Finite Group Databasebeta

← #200418

Subgroups — #200418

Order 512.

1395 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#35010C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4C2×#1520C212yesmaximal, complemented
5#9385C218yesmaximal, complemented
6#8023C212yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2973—24no—
10#2973—22no—
11#2973C2214yes—
12C2×#435—24no—
13C2×#432—21no—
14C22×#99—21no—
15C2×#407—21no—
16C22×C22⋊C8C2212yes—
17#2855—21no—
18#2848—21no—
19#2627C412yescomplemented
20#2626C412yescomplemented
21#2579—22no—
22#2564C2214yes—
23#2487C412yescomplemented
24#2417—22no—
25#2409C412yescomplemented
26#2315—21no—
27#2308—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30#1799—22no—
31#1799C2214yes—
32C2×#342—21no—
33#1520—24no—
34#1520C2218yescomplemented
35C2×#328—24no—
36#1294C2218yescomplemented
37C23×Q8C2×C411yescharacteristic
38C23×D4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C23×C8—22no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×C4×D4C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42D411yescharacteristic
71#435—216no—
72#435—216no—
73#432—24no—
74#432—24no—
75C2×C4⋊C8—22no—
76C2×#99—24no—
77C2×#99—44no—
78#407—24no—
79#407—24no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8D414yes—
85C2×C22⋊C8D414yes—
86C2×C22⋊C8C2314yes—
87C2×#97—22no—
88#400C2×C414yescomplemented
89#399C2×C414yescomplemented
90#398—24no—
91#396—24no—
92#395C2×C414yescomplemented
93#393C2×C414yescomplemented
94#391—22no—
95#389—22no—
96#387—24no—
97#386—22no—
98#386—22no—
99#386—24no—
100#385—22no—
101#385—22no—
102#385—24no—
103#385—22no—
104#385—22no—
105#381—22no—
106#380—22no—
107#380—21no—
108#380—21no—
109#380—24no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380C2×C412yes—
114#380C2×C412yes—
115#380C2314yes—
116#379—22no—
117C2×#95—22no—
118C2×#95C2312yes—
119#342—24no—
120#328—216no—
121C25C22⋊C411yescharacteristic
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—82no—
127C22×Q8C22×C412yes—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4C22⋊C412yes—
134C22×D4C2×D412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×M16—28no—
161C2×M16—84no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8—84no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—84no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22×C412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C22⋊C412yes—
232C2×C42C2×D412yes—
233C4⋊C8—28no—
234#99—48no—
235C22⋊C8—28no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8C2×D418yes—
240#97—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—24no—
247#95—42no—
248#95C22×C414yes—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—81no—
254C24—81no—
255C24—81no—
256C24—81no—
257C24C2×C22⋊C412yes—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260C24#12112yes—
261C24#12112yes—
262C24AGL(1,ℤ/8)12yes—
263C24#13712yes—
264C24#13714yes—
265C24C22×D411yescharacteristic
266C2×Q8—24no—
267C2×Q8—44no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—84no—
271C2×Q8—82no—
272C2×Q8—84no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—82no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—82no—
319C22×C4—81no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12012yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12311yescharacteristic
344C22×C4#12311yescharacteristic
345C22×C4#12312yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4112yes—
350M16—88no—
351C2×C8—28no—
352C2×C8—48no—
353C2×C8—84no—
354C2×C8—84no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—44no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—81no—
407C23—81no—
408C23—82no—
409C23—82no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23#38612yes—
414C23#38611yescharacteristic
415C23#38611yescharacteristic
416C23#41712yes—
417C23#41712yes—
418C23#41912yes—
419C23#41912yes—
420C23#46814yes—
421C23#46814yes—
422C23#47014yes—
423C23#47014yes—
424C23C22×C22⋊C411yescharacteristic
425C23C2×#12112yes—
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427C23C2×#13712yes—
428C23C2×#13711yescharacteristic
429Q8—84no—
430Q8—82no—
431Q8—82no—
432D4—48no—
433D4—44no—
434D4—44no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—162no—
483C2×C4—162no—
484C2×C4—168no—
485C2×C4#38611yescharacteristic
486C2×C4#38612yes—
487C2×C4#38611yescharacteristic
488C2×C4C2×C4×D412yes—
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C8—48no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—82no—
518C22—84no—
519C22—84no—
520C22#189412yes—
521C22#189412yes—
522C22#189414yes—
523C22C2×#38611yescharacteristic
524C22C2×#41911yescharacteristic
525C22C2×#41711yescharacteristic
526C22C2×#46812yes—
527C22C2×#47012yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—22no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—84no—
537C4—82no—
538C4—82no—
539C4—164no—
540C4#191312yes—
541C2—21no—
542C2—22no—
543C2—21no—
544C2—22no—
545C2—22no—
546C2—24no—
547C2—22no—
548C2—82no—
549C2#2094711yescharacteristic
550C2C2×#189411yescharacteristic
551C2C2×#191311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1520C2semidirect
5. #9385C2semidirect
6. #8023C2semidirect
19. #2627C4semidirect
20. #2626C4semidirect
23. #2487C4semidirect
25. #2409C4semidirect
34. #1520C22semidirect
36. #1294C22semidirect
88. #400C2×C4semidirect
89. #399C2×C4semidirect
92. #395C2×C4semidirect
93. #393C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#350102
C2×#17992
C2×#15202
#93852
#80232