Finite Group Databasebeta

← #192370

Subgroups — #192370

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21386C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic, complemented
6#8873C214yesmaximal, complemented
7#8870C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#7908C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2367—22no—
15#2366—21no—
16#2365—21no—
17#2354—21no—
18#2352—22no—
19D4×C4⋊C4—21no—
20#2325C2214yes—
21#2311—21no—
22C4×#121—22no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1988—21no—
29#1982—21no—
30#1948—21no—
31#1948—21no—
32#1874C2212yes—
33#1874C2212yes—
34#1853C2212yes—
35#1853C2212yes—
36C4×#106—22no—
37C4×#102—22no—
38#1517C418yescomplemented
39C2×#327C2214yes—
40#1444C2212yes—
41#1442C2212yes—
42#1436—22no—
43C4⋊#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46C22×#41—21no—
47#532—22no—
48#532—22no—
49#532C2×C414yes—
50#532Q814yes—
51C2×C4⋊Q8D411yescharacteristic
52C2×#127D411yescharacteristic
53C2×#124—21no—
54C2×#121—21no—
55C2×#121C2312yes—
56#520—44no—
57#520—44no—
58C2×C4×Q8—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4D412yes—
64C2×#117—22no—
65C2×#117—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yescomplemented
80C2×#106—22no—
81C2×#106—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#391—22no—
91#390—22no—
92#390—22no—
93#389—22no—
94#389—44no—
95#388—24no—
96#388—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#385—24no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—21no—
104#382—22no—
105#382C2×C414yescomplemented
106#380—22no—
107#380—22no—
108#380—21no—
109#380C2312yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D412yes—
121C4×C22⋊C4—24no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#327C2×C4116yescomplemented
134C2×#41—24no—
135C2×#41—24no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×D412yes—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—22no—
158C4⋊Q8C4⋊C414yescomplemented
159#127—22no—
160#127C4⋊C414yescomplemented
161#124—24no—
162#124—24no—
163#121—24no—
164#121—24no—
165#121—22no—
166#121—22no—
167#121C22×C414yes—
168#121C2×Q814yes—
169C4×Q8—48no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—48no—
174C4×D4—48no—
175C4×D4—48no—
176C4×D4—84no—
177C4×D4C4⋊C418yescomplemented
178#117—48no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4SD1614yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234#106—44no—
235#106—44no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238#102—44no—
239#102—44no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8C22×C414yes—
244C4×C8—24no—
245C4×C8—42no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—44no—
255#95—44no—
256#95—41no—
257#95—41no—
258#95—82no—
259#95C22×C414yes—
260C24—22no—
261C24C22×D411yescharacteristic
262#41—48no—
263#41—48no—
264#41—48no—
265#41—48no—
266C2×Q8—24no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4C2×C4⋊C414yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—28no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—48no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4C22⋊Q812yes—
335C22×C4C22⋊Q812yes—
336C22×C4C2×SD1614yes—
337C22×C4C2×SD1612yes—
338C22×C4#13512yes—
339C22×C4AGL(1,ℤ/8)12yes—
340C22×C4AGL(1,ℤ/8)12yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q811yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—88no—
376C4⋊C4C2×C4⋊C414yes—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—44no—
386C42—48no—
387C42—44no—
388C42—82no—
389C42—82no—
390C42—82no—
391C42C2×C4⋊C412yes—
392C42C2×C4⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42C22⋊Q812yes—
396C42C22⋊Q812yes—
397C42#12312yes—
398C42#12312yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23C2×C4×D411yescharacteristic
411C23C2×#12011yescharacteristic
412C23C2×C22⋊Q811yescharacteristic
413C23C22×SD1611yescharacteristic
414C23C2×#13511yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23C2×AGL(1,ℤ/8)11yescharacteristic
417Q8—48no—
418D4—44no—
419D4—48no—
420D4—44no—
421D4—44no—
422D4—44no—
423D4—44no—
424D4—48no—
425D4—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—168no—
482C2×C4#38512yes—
483C2×C4#38512yes—
484C2×C4#38512yes—
485C2×C4#38512yes—
486C2×C4C4×SD1614yes—
487C2×C4#44214yes—
488C2×C4#44912yes—
489C2×C4#44912yes—
490C2×C4#45012yes—
491C2×C4#45012yes—
492C2×C4#47614yes—
493C2×C4#47814yes—
494C2×C4C22×C4⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C8—44no—
499C8—82no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22C2×#38511yescharacteristic
520C22C2×C4×SD1611yescharacteristic
521C22C2×#44211yescharacteristic
522C22C2×#45011yescharacteristic
523C22C2×#44911yescharacteristic
524C22#301911yescharacteristic
525C22#302011yescharacteristic
526C22#302511yescharacteristic
527C22#302511yescharacteristic
528C22C2×#47611yescharacteristic
529C22C2×#47811yescharacteristic
530C4—22no—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—44no—
538C4—84no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4#186414yes—
545C4#186414yes—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—84no—
551C2#1821111yescharacteristic
552C2#1821411yescharacteristic
553C2#1821911yescharacteristic
554C2#1822211yescharacteristic
555C2#2086411yescharacteristic
556C2C2×#186411yescharacteristic
557C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21359C2semidirect
5. #14150C2semidirect
6. #8873C2semidirect
7. #8870C2semidirect
8. #8026C2semidirect
9. #7908C2semidirect
38. #1517C4semidirect
79. #431C2×C4semidirect
105. #382C2×C4semidirect
133. #327C2×C4semidirect
158. C4⋊Q8C4⋊C4semidirect
160. #127C4⋊C4semidirect
177. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#213862
#213592
C2×#15172
#141502
#88732
#88702
#80262
#79082