Finite Group Databasebeta

← #201432

Subgroups — #201432

Order 512.

1209 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21841C212yesmaximal, complemented
4#20723C211yesmaximal, characteristic
5#9473C214yesmaximal, complemented
6#9469C214yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2990—24no—
11#2983—21no—
12#2973—21no—
13#2973C2212yes—
14#2972C2212yes—
15#2970—24no—
16#2920—21no—
17C2×#432—21no—
18C2×#432C2212yes—
19#2857—21no—
20#2857—21no—
21#2628C412yescomplemented
22#2628C412yescomplemented
23#2568C412yescomplemented
24#2567—21no—
25#2563—21no—
26#2542C412yescomplemented
27#2487C2212yes—
28#2487C2212yes—
29#2409—21no—
30#2409—21no—
31#2317—21no—
32#2311—21no—
33#2285C2211yescharacteristic
34#2283C2212yes—
35#1993—24no—
36#1988—24no—
37#1824C2218yes—
38#1820—21no—
39#1799C2212yes—
40#1799C2212yes—
41#1782—21no—
42C4×#99—21no—
43C4×#99—21no—
44#1289C2218yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2×C412yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78#446—28no—
79#435—28no—
80#435—28no—
81C8×D4—28no—
82#432—22no—
83#432—22no—
84#432—22no—
85#432—22no—
86#432—22no—
87#432—22no—
88#432—22no—
89#432—22no—
90#431—22no—
91#431—22no—
92#423—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8D414yes—
95C2×#99—22no—
96C2×#99—22no—
97#407—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100C4×M16—22no—
101C2×#97—22no—
102C2×#97D414yes—
103#400C2×C414yes—
104#400C2×C414yes—
105#399C2×C414yes—
106#399C2×C414yes—
107#395—22no—
108#395—22no—
109#393—22no—
110#393—22no—
111#391—22no—
112#391—22no—
113#391—22no—
114#390—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#386—22no—
120#386—22no—
121#385—22no—
122#385—22no—
123#384—22no—
124#384—21no—
125#383—22no—
126#382—22no—
127#382—22no—
128#382C2×C412yes—
129#381—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—21no—
135#380C2314yes—
136#380C2312yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4C2×C412yes—
140C4×C22⋊C4—22no—
141#376—22no—
142#342—22no—
143#342—22no—
144#340—216no—
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—84no—
154C22×Q8C22⋊C411yescharacteristic
155C22×D4C22⋊C411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—22no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C2×M16—24no—
169C22×C8—24no—
170C22×C8C2×D418yes—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—41no—
176C4⋊Q8—42no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—42no—
181C22⋊Q8—84no—
182C22⋊Q8—84no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—44no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22×C411yescharacteristic
233C2×C22⋊C4C22×C411yescharacteristic
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2×D411yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4112yes—
259C2×C42#4112yes—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264#99—44no—
265#99—44no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270#97—24no—
271#97—24no—
272#97—24no—
273C4×C8—24no—
274#95—22no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95C22×C414yes—
284C24C2×C22⋊C411yescharacteristic
285C24C2×C22⋊C411yescharacteristic
286C24C22×D411yescharacteristic
287#41—48no—
288#41—48no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—88no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4#13714yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—84no—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4112yes—
356M16—84no—
357C2×C8—28no—
358C2×C8—48no—
359C2×C8—84no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—28no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—84no—
388C22⋊C4—84no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—44no—
402C42—84no—
403C42C4×D412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42C4×D412yes—
407C42#12011yescharacteristic
408C42#12012yes—
409C42#12011yescharacteristic
410C42#12311yescharacteristic
411C42#12311yescharacteristic
412C42#12312yes—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23#41912yes—
425C23C22×C22⋊C411yescharacteristic
426C23C2×#12112yes—
427C23C2×#13712yes—
428C23#57512yes—
429Q8—44no—
430Q8—84no—
431Q8—88no—
432Q8—84no—
433D4—28no—
434D4—44no—
435D4—44no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—88no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4#38611yescharacteristic
486C2×C4#38611yescharacteristic
487C2×C4#38611yescharacteristic
488C2×C4#38612yes—
489C2×C4#38612yes—
490C2×C4#38611yescharacteristic
491C2×C4#41912yes—
492C2×C4C2×C4×D412yes—
493C2×C4C2×#12011yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C8—48no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—81no—
511C22—81no—
512C22C2×#38611yescharacteristic
513C22C2×#41911yescharacteristic
514C22#289611yescharacteristic
515C22#305614yes—
516C4—21no—
517C4—22no—
518C4—21no—
519C4—22no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—84no—
530C4—84no—
531C4—164no—
532C4#191012yes—
533C4#302812yes—
534C2—21no—
535C2—24no—
536C2—21no—
537C2—42no—
538C2—81no—
539C2#2096211yescharacteristic
540C2#2097211yescharacteristic
541C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21841C2semidirect
5. #9473C2semidirect
6. #9469C2semidirect
7. #7984C2semidirect
21. #2628C4semidirect
22. #2628C4semidirect
23. #2568C4semidirect
26. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#602822
#352792
#218412
#207232
#94732
#94692
#79842