Finite Group Databasebeta

← #189857

Subgroups — #189857

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34180C211yesmaximal, characteristic
2#20685C211yesmaximal, characteristic
3#13958C211yesmaximal, characteristic
4#8717C212yesmaximal, complemented
5#8716C212yesmaximal, complemented
6#8714C212yesmaximal, complemented
7#8714C212yesmaximal, complemented
8#8024C211yesmaximal, characteristic
9#8020C211yesmaximal, characteristic
10#8015C211yesmaximal, characteristic
11C2×#1294C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2347—21no—
16#2346—21no—
17#2345—21no—
18#2344—21no—
19#2343—22no—
20#2342—22no—
21#2317C412yescomplemented
22#2314C412yescomplemented
23#2314C412yescomplemented
24#2285C2212yes—
25#2283C2212yes—
26#2280C2211yescharacteristic
27C2×#381C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#2274C412yescomplemented
30C22×#95C2211yescharacteristic
31#1818—21no—
32#1818—21no—
33#1817C2212yes—
34#1817C2212yes—
35#1729—22no—
36#1445—21no—
37#1445—21no—
38#1434C2212yes—
39#1434C2212yes—
40C22⋊#95—22no—
41#1294C2212yes—
42#1294C2211yescharacteristic
43#1294C2211yescharacteristic
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2211yescharacteristic
49#1294C2212yes—
50#1294C2211yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117C2×C412yes—
61C2×#117C2×C412yes—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117D412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42C2312yes—
74#407—22no—
75#407—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8C2311yescharacteristic
81C2×C22⋊C8C2311yescharacteristic
82C2×C22⋊C8C2312yes—
83#389—22no—
84#389—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C412yescomplemented
96#384C2×C412yescomplemented
97#383—22no—
98#382—22no—
99#382C2×C412yescomplemented
100#382C2×C412yescomplemented
101#381—22no—
102#381—21no—
103#381—22no—
104#381—22no—
105#380—22no—
106#380—21no—
107#380—21no—
108#380C2×C411yescharacteristic
109#380C2×C411yescharacteristic
110#380C2×C412yes—
111#380C2312yes—
112#380C2311yescharacteristic
113#380C2311yescharacteristic
114#379—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yescomplemented
119C4×C4⋊C4C2×C412yescomplemented
120C4×C4⋊C4C2×C412yescomplemented
121C4×C4⋊C4C2×C412yescomplemented
122C4×C22⋊C4—22no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127#376—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C25C2×D411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C2×M16—44no—
168C2×M16—42no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—44no—
173C22×C8—42no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—44no—
187#117—82no—
188#117—82no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—81no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—82no—
272#95—82no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—24no—
278C24—42no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24#12312yes—
282C24#12312yes—
283C24C22×D411yescharacteristic
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4#11712yes—
340C22×C4#11712yes—
341C22×C4#11712yes—
342C22×C4#11712yes—
343C22×C4#12312yes—
344C22×C4#12312yes—
345C22×C4#12312yes—
346C22×C4#12312yes—
347C22×C4#12312yes—
348C22×C4#12312yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355M16—84no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—44no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—84no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—42no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—44no—
389C22⋊C4—82no—
390C22⋊C4—81no—
391C22⋊C4—81no—
392C22⋊C4—84no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—44no—
402C42—82no—
403C42—82no—
404C42—82no—
405C42—82no—
406C23—22no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23#38112yes—
428C23#38112yes—
429C23#41014yes—
430C23C22×C22⋊C411yescharacteristic
431C23C2×#12311yescharacteristic
432C23C2×#12311yescharacteristic
433C23#57611yescharacteristic
434C23#57711yescharacteristic
435C23#57711yescharacteristic
436C23#57811yescharacteristic
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—816no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—164no—
504C2×C4#38112yes—
505C2×C4#38112yes—
506C2×C4C2×#11711yescharacteristic
507C2×C4C2×#11711yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C8—81no—
511C8—81no—
512C8—82no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—22no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22#182014yes—
535C22C2×#38111yescharacteristic
536C22C2×#41012yes—
537C22#289711yescharacteristic
538C22#289811yescharacteristic
539C22#290112yes—
540C22#309611yescharacteristic
541C22#309711yescharacteristic
542C22#309811yescharacteristic
543C22#309911yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—22no—
549C4—22no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—161no—
560C4—161no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—24no—
566C2—42no—
567C2#1760412yes—
568C2#1760912yes—
569C2C2×#182011yescharacteristic
570C2#2070311yescharacteristic
571C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8717C2semidirect
5. #8716C2semidirect
6. #8714C2semidirect
7. #8714C2semidirect
21. #2317C4semidirect
22. #2314C4semidirect
23. #2314C4semidirect
29. #2274C4semidirect
95. #384C2×C4semidirect
96. #384C2×C4semidirect
99. #382C2×C4semidirect
100. #382C2×C4semidirect
118. C4×C4⋊C4C2×C4semidirect
119. C4×C4⋊C4C2×C4semidirect
120. C4×C4⋊C4C2×C4semidirect
121. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341802
#206852
#139582
#87172
#87162
#87142
#80242
#80202
#80152
C2×#12942