Finite Group Databasebeta

← #239882

Subgroups — #239882

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10422C214yesmaximal, complemented
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#427—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2862—21no—
17#2862—21no—
18#2642—22no—
19#2568—22no—
20#2568C2214yes—
21#2503—22no—
22#2503—22no—
23#2310—21no—
24#2310—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1990C2214yes—
28#1988—22no—
29#1987—21no—
30#1981—21no—
31#1949—22no—
32#1948C2212yes—
33#1947C2212yes—
34#1892—22no—
35#1823—22no—
36#1803C2214yes—
37#1735—21no—
38C2×#332—24no—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1481—24no—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#427—44no—
77C2×#107—22no—
78C2×#106—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D414yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103—24no—
90C2×C22⋊C8—21no—
91C2×C4×C8C2312yes—
92#400—28no—
93#400—28no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391—24no—
99#391—24no—
100#391—24no—
101#391—24no—
102#384C2312yes—
103#382—24no—
104#382—24no—
105#382—21no—
106#381—24no—
107#381—24no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116#376—21no—
117C43—21no—
118#340—24no—
119#340—28no—
120#340—22no—
121#340—22no—
122#332—24no—
123#332—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8C2×D412yes—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C22×C8—24no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—82no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163#125—42no—
164#125—42no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—42no—
180#117—42no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—41no—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220#107—44no—
221#106—44no—
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8C2×D412yes—
230C4⋊C8C2×D412yes—
231#103—84no—
232C22⋊C8—42no—
233C4×C8—22no—
234C4×C8—22no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—22no—
240#95—22no—
241#95—41no—
242#95—41no—
243C24#12011yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—84no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—84no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12112yes—
279C22×C4#12112yes—
280C22×C4#12112yes—
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12412yes—
284C22×C4#13512yes—
285C22×C4#13512yes—
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4#13712yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—48no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—44no—
302C2×C8—84no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—82no—
330C42—44no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—82no—
338C42#12112yes—
339C42#12112yes—
340C42#12112yes—
341C42#12312yes—
342C42#12312yes—
343C42#12412yes—
344C42C22×D411yescharacteristic
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C23—22no—
349C23—22no—
350C23#39412yes—
351C23#44912yes—
352C23#45212yes—
353C23C2×#12011yescharacteristic
354Q8—44no—
355Q8—42no—
356Q8—42no—
357Q8—84no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362Q8—88no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#39412yes—
389C2×C4#39412yes—
390C2×C4#39412yes—
391C2×C4#44912yes—
392C2×C4#45212yes—
393C2×C4#46314yes—
394C2×C4#46414yes—
395C2×C4#48814yes—
396C2×C4#48914yes—
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12311yescharacteristic
401C2×C4C2×#12311yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4#57712yes—
408C2×C4#57812yes—
409C8—42no—
410C8—42no—
411C8—82no—
412C8—82no—
413C22—22no—
414C22—24no—
415C22—22no—
416C22—24no—
417C22#204214yes—
418C22C2×#39411yescharacteristic
419C22C2×#44911yescharacteristic
420C22C2×#45211yescharacteristic
421C22#301312yes—
422C22#302012yes—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—84no—
432C4—162no—
433C4#204214yes—
434C4C2×#46311yescharacteristic
435C4C2×#46411yescharacteristic
436C4#305711yescharacteristic
437C4#305911yescharacteristic
438C4#309712yes—
439C4C2×#48811yescharacteristic
440C4C2×#48911yescharacteristic
441C2—24no—
442C2#1888612yes—
443C2#1888712yes—
444C2#2348711yescharacteristic
445C2#2349411yescharacteristic
446C2C2×#204211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10423C2semidirect
8. #10422C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#218852
#218332
#211692
C22×#3322
C2×#14812
#104232
#104222
#78912