Finite Group Databasebeta

← #194838

Subgroups — #194838

Order 512.

1782 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#21759C212yesmaximal, complemented
4C22×#323C211yesmaximal, characteristic
5#9024C214yesmaximal, complemented
6#9016C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2626—24no—
12#2626C418yescomplemented
13C2×#399C2212yes—
14C2×#396—22no—
15C2×#393C2212yes—
16#2312—24no—
17#2312—22no—
18#2312C2214yes—
19C2×#388—24no—
20C2×#385—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2212yes—
23C2×#379—21no—
24C2×#379—21no—
25C2×C4×C22⋊C4—21no—
26#1892—24no—
27#1872—24no—
28#1824C2214yescomplemented
29#1787C2214yescomplemented
30C2×#323—24no—
31C2×#323C2214yes—
32#1294C2218yescomplemented
33C26D411yescharacteristic
34C23×Q8C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4—22no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—24no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4C2×C414yes—
59C22×C22⋊C4D412yes—
60C22×C22⋊C4D414yes—
61C22×C22⋊C4C2312yes—
62C22×C42—22no—
63C2×#103—28no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#399—24no—
69#399C2×C418yescomplemented
70#396—216no—
71#393—24no—
72#393C2×C418yescomplemented
73#388—216no—
74#388—216no—
75#385—28no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380C2×C414yes—
80#380D414yes—
81#380C2314yes—
82#379—24no—
83#379—28no—
84#379—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#323—28no—
93C25—21no—
94C25—21no—
95C25—22no—
96C25—24no—
97C25—24no—
98C25—82no—
99C25—82no—
100C25C22⋊C414yes—
101C25C2×D412yes—
102C25C2×D411yescharacteristic
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—82no—
107C22×Q8C22×C412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C22⋊C414yes—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4#4114yes—
125C23×C4C2411yescharacteristic
126C22×C8—24no—
127C22×C8—44no—
128C22×C8—84no—
129C22⋊Q8—28no—
130C22⋊Q8—28no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—84no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—84no—
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C2×D418yes—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—84no—
170C2×C22⋊C4—84no—
171C2×C22⋊C4—84no—
172C2×C22⋊C4—84no—
173C2×C22⋊C4C22⋊C418yes—
174C2×C22⋊C4C22×C414yes—
175C2×C22⋊C4C2×D414yes—
176C2×C42—28no—
177C2×C42—84no—
178#103—88no—
179C22⋊C8—24no—
180C22⋊C8—24no—
181C22⋊C8—44no—
182#95—28no—
183#95—28no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95C22×C414yes—
188C24—22no—
189C24—24no—
190C24—24no—
191C24—22no—
192C24—24no—
193C24—24no—
194C24—28no—
195C24—28no—
196C24—42no—
197C24—42no—
198C24—44no—
199C24—44no—
200C24—44no—
201C24—44no—
202C24—44no—
203C24—44no—
204C24—48no—
205C24—82no—
206C24—82no—
207C24—84no—
208C24—84no—
209C24—84no—
210C24—84no—
211C24—816no—
212C24—816no—
213C24C2×C22⋊C414yes—
214C24C2×C22⋊C412yes—
215C24#12011yescharacteristic
216C24#12011yescharacteristic
217C24#12312yes—
218C24#13514yes—
219C24#13714yes—
220C24C22×D411yescharacteristic
221C2×Q8—24no—
222C2×Q8—48no—
223C2×Q8—42no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×Q8—84no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—82no—
258C22×C4—84no—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C4×D414yes—
262C22×C4C4×D414yes—
263C22×C4#12012yes—
264C22×C4#12114yes—
265C22×C4#12114yes—
266C22×C4#12312yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271C2×C8—48no—
272C2×C8—44no—
273C2×C8—88no—
274C4⋊C4—24no—
275C4⋊C4—48no—
276C4⋊C4—42no—
277C4⋊C4—48no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—88no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—216no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—88no—
291C22⋊C4—88no—
292C22⋊C4—88no—
293C22⋊C4—88no—
294C42—88no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—24no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—48no—
309C23—48no—
310C23—44no—
311C23—44no—
312C23—416no—
313C23—48no—
314C23—48no—
315C23—44no—
316C23—48no—
317C23—48no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—82no—
322C23—82no—
323C23—84no—
324C23—84no—
325C23—88no—
326C23—816no—
327C23—816no—
328C23—816no—
329C23—816no—
330C23—832no—
331C23#38612yes—
332C23#38612yes—
333C23#41614yes—
334C23#41914yes—
335C23#45214yes—
336C23#45214yes—
337C23#48518yes—
338C23C22×C22⋊C411yescharacteristic
339C23C2×#12011yescharacteristic
340C23C2×#12311yescharacteristic
341C23C2×#13512yes—
342C23C2×#13712yes—
343Q8—48no—
344Q8—84no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—88no—
373C2×C4—816no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#38614yes—
379C2×C4C2×C4×D412yes—
380C2×C4C2×#12112yes—
381C8—84no—
382C22—24no—
383C22—28no—
384C22—24no—
385C22—28no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—44no—
395C22—44no—
396C22—48no—
397C22—48no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—84no—
402C22—88no—
403C22—816no—
404C22—816no—
405C22—88no—
406C22#187314yes—
407C22#187314yes—
408C22C2×#38611yescharacteristic
409C22C2×#41611yescharacteristic
410C22C2×#41911yescharacteristic
411C22C2×#45212yes—
412C22#301912yes—
413C22#302612yes—
414C22C2×#48512yes—
415C4—22no—
416C4—22no—
417C4—24no—
418C4—44no—
419C4—84no—
420C4—88no—
421C4—82no—
422C4—82no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—24no—
427C2—24no—
428C2—84no—
429C2#1829714yes—
430C2C2×#187311yescharacteristic
431C2#2095311yescharacteristic
432C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21759C2semidirect
5. #9024C2semidirect
6. #9016C2semidirect
7. #8023C2semidirect
12. #2626C4semidirect
28. #1824C22semidirect
29. #1787C22semidirect
32. #1294C22semidirect
69. #399C2×C4semidirect
72. #393C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23122
#217592
C22×#3232
#90242
#90162
#80232