Finite Group Databasebeta

← #232302

Subgroups — #232302

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic
3#35019C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic, complemented
5C2×#1519C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic, complemented
7#9971C218yesmaximal, complemented
8#7907C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#483—24no—
13C2×#483—22no—
14C2×#483—24no—
15C2×#483C2214yescomplemented
16C2×#480—24no—
17C2×#480—22no—
18C2×#480—24no—
19C2×#480C2214yes—
20C22×#107—21no—
21C22×#106—21no—
22C22×#103—21no—
23C22×#102—21no—
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2850—22no—
27#2639C2214yes—
28#2591—22no—
29#2561—22no—
30#2504—21no—
31#2503—21no—
32#2436—21no—
33#2435—21no—
34#2308—22no—
35C2×#381—22no—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1520—24no—
39#1519—24no—
40C2×#328C2214yescomplemented
41C2×#327C2214yes—
42#1445C2214yescomplemented
43C4⋊#95—22no—
44#1280C2214yescomplemented
45C23×Q8D411yescharacteristic
46C23×D4—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—21no—
51C23×C8—21no—
52C2×#124—22no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8D412yes—
60C2×C22⋊Q8C2312yes—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121D412yes—
68C2×#121C2312yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D412yes—
87C22×C42D412yes—
88#483—216no—
89#483—28no—
90#483—28no—
91#480—216no—
92#480—28no—
93#480—28no—
94C2×#107—28no—
95C2×#107—24no—
96C2×#106—28no—
97C2×#106—24no—
98C2×#103—28no—
99C2×#103—24no—
100C2×#102—28no—
101C2×#102—24no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8D414yes—
107C2×C22⋊C8D414yes—
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2312yes—
110#400—24no—
111#397—24no—
112#394—24no—
113#393—24no—
114#391—22no—
115#390—22no—
116#388—28no—
117#386—24no—
118#386—24no—
119#386—28no—
120#382—22no—
121#381—24no—
122#381—48no—
123#380—22no—
124#380—22no—
125#379—24no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C2×#95—28no—
131C2×#95C2312yes—
132C2×#95C2312yes—
133#328—28no—
134#327—28no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—44no—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4C2×D412yes—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D414yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—44no—
181C22×C8—44no—
182#124—84no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193#121—24no—
194#121—24no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—42no—
200#121—42no—
201#121—42no—
202#121—42no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
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—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—44no—
254#107—216no—
255#106—216no—
256#103—216no—
257#102—216no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8C2×D414yes—
263C22⋊C8C2×D414yes—
264#95—28no—
265#95—28no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—416no—
271C24—22no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—44no—
276C24—82no—
277C24—82no—
278C24#12011yescharacteristic
279C24#12011yescharacteristic
280C24#12112yes—
281C24#12712yes—
282C24#13514yes—
283C24AGL(1,ℤ/8)14yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—44no—
288C2×Q8—48no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—84no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—44no—
295C2×D4—48no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—84no—
305C2×D4—88no—
306C2×D4—84no—
307C2×D4—84no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—416no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4#12011yescharacteristic
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4#12011yescharacteristic
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—48no—
371C2×C8—48no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—84no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—84no—
399C42—44no—
400C42—82no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—48no—
412C23—48no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—168no—
421C23#39212yes—
422C23#39212yes—
423C23#44914yes—
424C23#44914yes—
425C23#46318yes—
426C23#49514yes—
427C23#49614yes—
428C23C2×#12011yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12711yescharacteristic
431C23C2×#13512yes—
432C23C2×AGL(1,ℤ/8)12yes—
433Q8—44no—
434Q8—82no—
435Q8—84no—
436D4—44no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—84no—
441D4—88no—
442D4—84no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—48no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—88no—
478C2×C4—88no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#39212yes—
486C2×C4#39212yes—
487C2×C4#45814yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C8—44no—
493C8—44no—
494C22—24no—
495C22—24no—
496C22—28no—
497C22—28no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—21no—
506C22—48no—
507C22—44no—
508C22—44no—
509C22—88no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—168no—
514C22#201214yes—
515C22#201214yes—
516C22C2×#39211yescharacteristic
517C22C2×#44912yes—
518C22#302212yes—
519C22C2×#45812yes—
520C22C2×#46312yes—
521C22C2×#49511yescharacteristic
522C22C2×#49611yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—22no—
528C4—84no—
529C4—82no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—162no—
534C4#202214yes—
535C2—21no—
536C2—24no—
537C2—21no—
538C2—22no—
539C2—24no—
540C2—162no—
541C2#1820614yes—
542C2#2340811yescharacteristic
543C2C2×#201211yescharacteristic
544C2C2×#202211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#483C2semidirect
4. C2×#1520C2semidirect
6. #13904C2semidirect
7. #9971C2semidirect
8. #7907C2semidirect
15. C2×#483C22semidirect
40. C2×#328C22semidirect
42. #1445C22semidirect
44. #1280C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4802
#350192
C2×#15202
C2×#15192
#139042
#99712
#79072