Finite Group Databasebeta

← #195346

Subgroups — #195346

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8014C211yesmaximal, characteristic, complemented
8#7908C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520—21no—
12#3227—24no—
13#3227C418yescomplemented
14C2×#498—21no—
15C2×#486—21no—
16C2×#481C2214yes—
17C2×#476—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2367—22no—
21#2366—21no—
22#2365—21no—
23#2354—21no—
24#2352—22no—
25D4×C4⋊C4—21no—
26#2325C2214yes—
27#2311—21no—
28C4×#121—22no—
29C2×#391—22no—
30#2285C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990—21no—
34#1990—21no—
35#1987—22no—
36#1824C2214yes—
37#1502—24no—
38C2×#327C2214yes—
39#1445C2214yes—
40C4⋊#95—21no—
41C4⋊#95—21no—
42#1294C2214yes—
43#1280C2214yes—
44C22×#41—21no—
45C23×Q8—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532D414yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yescomplemented
58C2×C4⋊Q8D411yescharacteristic
59C2×#127D411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121C2312yes—
63#520—44no—
64#520—44no—
65C2×C4×Q8—22no—
66C2×C4×D4—22no—
67C2×C4×D4—24no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4D412yes—
71C2×#117—22no—
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C42—21no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82#498—28no—
83#486—28no—
84#481—28no—
85#481C2×C4116yescomplemented
86#476—216no—
87#431—22no—
88#431—22no—
89#431—22no—
90#431C2×C414yescomplemented
91C2×#106—24no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×#102—24no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99#391—22no—
100#391—48no—
101#390—22no—
102#390—22no—
103#389—22no—
104#388—24no—
105#388—24no—
106#387—24no—
107#386—24no—
108#386—24no—
109#385—24no—
110#384D411yescharacteristic
111#384D411yescharacteristic
112#382—21no—
113#380—22no—
114#380—22no—
115#380—21no—
116#380C2312yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4D412yes—
126C4×C22⋊C4—24no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95C2312yes—
130C2×#95C2312yes—
131#340—24no—
132#340—24no—
133#327—28no—
134C2×#41—24no—
135C2×#41—24no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8—24no—
141C22×Q8—21no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—42no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4C2×D412yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8C22⋊C414yes—
172#127—22no—
173#127C22⋊C414yes—
174#124—24no—
175#124—24no—
176#121—24no—
177#121—24no—
178#121—22no—
179#121—22no—
180#121C22×C414yes—
181#121C2×D414yes—
182C4×Q8—48no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—48no—
189C4×D4—84no—
190C4×D4C22⋊C418yes—
191#117—48no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4SD1614yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247#106—216no—
248C4⋊C8—24no—
249C4⋊C8—42no—
250#102—216no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8C22×C414yes—
255C4×C8—24no—
256C4×C8—42no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—48no—
266#95—48no—
267C24—22no—
268C24C22×D411yescharacteristic
269#41—48no—
270#41—48no—
271#41—48no—
272#41—48no—
273C2×Q8—24no—
274C2×Q8—416no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—88no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C2×C22⋊C414yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—48no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4C2×SD1614yes—
343C22×C4C2×SD1612yes—
344C22×C4#13512yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4C2×C22⋊C414yes—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—28no—
383C22⋊C4—48no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—48no—
389C42—44no—
390C42—44no—
391C42—82no—
392C42—84no—
393C42C2×C22⋊C412yes—
394C42C2×C22⋊C412yes—
395C42C4×D412yes—
396C42C4×D412yes—
397C42#12112yes—
398C42#12112yes—
399C42#12312yes—
400C42#12312yes—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23C2×C4×D411yescharacteristic
413C23C2×#12011yescharacteristic
414C23C2×#12111yescharacteristic
415C23C22×SD1611yescharacteristic
416C23C2×#13511yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23C2×#13711yescharacteristic
419Q8—48no—
420Q8—88no—
421Q8—88no—
422D4—44no—
423D4—44no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—48no—
428D4—48no—
429D4—82no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—416no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—42no—
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—44no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—168no—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4#38612yes—
478C2×C4#38612yes—
479C2×C4C4×SD1614yes—
480C2×C4#44014yes—
481C2×C4#45012yes—
482C2×C4#45012yes—
483C2×C4#45212yes—
484C2×C4#45212yes—
485C2×C4#45312yes—
486C2×C4#45412yes—
487C2×C4#46114yes—
488C2×C4#46314yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C8—44no—
494C8—82no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22C2×#38611yescharacteristic
515C22C2×C4×SD1611yescharacteristic
516C22C2×#44011yescharacteristic
517C22C2×#45011yescharacteristic
518C22C2×#45211yescharacteristic
519C22C2×#45311yescharacteristic
520C22C2×#45411yescharacteristic
521C22#302811yescharacteristic
522C22#302811yescharacteristic
523C22C2×#46111yescharacteristic
524C22C2×#46311yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—44no—
531C4—44no—
532C4—84no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#187714yes—
538C4#187714yes—
539C4#188314yes—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—84no—
545C2#1839511yescharacteristic
546C2#1839911yescharacteristic
547C2#1840911yescharacteristic
548C2#1841311yescharacteristic
549C2C2×#187711yescharacteristic
550C2C2×#188311yescharacteristic
551C2#2096211yescharacteristic
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. #21834C2semidirect
5. #14155C2semidirect
6. #9045C2semidirect
7. #8014C2semidirect
8. #7908C2semidirect
13. #3227C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
85. #481C2×C4semidirect
90. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#342062
#218342
C2×#15022
#141552
#90452
#80142
#79082