Finite Group Databasebeta

← #192304

Subgroups — #192304

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34083C211yesmaximal, characteristic
2#21388C211yesmaximal, characteristic
3#21361C211yesmaximal, characteristic, complemented
4C2×#1500C211yesmaximal, characteristic
5#13856C211yesmaximal, characteristic, complemented
6#8873C212yesmaximal
7#8871C212yesmaximal, complemented
8#8868C212yesmaximal, complemented
9#8866C212yesmaximal
10#8013C211yesmaximal, characteristic, complemented
11#7908C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2337—21no—
16#2336—21no—
17#2334—22no—
18#2329C2212yes—
19#2325C2212yes—
20#2322—22no—
21#2298—21no—
22#2296—21no—
23#2295—21no—
24#2290—21no—
25#2290—21no—
26#2283C2211yescharacteristic
27C42×D4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1989—21no—
34#1983—21no—
35#1949—22no—
36#1874C2212yes—
37#1872C2212yes—
38#1853C2212yes—
39#1852C2212yes—
40C4×#107—21no—
41C4×#106—21no—
42C4×#103—21no—
43C4×#102—21no—
44#1500C418yescomplemented
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2211yescharacteristic
49#1444—22no—
50#1444C2212yes—
51#1444C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1280C2212yes—
55#1280C2212yes—
56#529—22no—
57#529—22no—
58#529C2×C414yes—
59#529Q814yes—
60C2×#126—21no—
61C2×#125D411yescharacteristic
62C2×#124D411yescharacteristic
63C2×#123—21no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4D411yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yescomplemented
86C2×#107—22no—
87C2×#106—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×#103—22no—
91C2×#102—22no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#386—22no—
112#386—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#383—21no—
118#382D412yes—
119#381—21no—
120#380C2312yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4—44no—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141#376—21no—
142#376—21no—
143#376—22no—
144#376C2×C414yescomplemented
145C2×#95—21no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C43—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#328C2×C414yescomplemented
158#328C2×C414yescomplemented
159#327C2×C414yes—
160#327C2×C414yes—
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177#126—24no—
178#126—24no—
179#125—22no—
180#125C4⋊C414yescomplemented
181#124—22no—
182#124C4⋊C414yescomplemented
183#123—24no—
184#123—24no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8C22×C412yes—
188C22⋊Q8C2×Q812yes—
189#121—21no—
190#121—21no—
191#121C22×C412yes—
192#121C2×Q812yes—
193C4×Q8—22no—
194C4×Q8C4⋊C414yescomplemented
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4C4⋊C414yescomplemented
201#117—24no—
202#117—24no—
203#117—82no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—82no—
257C2×C42C22×C412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×Q812yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265#107—44no—
266#106—44no—
267C4⋊C8—24no—
268C4⋊C8—42no—
269#103—44no—
270#102—44no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8C22×C412yes—
276C22⋊C8C22×C412yes—
277C4×C8—24no—
278C4×C8—42no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—44no—
296#95—44no—
297#95—41no—
298#95—41no—
299#95C22×C412yes—
300#95C22×C412yes—
301C24—21no—
302C24C22×D411yescharacteristic
303C2×Q8—21no—
304C2×Q8—42no—
305C2×Q8C2×C4⋊C412yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4C2×C4⋊C412yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4#12011yescharacteristic
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4#13514yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—416no—
394C4⋊C4—416no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4C2×C4⋊C412yes—
398C4⋊C4C2×C4⋊C412yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—216no—
412C42—24no—
413C42—44no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—81no—
419C42—81no—
420C42—82no—
421C42—82no—
422C42—82no—
423C42—82no—
424C42C2×C4⋊C412yes—
425C42C2×C4⋊C412yes—
426C42C4×D412yes—
427C42C4×D412yes—
428C42C22⋊Q812yes—
429C42C22⋊Q812yes—
430C42#12312yes—
431C42#12312yes—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23C2×C4×D411yescharacteristic
441C23C2×#12011yescharacteristic
442C23C2×C22⋊Q811yescharacteristic
443C23C2×#13512yes—
444C23#57512yes—
445Q8—42no—
446Q8—81no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—81no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—88no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#38512yes—
510C2×C4#38512yes—
511C2×C4#38512yes—
512C2×C4#38512yes—
513C2×C4#45312yes—
514C2×C4#45612yes—
515C2×C4C22×C4⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C22⋊Q811yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C8—42no—
520C8—42no—
521C8—82no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—82no—
535C22C2×#38511yescharacteristic
536C22C4×#13511yescharacteristic
537C22#294311yescharacteristic
538C22#300212yes—
539C22C2×#45311yescharacteristic
540C22C2×#45611yescharacteristic
541C22#302512yes—
542C22#307612yes—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—82no—
557C4—84no—
558C4—82no—
559C4—82no—
560C4#185914yes—
561C2—21no—
562C2—21no—
563C2—22no—
564C2—82no—
565C2#1812812yes—
566C2#1813112yes—
567C2C2×#185911yescharacteristic
568C2#2087211yescharacteristic
569C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21361C2semidirect
5. #13856C2semidirect
7. #8871C2semidirect
8. #8868C2semidirect
10. #8013C2semidirect
11. #7908C2semidirect
44. #1500C4semidirect
85. #431C2×C4semidirect
144. #376C2×C4semidirect
157. #328C2×C4semidirect
158. #328C2×C4semidirect
180. #125C4⋊C4semidirect
182. #124C4⋊C4semidirect
194. C4×Q8C4⋊C4semidirect
200. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340832
#213882
#213612
C2×#15002
#138562
#88732
#88712
#88682
#88662
#80132
#79082