Finite Group Databasebeta

← #192828

Subgroups — #192828

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic
6#8894C214yesmaximal, complemented
7#8885C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×#106—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2367—22no—
17#2366—21no—
18#2365—21no—
19#2354C2212yes—
20#2352—22no—
21D4×C4⋊C4C2212yes—
22#2325—22no—
23#2311—21no—
24C4×#121—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990—21no—
29#1957—21no—
30#1948—22no—
31#1948C2214yes—
32#1947—22no—
33#1874—22no—
34#1852—22no—
35#1845C2212yes—
36#1834C2212yes—
37#1802C2214yes—
38C2×#340—21no—
39#1480C418yescomplemented
40C2×D4⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C22×#41—21no—
45C23×C8—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×#127D411yescharacteristic
52C2×#124—21no—
53C2×#121—21no—
54C2×#121D412yes—
55#520—44no—
56#520—44no—
57C2×C4×Q8—22no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2312yes—
63C2×#117—21no—
64C2×#117—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431C2×C414yescomplemented
77C2×#107—24no—
78C2×#106—24no—
79C2×#106—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2×C414yescomplemented
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#102—24no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#391—22no—
92#390—22no—
93#390—22no—
94#389—22no—
95#388—24no—
96#388—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#385—24no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2312yes—
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#382—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380D412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—24no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—24no—
124D4⋊C8C2×C4116yescomplemented
125C2×#41—24no—
126C2×#41—24no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C22×Q8—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—42no—
149C22×C8—42no—
150C4⋊Q8—22no—
151C4⋊Q8C4⋊C414yes—
152#127—22no—
153#127C4⋊C414yes—
154#124—24no—
155#124—24no—
156#121—24no—
157#121—24no—
158#121—24no—
159#121C4⋊C418yes—
160C4×Q8—48no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—48no—
166C4×D4—416no—
167C4×D4—84no—
168C4×D4C22×C414yes—
169C4×D4C2×Q814yes—
170#117—44no—
171#117—42no—
172#117—42no—
173#117—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4D814yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—82no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#107—84no—
223#106—48no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8C22×C414yes—
230#102—84no—
231C22⋊C8—24no—
232C22⋊C8—42no—
233C4×C8—22no—
234C4×C8—42no—
235C4×C8C22×C414yes—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24—22no—
243C24#12011yescharacteristic
244#41—48no—
245#41—48no—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—28no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—81no—
262C2×D4—81no—
263C2×D4C2×C4⋊C414yes—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C4⋊C412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4#12312yes—
301C22×C4C2×D812yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—48no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—48no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4C2×C4⋊C414yes—
342C4⋊C4C2×D814yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—28no—
346C22⋊C4—48no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C42—48no—
351C42—44no—
352C42—44no—
353C42—82no—
354C42C2×C4⋊C412yes—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C22⋊Q812yes—
358C42C22⋊Q812yes—
359C42#12312yes—
360C42C23×C411yescharacteristic
361C42C22×Q811yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#38512yes—
376C23#44712yes—
377C23#45212yes—
378C23C2×#12011yescharacteristic
379Q8—48no—
380D4—28no—
381D4—416no—
382D4—44no—
383D4—48no—
384D4—84no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—168no—
420C2×C4—164no—
421C2×C4#38512yes—
422C2×C4#38512yes—
423C2×C4#38512yes—
424C2×C4C4×D814yes—
425C2×C4#44014yes—
426C2×C4#44712yes—
427C2×C4#45212yes—
428C2×C4#47414yes—
429C2×C4#47814yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C22×D811yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57712yes—
441C2×C4#57712yes—
442C8—44no—
443C8—44no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22#186214yes—
459C22C2×#38511yescharacteristic
460C22C2×#44711yescharacteristic
461C22C2×#45211yescharacteristic
462C22#300911yescharacteristic
463C22#301011yescharacteristic
464C22#302111yescharacteristic
465C22#302211yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—44no—
471C4—44no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—162no—
476C4#186214yes—
477C4#291212yes—
478C4C2×C4×D811yescharacteristic
479C4C2×#44011yescharacteristic
480C4C2×#47411yescharacteristic
481C4C2×#47811yescharacteristic
482C4#309912yes—
483C2—24no—
484C2—84no—
485C2#1890211yescharacteristic
486C2#1890911yescharacteristic
487C2#1891211yescharacteristic
488C2#1891911yescharacteristic
489C2#2084811yescharacteristic
490C2#2086511yescharacteristic
491C2C2×#186211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21982C2semidirect
4. C2×#1948C2semidirect
6. #8894C2semidirect
7. #8885C2semidirect
8. #8039C2semidirect
9. #7892C2semidirect
39. #1480C4semidirect
76. #431C2×C4semidirect
83. C2×C4⋊C8C2×C4semidirect
124. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219822
#218592
C2×#19482
C2×#14802
#88942
#88852
#80392
#78922