C2×C4C22×SD1611yescharacteristic
460C2×C4C22×Q1611yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4C2×#13712yes—
464C2×C4#57512yes—
465C8—44no—
466C8—82no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22#187214yes—
475C22C2×#38611yescharacteristic
476C22C2×#45011yescharacteristic
477C22C2×#45111yescharacteristic
478C22#300411yescharacteristic
479C22#300511yescharacteristic
480C22#301911yescharacteristic
481C22#302011yescharacteristic
482C4—28no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—22no—
487C4—44no—
488C4—44no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4#187214yes—
494C4C22×#10311yescharacteristic
495C4C2×#41711yescharacteristic
496C4C4×#13712yes—
497C4#303712yes—
498C4C2×#48311yescharacteristic
499C4C2×#48411yescharacteristic
500C2—24no—
501C2#1876711yescharacteristic
502C2#1876911yescharacteristic
503C2#1877711yescharacteristic
504C2#1877911yescharacteristic
505C2C2×#187211yescharacteristic
506C2#2092611yescharacteristic
507C2#2094311yescharacteristic
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. #8975C2semidirect
14. #3078C4semidirect
87. #477C2×C4semidirect
89. #475C2×C4semidirect
285. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×Q8×C4⋊C42
#218602
#213852
C2×#14792
#89832
#89752
#80432
#79012