Finite Group Databasebeta

← #195347

Subgroups — #195347

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#9044C218yesmaximal
7#8014C211yesmaximal, characteristic
8#7908C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3229—24no—
12#3229C418yescomplemented
13C2×#498—21no—
14C2×#487—21no—
15C2×#483C2214yes—
16C2×#475—22no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2370—22no—
20#2369—21no—
21#2365—21no—
22#2355—21no—
23#2355—22no—
24Q8×C4⋊C4—21no—
25#2329C2214yes—
26#2317—21no—
27C4×C22⋊Q8—22no—
28C2×#391—21no—
29C2×#391—21no—
30#2285C2211yescharacteristic
31#2274—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1990—21no—
35#1990—21no—
36#1988—22no—
37#1823C2214yes—
38#1503—24no—
39C2×#328C2214yes—
40#1445C2214yes—
41C4⋊#95—22no—
42#1294C2214yes—
43#1280C2214yes—
44C23×Q8—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533D414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2312yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C42—21no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#498—28no—
76#487—28no—
77#483—28no—
78#483C2×C4116yes—
79#475—216no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2×C414yes—
84C2×#106—24no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×#103—24no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#391—22no—
93#391—24no—
94#391—44no—
95#391—44no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#384—22no—
105#384D411yescharacteristic
106#384D411yescharacteristic
107#382—22no—
108#382—21no—
109#380—21no—
110#380C2312yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95C2312yes—
129C2×#95C2312yes—
130#340—24no—
131#340—24no—
132#328—28no—
133C22×Q8—24no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8C22⋊C414yes—
163C4⋊Q8C22⋊C414yes—
164#125—24no—
165#125—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8C22×C414yes—
171C22⋊Q8C2×D414yes—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—84no—
176C4×Q8C22⋊C418yes—
177#117—24no—
178#117—24no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22×C412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4112yes—
235C2×C42#4112yes—
236#106—216no—
237C4⋊C8—24no—
238C4⋊C8—42no—
239#103—216no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8C22×C414yes—
244C4×C8—24no—
245C4×C8—42no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—48no—
257#95—48no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8C2×C22⋊C414yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4C2×SD1614yes—
316C22×C4C2×SD1612yes—
317C22×C4#13512yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4#13712yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4C2×C22⋊C414yes—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C42—28no—
364C42—44no—
365C42—44no—
366C42—82no—
367C42—84no—
368C42—84no—
369C42—84no—
370C42C2×C22⋊C412yes—
371C42C2×C22⋊C412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42#12112yes—
375C42#12112yes—
376C42#12312yes—
377C42#12312yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23C2×C4×D411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×#12111yescharacteristic
385C23C22×SD1611yescharacteristic
386C23C2×#13511yescharacteristic
387C23C2×AGL(1,ℤ/8)11yescharacteristic
388C23C2×#13711yescharacteristic
389Q8—44no—
390Q8—48no—
391Q8—82no—
392Q8—88no—
393Q8—88no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#38612yes—
443C2×C4#38612yes—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4C4×SD1614yes—
447C2×C4#44014yes—
448C2×C4#44812yes—
449C2×C4#44812yes—
450C2×C4#44912yes—
451C2×C4#44912yes—
452C2×C4#45512yes—
453C2×C4#45612yes—
454C2×C4#46014yes—
455C2×C4#46414yes—
456C2×C4C22×C22⋊C411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12311yescharacteristic
460C8—44no—
461C8—82no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22C2×#38611yescharacteristic
475C22C2×C4×SD1611yescharacteristic
476C22C2×#44011yescharacteristic
477C22C2×#44811yescharacteristic
478C22C2×#44911yescharacteristic
479C22C2×#45511yescharacteristic
480C22C2×#45611yescharacteristic
481C22#302811yescharacteristic
482C22#302811yescharacteristic
483C22C2×#46011yescharacteristic
484C22C2×#46411yescharacteristic
485C4—22no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—44no—
491C4—44no—
492C4—44no—
493C4—84no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4#188014yes—
499C4#188014yes—
500C4#188214yes—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2#1839611yescharacteristic
506C2#1839811yescharacteristic
507C2#1841011yescharacteristic
508C2#1841211yescharacteristic
509C2C2×#188011yescharacteristic
510C2C2×#188211yescharacteristic
511C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#342132
#218332
C2×#15032
#141552
#90442
#80142
#79082