Finite Group Databasebeta

← #194109

Subgroups — #194109

Order 512.

1252 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2214yes—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2360—22no—
22#2360—22no—
23#2355—21no—
24#2355C2212yes—
25Q8×C4⋊C4—21no—
26Q8×C4⋊C4C2212yes—
27#2330—22no—
28#2317—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C2×#384C2211yescharacteristic
32#2274—21no—
33C4×#117C2211yescharacteristic
34#1990—22no—
35#1990C2214yes—
36#1988C2212yes—
37#1988C2212yes—
38#1987—21no—
39#1987—21no—
40#1736—21no—
41#1736—21no—
42C2×#340—22no—
43#1493—24no—
44C2×Q8⋊C8C2214yes—
45#1296C2214yes—
46#1279C2212yes—
47#1279C2212yes—
48C23×Q8—21no—
49C22×Q16—24no—
50C22×M16—21no—
51C23×C8—21no—
52#533—22no—
53#533—22no—
54#533C2×C414yes—
55#533D414yes—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8D412yes—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2312yes—
73C2×#117—21no—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#494—28no—
83#487—28no—
84#462—28no—
85#462C2×C4116yescomplemented
86#451—216no—
87#431—22no—
88#431C2×C414yes—
89#423—22no—
90#423—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×#103—24no—
95C2×C22⋊C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#391—22no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—24no—
107#389—22no—
108#389—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#384—22no—
113#384—21no—
114#384—21no—
115#384D412yes—
116#384C2312yes—
117#382—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—24no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—21no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—28no—
138Q8⋊C8—28no—
139C22×Q8—21no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—41no—
143C22×Q8—42no—
144C22×Q8—82no—
145C22×Q8—82no—
146C22×Q8C2×D412yes—
147C23×C4—22no—
148C23×C4—41no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×Q16—216no—
153C2×Q16—84no—
154C2×Q16—84no—
155C2×M16—42no—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8—24no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—42no—
163C22×C8—42no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22×C414yes—
178#125—24no—
179#125—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8C22⋊C418yes—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—84no—
189C4×Q8—84no—
190C4×Q8C22×C414yes—
191C4×Q8C2×D414yes—
192#117—24no—
193#117—24no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117Q1614yescomplemented
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42C22×C412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252C4⋊C8C22×C414yes—
253#103—216no—
254C22⋊C8—24no—
255C4×C8—24no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—42no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C24#12011yescharacteristic
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—88no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—88no—
283C2×Q8C2×C22⋊C414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C2×C22⋊C412yes—
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4#12312yes—
311C22×C4C2×Q1612yes—
312C22×C4#13512yes—
313C22×C4#13712yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318Q16—816no—
319M16—84no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—48no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4C2×C22⋊C414yes—
359C4⋊C4C2×Q1612yes—
360C4⋊C4C2×Q1612yes—
361C22⋊C4—28no—
362C22⋊C4—24no—
363C22⋊C4—42no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—84no—
373C42—84no—
374C42C2×C22⋊C412yes—
375C42C4×D412yes—
376C42C4×D412yes—
377C42#12112yes—
378C42#12112yes—
379C42#12312yes—
380C42C23×C411yescharacteristic
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#38612yes—
389C23#45112yes—
390C23#45212yes—
391C23C2×#12011yescharacteristic
392Q8—28no—
393Q8—44no—
394Q8—48no—
395Q8—48no—
396Q8—88no—
397Q8—84no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—88no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—164no—
432C2×C4#38612yes—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4C4×Q1614yes—
436C2×C4#44114yes—
437C2×C4#45112yes—
438C2×C4#45212yes—
439C2×C4#45512yes—
440C2×C4#45612yes—
441C2×C4#46214yes—
442C2×C4#46414yes—
443C2×C4C22×C22⋊C411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4C22×Q1611yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57612yes—
454C2×C4#57712yes—
455C8—44no—
456C8—44no—
457C8—82no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#187814yes—
465C22C2×#38611yescharacteristic
466C22C2×#45111yescharacteristic
467C22C2×#45211yescharacteristic
468C22#300811yescharacteristic
469C22#301011yescharacteristic
470C22C2×#45511yescharacteristic
471C22C2×#45611yescharacteristic
472C4—22no—
473C4—21no—
474C4—21no—
475C4—44no—
476C4—42no—
477C4—44no—
478C4—42no—
479C4—84no—
480C4—81no—
481C4—84no—
482C4—81no—
483C4#187814yes—
484C4#188214yes—
485C4#289712yes—
486C4C2×C4×Q1611yescharacteristic
487C4C2×#44111yescharacteristic
488C4C2×#46211yescharacteristic
489C4C2×#46411yescharacteristic
490C4#309612yes—
491C2—24no—
492C2#1863311yescharacteristic
493C2#1863511yescharacteristic
494C2#1865311yescharacteristic
495C2#1865511yescharacteristic
496C2#2091411yescharacteristic
497C2C2×#187811yescharacteristic
498C2C2×#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. #8986C2semidirect
13. #3209C4semidirect
85. #462C2×C4semidirect
197. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#342142
C2×#19902
#213132
C2×#14932
#89872
#89862
#80432
#78902