Finite Group Databasebeta

← #207280

Subgroups — #207280

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3#13749C211yesmaximal, characteristic
4#9910C212yesmaximal, complemented
5#9910C212yesmaximal, complemented
6#9909C212yesmaximal, complemented
7#9908C212yesmaximal, complemented
8#8020C211yesmaximal, characteristic
9#8019C211yesmaximal, characteristic
10#8012C211yesmaximal, characteristic
11C2×#1294C211yesmaximal, characteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2667—21no—
17#2665—21no—
18#2569—21no—
19#2549—21no—
20#2442C412yescomplemented
21#2441C412yescomplemented
22#2440C412yescomplemented
23#2439C412yescomplemented
24#2437—22no—
25#2329—22no—
26#2280C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4—22no—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C22×#95C2211yescharacteristic
32#1851—21no—
33#1851—21no—
34#1850—22no—
35#1797C2212yes—
36#1797C2212yes—
37C4×#95—21no—
38C4×#95—21no—
39C4×#95C2212yes—
40C4×#95C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C412yescomplemented
56C2×#125C2×C412yescomplemented
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×#117C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42C2312yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70#407—22no—
71#407—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8C2311yescharacteristic
77C2×C22⋊C8C2311yescharacteristic
78C2×C22⋊C8C2312yes—
79#400—22no—
80#399—22no—
81#398—22no—
82#398—22no—
83#396—22no—
84#396—22no—
85#395—22no—
86#393—22no—
87#391—22no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389C2×C412yescomplemented
92#389C2×C412yescomplemented
93#389C2×C412yescomplemented
94#389C2×C412yescomplemented
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384C2×C412yescomplemented
104#382C2×C412yescomplemented
105#381—21no—
106#380—22no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380C2×C411yescharacteristic
111#380C2×C412yes—
112#380C2×C411yescharacteristic
113#380C2311yescharacteristic
114#380C2312yes—
115#380C2311yescharacteristic
116#379—21no—
117C4×C22⋊C4—216no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C411yescharacteristic
123C4×C22⋊C4C2×C411yescharacteristic
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D411yescharacteristic
126C4×C22⋊C4D411yescharacteristic
127C4×C22⋊C4Q814yescomplemented
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95Q814yescomplemented
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139#336—22no—
140#336—22no—
141C25C2×D411yescharacteristic
142C22×Q8—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—28no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—82no—
150C23×C4—81no—
151C23×C4—81no—
152C23×C4—82no—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×Q812yes—
161C23×C4C2×Q812yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—42no—
166C2×M16—82no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—42no—
171C22×C8—82no—
172C4⋊Q8—82no—
173#125—82no—
174C22⋊Q8—48no—
175C22⋊Q8—82no—
176C22⋊Q8—82no—
177#117—82no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—82no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42—82no—
217C2×C42—82no—
218C2×C42—82no—
219C2×C42C22×C412yes—
220C2×C42C22×C412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C4⋊C8—82no—
224C4⋊C8—82no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233#95—28no—
234#95—28no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—24no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C411yescharacteristic
254#95C22⋊C411yescharacteristic
255#95C22⋊C411yescharacteristic
256#95C22⋊C411yescharacteristic
257#95C2×Q814yes—
258#95C2×Q812yes—
259#95C2×Q812yes—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—24no—
265C24—42no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24C22⋊Q812yes—
269C24C22⋊Q812yes—
270C24C22×D411yescharacteristic
271C2×Q8—44no—
272C2×Q8—82no—
273C2×Q8—82no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—216no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—44no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4—88no—
317C22×C4—88no—
318C22×C4—84no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C4×Q812yes—
328C22×C4C4×Q812yes—
329C22×C4C4×Q812yes—
330C22×C4C4×Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4C22⋊Q812yes—
333C22×C4#12412yes—
334C22×C4#12412yes—
335C22×C4C4⋊Q812yes—
336C22×C4C4⋊Q812yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343M16—84no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—84no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—216no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—44no—
377C22⋊C4—81no—
378C22⋊C4—81no—
379C22⋊C4—82no—
380C22⋊C4#10418yes—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—82no—
386C42—84no—
387C42—88no—
388C42—88no—
389C23—22no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23#39112yes—
411C23#39112yes—
412C23C2×#10414yes—
413C23C22×C22⋊C411yescharacteristic
414C23C2×C22⋊Q811yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23#57611yescharacteristic
417C23#57711yescharacteristic
418C23#57711yescharacteristic
419C23#57811yescharacteristic
420Q8—164no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
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—816no—
483C2×C4—164no—
484C2×C4#39112yes—
485C2×C4#39112yes—
486C2×C4C2×#10414yes—
487C2×C4C2×#10414yes—
488C2×C4C2×C4×Q811yescharacteristic
489C2×C4C2×C4×Q811yescharacteristic
490C2×C4C2×#12411yescharacteristic
491C2×C4C2×C4⋊Q811yescharacteristic
492C8—81no—
493C8—81no—
494C8—82no—
495C22—22no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—24no—
504C22—22no—
505C22—22no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22#199414yes—
517C22C2×#39111yescharacteristic
518C22#288312yes—
519C22#289711yescharacteristic
520C22#289811yescharacteristic
521C22C22×#10412yes—
522C22#308111yescharacteristic
523C22#308211yescharacteristic
524C22#308311yescharacteristic
525C22#308411yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—22no—
531C4—22no—
532C4—24no—
533C4—24no—
534C4—44no—
535C4—44no—
536C4—48no—
537C4—42no—
538C4—42no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2—24no—
548C2—42no—
549C2#1754712yes—
550C2#1754812yes—
551C2#2131511yescharacteristic
552C2C2×#199411yescharacteristic
553C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9910C2semidirect
5. #9910C2semidirect
6. #9909C2semidirect
7. #9908C2semidirect
20. #2442C4semidirect
21. #2441C4semidirect
22. #2440C4semidirect
23. #2439C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
56. C2×#125C2×C4semidirect
91. #389C2×C4semidirect
92. #389C2×C4semidirect
93. #389C2×C4semidirect
94. #389C2×C4semidirect
103. #384C2×C4semidirect
104. #382C2×C4semidirect
127. C4×C22⋊C4Q8semidirect
135. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#350792
#219322
#137492
#99102
#99092
#99082
#80202
#80192
#80122
C2×#12942