Finite Group Databasebeta

← #239876

Subgroups — #239876

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5C22×#333C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7#10415C214yesmaximal, complemented
8#10414C214yesmaximal, complemented
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#427—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2868—21no—
17#2862—21no—
18#2660—22no—
19#2567C2214yes—
20#2546—22no—
21#2502—22no—
22#2434—22no—
23#2317—21no—
24#2317—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1987—21no—
28#1981—22no—
29#1981—21no—
30#1956C2214yes—
31#1949—22no—
32#1948C2212yes—
33#1947C2212yes—
34#1892—22no—
35#1803C2214yes—
36#1787—22no—
37#1735—21no—
38C2×#333—24no—
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1481—24no—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44#1279C2214yes—
45C23×Q8D412yes—
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#427—44no—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D414yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—24no—
91C2×#103—28no—
92C2×C22⋊C8—21no—
93C2×C4×C8C2312yes—
94#398—28no—
95#395—28no—
96#391—24no—
97#391—24no—
98#389—24no—
99#389—24no—
100#385—24no—
101#385—24no—
102#384—22no—
103#384—22no—
104#384—24no—
105#384C2312yes—
106#382—22no—
107#382—22no—
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#376—21no—
118C43—21no—
119#340—24no—
120#340—22no—
121#340—22no—
122#333—24no—
123#333—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—84no—
131C22×Q8C2×D412yes—
132C22×Q8C2×D414yes—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—24no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—82no—
146C4⋊Q8—22no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—88no—
161#125—42no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—42no—
172C4×Q8—42no—
173#117—22no—
174#117—22no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—42no—
179#117SD1614yes—
180#117Q1614yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
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—
232#103—88no—
233C22⋊C8—42no—
234C4×C8—22no—
235C4×C8—22no—
236#95—28no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—41no—
241#95—41no—
242C24#12011yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—42no—
253C2×Q8—44no—
254C2×Q8—88no—
255C2×Q8—88no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—84no—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12112yes—
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#12412yes—
276C22×C4C2×SD1612yes—
277C22×C4C2×Q1612yes—
278C22×C4#13712yes—
279C22×C4#13712yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—48no—
290C2×C8—44no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—84no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C4⋊C4C2×SD1614yes—
319C4⋊C4C2×Q1614yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C42—24no—
325C42—24no—
326C42—44no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—82no—
332C42#12112yes—
333C42#12112yes—
334C42#12112yes—
335C42#12312yes—
336C42#12312yes—
337C42#12412yes—
338C42C22×D411yescharacteristic
339C42C2×#4111yescharacteristic
340C42C2×#4111yescharacteristic
341C42C2×#4111yescharacteristic
342C23—22no—
343C23—22no—
344C23#39412yes—
345C23#44812yes—
346C23#45112yes—
347C23C2×#12011yescharacteristic
348Q8—48no—
349Q8—44no—
350Q8—42no—
351Q8—42no—
352Q8—84no—
353Q8—84no—
354Q8—88no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—88no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4#39412yes—
380C2×C4#39412yes—
381C2×C4#39412yes—
382C2×C4#44812yes—
383C2×C4#45112yes—
384C2×C4#46114yes—
385C2×C4#46214yes—
386C2×C4#48714yes—
387C2×C4#49114yes—
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C2×#12411yescharacteristic
394C2×C4C22×SD1611yescharacteristic
395C2×C4C22×Q1611yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57712yes—
399C2×C4#57812yes—
400C8—42no—
401C8—42no—
402C8—82no—
403C8—82no—
404C22—24no—
405C22—24no—
406C22—22no—
407C22—22no—
408C22#204314yes—
409C22C2×#39411yescharacteristic
410C22C2×#44811yescharacteristic
411C22C2×#45111yescharacteristic
412C22#301312yes—
413C22#301912yes—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—44no—
419C4—44no—
420C4—82no—
421C4—84no—
422C4—82no—
423C4—162no—
424C4#204314yes—
425C4C2×#46111yescharacteristic
426C4C2×#46211yescharacteristic
427C4#305711yescharacteristic
428C4#305911yescharacteristic
429C4#309712yes—
430C4C2×#48711yescharacteristic
431C4C2×#49111yescharacteristic
432C2—24no—
433C2#1887212yes—
434C2#1887312yes—
435C2#2348711yescharacteristic
436C2#2349311yescharacteristic
437C2C2×#204311yescharacteristic
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. #10415C2semidirect
8. #10414C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#218852
#218042
#211692
C22×#3332
C2×#14812
#104152
#104142
#78912