Finite Group Databasebeta

← #192144

Subgroups — #192144

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21987C212yesmaximal
3C2×#1467C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#8857C212yesmaximal
6#8856C212yesmaximal
7#8855C214yesmaximal
8#8021C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2320—22no—
12#2320—22no—
13#2316C2211yescharacteristic
14#2316C2211yescharacteristic
15#2313—21no—
16#2313C2212yes—
17#2310—21no—
18#2295—22no—
19#2295—22no—
20C2×#391—21no—
21C2×#384C2211yescharacteristic
22#2283—21no—
23#2283C2212yes—
24C4×#117—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948—22no—
27#1947—22no—
28#1853C2214yes—
29#1852C2214yes—
30#1847—22no—
31#1836—22no—
32#1467C418yescomplemented
33C2×#324C2212yes—
34C2×#324C2212yes—
35#1441C2212yes—
36#1441C2212yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8D411yescharacteristic
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—24no—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8Q814yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—24no—
62#432C2×C418yes—
63C2×#107—24no—
64C2×#106—24no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#391—22no—
72#391—44no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#384—22no—
86#384—22no—
87#383—22no—
88#383—22no—
89#382—21no—
90#382—21no—
91#382D412yes—
92#382D412yes—
93#381—22no—
94#381—22no—
95#380—21no—
96#380—21no—
97#380C2312yes—
98#380C2312yes—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110#376—22no—
111#376—22no—
112#376—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95C2312yes—
118C43—22no—
119#340—24no—
120#340—24no—
121#324C2×C418yes—
122#324C2×C418yes—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8C4⋊C414yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C4×Q8—28no—
141C4×Q8—28no—
142C4×Q8—28no—
143C4×Q8—28no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C4⋊C414yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×Q812yes—
182C2×C4⋊C4C2×Q812yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42C4⋊C414yes—
201C2×C42C4⋊C414yes—
202C2×C42C22×C412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×Q812yes—
206C2×C42#4114yes—
207#107—84no—
208#106—84no—
209C4⋊C8—28no—
210C22⋊C8—24no—
211C22⋊C8C22×C418yes—
212#97—28no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—82no—
233#95—82no—
234C24C22×D411yescharacteristic
235C2×Q8—28no—
236C2×Q8—28no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C4⋊C412yes—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4C2×C4⋊C412yes—
291C22×C4C2×C4⋊C412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12012yes—
294C22×C4#12011yescharacteristic
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4#13712yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C22×Q811yescharacteristic
302C22×C4C2×#4112yes—
303C2×C8—28no—
304C2×C8—44no—
305C2×C8—44no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—84no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C4⋊C4—82no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—44no—
338C42—44no—
339C42—44no—
340C42—44no—
341C42—84no—
342C42—82no—
343C42—82no—
344C42C4×D414yes—
345C42C4×D414yes—
346C42C22⋊Q814yes—
347C42C22⋊Q814yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23C22×C4⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23C2×AGL(1,ℤ/8)11yescharacteristic
356C23C2×#13711yescharacteristic
357C23#57511yescharacteristic
358C23#57511yescharacteristic
359Q8—416no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38512yes—
425C2×C4#38514yes—
426C2×C4#38512yes—
427C2×C4#42814yes—
428C2×C4#45412yes—
429C2×C4#45612yes—
430C2×C4C2×C4×D412yes—
431C2×C4C2×C22⋊Q812yes—
432C8—48no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22C2×#38511yescharacteristic
446C22C2×#42811yescharacteristic
447C22#291111yescharacteristic
448C22#300611yescharacteristic
449C22#300711yescharacteristic
450C22C2×#45411yescharacteristic
451C22C2×#45611yescharacteristic
452C22#302711yescharacteristic
453C22#302811yescharacteristic
454C22#309411yescharacteristic
455C22#309511yescharacteristic
456C4—24no—
457C4—24no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—162no—
470C4#185714yes—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—21no—
475C2#1833711yescharacteristic
476C2#1833811yescharacteristic
477C2#1834411yescharacteristic
478C2#1834511yescharacteristic
479C2#2084611yescharacteristic
480C2C2×#185711yescharacteristic
481C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
32. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#340592
#219872
C2×#14672
#139262
#88572
#88562
#88552
#80212