Finite Group Databasebeta

← #241222

Subgroups — #241222

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#14348C211yesmaximal, characteristic
5#10571C214yesmaximal
6#10568C214yesmaximal
7#8022C211yesmaximal, characteristic
8#8013C211yesmaximal, characteristic
9#7914C211yesmaximal, characteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2845—21no—
13#2844—21no—
14#2666—22no—
15#2660—21no—
16#2659C2212yes—
17#2647—21no—
18#2646C2212yes—
19#2570—21no—
20#2550—21no—
21#2442—21no—
22#2440—21no—
23#2438—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2283C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1948—22no—
30#1948—21no—
31#1947—21no—
32#1853C2212yes—
33#1853C2214yes—
34#1852C2212yes—
35#1847—21no—
36#1844—21no—
37#1836—21no—
38#1834—21no—
39#1442—22no—
40#1442C2214yes—
41C4⋊#95—21no—
42C4⋊#95—21no—
43#1294C2212yes—
44#1294C2214yes—
45#1294C2212yes—
46#1280C2214yes—
47C2×C4⋊Q8—21no—
48C2×#125—21no—
49C2×C22⋊Q8—22no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432Q814yescomplemented
60#431—22no—
61#431Q814yescomplemented
62C2×#107—22no—
63C2×#106—24no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C22⋊C8C2312yes—
68C2×C22⋊C8C2312yes—
69C2×#97—21no—
70C2×C4×C8—21no—
71#400—22no—
72#400—22no—
73#400—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#395—22no—
84#395—22no—
85#395—22no—
86#395—22no—
87#391—22no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#385—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#383—21no—
105#382—21no—
106#382—21no—
107#382—21no—
108#382—21no—
109#382—21no—
110#382—21no—
111#382D411yescharacteristic
112#382D411yescharacteristic
113#381—21no—
114#380—22no—
115#380—21no—
116#380C2311yescharacteristic
117#380C2312yes—
118#380C2311yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124#376—21no—
125#376—22no—
126#376Q814yescomplemented
127C2×#95—21no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95D412yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133#340—24no—
134#340—22no—
135#340—22no—
136C22×Q8—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—82no—
147#125—82no—
148C22⋊Q8—84no—
149#117—82no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q812yes—
198C2×C42C2×Q812yes—
199C2×C42C2×Q812yes—
200C2×C42#4112yes—
201#107—82no—
202#106—84no—
203#106—82no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8C2×Q814yes—
209C22⋊C8C2×Q814yes—
210#97—24no—
211C4×C8—24no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—84no—
235#95—84no—
236#95C2×Q814yes—
237C24C22×D411yescharacteristic
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—82no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C4⋊Q812yes—
279C22×C4C4⋊Q812yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q811yescharacteristic
283C22×C4C22×Q811yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C42—48no—
326C42—42no—
327C42—42no—
328C42—82no—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42C22⋊Q812yes—
332C42C22⋊Q812yes—
333C42C4⋊Q812yes—
334C42C4⋊Q812yes—
335C42C4⋊Q812yes—
336C42C4⋊Q812yes—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23C2×#12011yescharacteristic
342C23C2×C22⋊Q811yescharacteristic
343C23C2×C4⋊Q811yescharacteristic
344C23#57712yes—
345C23#57712yes—
346Q8—164no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#39512yes—
403C2×C4#39512yes—
404C2×C4#39512yes—
405C2×C4#39512yes—
406C2×C4#45412yes—
407C2×C4#45512yes—
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C4⋊Q811yescharacteristic
411C2×C4C2×C4⋊Q811yescharacteristic
412C8—44no—
413C8—44no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22C2×#39511yescharacteristic
423C22#300911yescharacteristic
424C22#301011yescharacteristic
425C22C2×#45511yescharacteristic
426C22C2×#45411yescharacteristic
427C22#302711yescharacteristic
428C22#302811yescharacteristic
429C22#308412yes—
430C22#316312yes—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—84no—
439C4—81no—
440C4—81no—
441C4—82no—
442C4—82no—
443C4—81no—
444C4—82no—
445C4—82no—
446C4—81no—
447C4—162no—
448C4#206114yes—
449C2—22no—
450C2—22no—
451C2#1842711yescharacteristic
452C2#1843211yescharacteristic
453C2#1843811yescharacteristic
454C2#1844111yescharacteristic
455C2#2355611yescharacteristic
456C2C2×#206111yescharacteristic
457C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
59. #432Q8semidirect
61. #431Q8semidirect
126. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#219872
#219812
#143482
#105712
#105682
#80222
#80132
#79142