Finite Group Databasebeta

← #192440

Subgroups — #192440

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic, complemented
6#8873C218yesmaximal, complemented
7#8014C211yesmaximal, characteristic, complemented
8#7908C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#520—21no—
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2367—22no—
14#2366—21no—
15#2365—21no—
16#2354—21no—
17#2352—22no—
18D4×C4⋊C4—21no—
19#2325C2214yes—
20#2311—21no—
21C4×#121—22no—
22C2×#389—22no—
23#2285C2211yescharacteristic
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—21no—
29#1957—21no—
30#1948—22no—
31#1874C2214yes—
32#1853C2214yes—
33#1844—21no—
34#1834—21no—
35#1802—22no—
36#1502C418yescomplemented
37C2×#327C2214yes—
38#1444C2214yes—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43#532—22no—
44#532—22no—
45#532C2×C414yes—
46#532Q814yes—
47C2×C4⋊Q8D411yescharacteristic
48C2×#127D411yescharacteristic
49C2×#124—21no—
50C2×#121—21no—
51C2×#121C2312yes—
52#520—44no—
53#520—44no—
54C2×C4×Q8—22no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4D412yes—
60C2×#117—22no—
61C2×#117—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431C2×C414yescomplemented
77C2×#106—24no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×#102—24no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#391—22no—
86#390—22no—
87#390—22no—
88#389—22no—
89#389—48no—
90#388—24no—
91#388—24no—
92#387—24no—
93#386—24no—
94#386—24no—
95#385—24no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—44no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—21no—
104#380—22no—
105#380—22no—
106#380—21no—
107#380C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yescomplemented
116C4×C4⋊C4D412yes—
117C4×C22⋊C4—24no—
118C2×#95—24no—
119C2×#95C2312yes—
120C2×#95C2312yes—
121#340—24no—
122#340—24no—
123#327C2×C4116yescomplemented
124C2×#41—24no—
125C2×#41—24no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C22×Q8—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×D412yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—22no—
147C4⋊Q8C4⋊C414yescomplemented
148#127—22no—
149#127C4⋊C414yescomplemented
150#124—24no—
151#124—24no—
152#121—24no—
153#121—24no—
154#121—22no—
155#121—22no—
156#121C22×C414yes—
157#121C2×Q814yes—
158C4×Q8—48no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—48no—
163C4×D4—48no—
164C4×D4—48no—
165C4×D4—84no—
166C4×D4C4⋊C418yescomplemented
167#117—48no—
168#117—42no—
169#117—42no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4SD1614yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—44no—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226#106—84no—
227C4⋊C8—24no—
228C4⋊C8—42no—
229#102—84no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8C22×C414yes—
234C4×C8—24no—
235C4×C8—42no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—48no—
244#95C22×C414yes—
245C24—22no—
246C24C22×D411yescharacteristic
247#41—48no—
248#41—48no—
249#41—48no—
250#41—48no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4C2×C4⋊C414yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4C2×SD1614yes—
313C22×C4C2×SD1612yes—
314C22×C4#13512yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—216no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—816no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4C2×C4⋊C414yes—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—28no—
354C22⋊C4—48no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C42—48no—
360C42—48no—
361C42—44no—
362C42—44no—
363C42—82no—
364C42C2×C4⋊C412yes—
365C42C2×C4⋊C412yes—
366C42C4×D412yes—
367C42C4×D412yes—
368C42C22⋊Q812yes—
369C42C22⋊Q812yes—
370C42#12312yes—
371C42#12312yes—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—24no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23C2×C4×D411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23C22×SD1611yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×AGL(1,ℤ/8)11yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390Q8—48no—
391D4—44no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—48no—
397D4—48no—
398D4—82no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—416no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—168no—
444C2×C4#38512yes—
445C2×C4#38512yes—
446C2×C4#38512yes—
447C2×C4#38512yes—
448C2×C4C4×SD1614yes—
449C2×C4#44214yes—
450C2×C4#44912yes—
451C2×C4#44912yes—
452C2×C4#45012yes—
453C2×C4#45012yes—
454C2×C4#45312yes—
455C2×C4#45612yes—
456C2×C4#47614yes—
457C2×C4#47814yes—
458C2×C4C22×C4⋊C411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×#12311yescharacteristic
462C8—44no—
463C8—82no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C2×#38511yescharacteristic
484C22C2×C4×SD1611yescharacteristic
485C22C2×#44211yescharacteristic
486C22C2×#45011yescharacteristic
487C22C2×#44911yescharacteristic
488C22C2×#45311yescharacteristic
489C22C2×#45611yescharacteristic
490C22#302711yescharacteristic
491C22#302811yescharacteristic
492C22C2×#47611yescharacteristic
493C22C2×#47811yescharacteristic
494C4—22no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—48no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—162no—
506C4#185914yes—
507C4#186414yes—
508C4#186414yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—84no—
514C2#1839511yescharacteristic
515C2#1839811yescharacteristic
516C2#1840211yescharacteristic
517C2#1840511yescharacteristic
518C2C2×#185911yescharacteristic
519C2C2×#186411yescharacteristic
520C2#2089011yescharacteristic
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
2. #21981C2semidirect
5. #13910C2semidirect
6. #8873C2semidirect
7. #8014C2semidirect
8. #7908C2semidirect
36. #1502C4semidirect
76. #431C2×C4semidirect
115. C4×C4⋊C4C2×C4semidirect
123. #327C2×C4semidirect
147. C4⋊Q8C4⋊C4semidirect
149. #127C4⋊C4semidirect
166. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219812
#218592
C2×#15022
#139102
#88732
#80142
#79082