Finite Group Databasebeta

← #201433

Subgroups — #201433

Order 512.

1285 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#21840C212yesmaximal, complemented
4#20723C211yesmaximal, characteristic
5#9474C214yesmaximal, complemented
6#9470C214yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C2×#536—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#2990—24no—
12#2983—21no—
13#2973—21no—
14#2973C2212yes—
15#2972C2212yes—
16#2970—24no—
17#2920—21no—
18C2×#432—21no—
19C2×#432C2212yes—
20#2857—21no—
21#2857—21no—
22#2621C412yescomplemented
23#2621C412yescomplemented
24#2547—21no—
25#2547C412yescomplemented
26#2546—21no—
27#2546C412yescomplemented
28#2486—21no—
29#2486—21no—
30#2486C2212yes—
31#2486C2212yes—
32#2310—21no—
33#2285C2211yescharacteristic
34#2283C2212yes—
35#1993—24no—
36#1987—24no—
37#1823C2218yes—
38#1820—21no—
39#1799C2212yes—
40#1799C2212yes—
41#1782—21no—
42C4×#99—21no—
43C4×#99—21no—
44#1289C2218yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47#536—48no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C412yes—
57C2×#124—22no—
58C2×#123—22no—
59C2×#123—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83#446—28no—
84#435—28no—
85#435—28no—
86C8×D4—28no—
87#432—22no—
88#432—22no—
89#432—22no—
90#432—22no—
91#432—22no—
92#432—22no—
93#432—22no—
94#432—22no—
95#431—22no—
96#431—22no—
97#423—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8D414yes—
100C2×#99—22no—
101C2×#99—22no—
102#407—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2314yes—
105C4×M16—22no—
106C2×#97—22no—
107C2×#97D414yes—
108#398—22no—
109#398—22no—
110#398C2×C414yes—
111#398C2×C414yes—
112#397—22no—
113#397—22no—
114#397C2×C414yes—
115#397C2×C414yes—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#384—21no—
125#383—22no—
126#382—22no—
127#382C2×C412yes—
128#381—22no—
129#381—22no—
130#381—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—21no—
136#380C2314yes—
137#380C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4C2×C412yes—
141C4×C22⋊C4—22no—
142#376—22no—
143#342—22no—
144#342—22no—
145#340—216no—
146C2×#41—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—42no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8C22⋊C411yescharacteristic
157C22×D4C22⋊C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—22no—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×M16—24no—
171C22×C8—24no—
172C22×C8C2×D418yes—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—48no—
178C4⋊Q8—41no—
179C4⋊Q8—42no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—42no—
184#124—48no—
185#123—48no—
186#123—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—44no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22×C411yescharacteristic
239C2×C22⋊C4C22×C411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—24no—
269C4⋊C8—24no—
270#99—44no—
271#99—44no—
272C22⋊C8—24no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276#97—24no—
277#97—24no—
278#97—24no—
279C4×C8—24no—
280#95—22no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95C22×C414yes—
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24C22×D411yescharacteristic
293#41—48no—
294#41—48no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—44no—
305C2×Q8—88no—
306C2×Q8—88no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4AGL(1,ℤ/8)14yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4112yes—
364M16—84no—
365C2×C8—28no—
366C2×C8—48no—
367C2×C8—84no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—28no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—48no—
393C22⋊C4—44no—
394C22⋊C4—48no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—44no—
410C42—84no—
411C42C4×D412yes—
412C42C4×D412yes—
413C42C4×D412yes—
414C42C4×D412yes—
415C42#12011yescharacteristic
416C42#12012yes—
417C42#12011yescharacteristic
418C42#12311yescharacteristic
419C42#12311yescharacteristic
420C42#12312yes—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23#41812yes—
433C23C22×C22⋊C411yescharacteristic
434C23C2×#12112yes—
435C23C2×AGL(1,ℤ/8)12yes—
436C23#57512yes—
437Q8—44no—
438Q8—84no—
439Q8—84no—
440Q8—88no—
441D4—28no—
442D4—44no—
443D4—44no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—88no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38611yescharacteristic
496C2×C4#38612yes—
497C2×C4#38612yes—
498C2×C4#38611yescharacteristic
499C2×C4#41812yes—
500C2×C4C2×C4×D412yes—
501C2×C4C2×#12011yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C8—48no—
504C22—21no—
505C22—22no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—81no—
519C22—81no—
520C22C2×#38611yescharacteristic
521C22C2×#41811yescharacteristic
522C22#289611yescharacteristic
523C22#305514yes—
524C4—21no—
525C4—22no—
526C4—21no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—22no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—84no—
538C4—84no—
539C4—164no—
540C4#190912yes—
541C4#302812yes—
542C2—21no—
543C2—24no—
544C2—21no—
545C2—42no—
546C2—81no—
547C2#2096211yescharacteristic
548C2#2097311yescharacteristic
549C2C2×#190911yescharacteristic
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. #21840C2semidirect
5. #9474C2semidirect
6. #9470C2semidirect
7. #7984C2semidirect
22. #2621C4semidirect
23. #2621C4semidirect
25. #2547C4semidirect
27. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#602822
#353162
#218402
#207232
#94742
#94702
#79842