Finite Group Databasebeta

← #238527

Subgroups — #238527

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35166C211yesmaximal, characteristic
2#21386C212yesmaximal, complemented
3C2×#1517C212yesmaximal, complemented
4#14150C211yesmaximal, characteristic, complemented
5#10403C214yesmaximal, complemented
6#10402C214yesmaximal, complemented
7#7908C211yesmaximal, characteristic, complemented
8C2×#537—21no—
9C2×#537—21no—
10C2×#532C2212yes—
11C2×#431C2212yes—
12#2856—22no—
13#2656C2214yes—
14#2542—22no—
15#2541—22no—
16#2540—24no—
17C2×#389—21no—
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1988—22no—
22#1982—22no—
23#1874C2214yes—
24#1874C2214yes—
25C4×#102—24no—
26#1517—28no—
27C2×#327C2218yes—
28#1444C2212yes—
29#1442C2212yes—
30#1436—22no—
31C4⋊#95—21no—
32#1280C2214yes—
33C22×#41—22no—
34#537—44no—
35#537—44no—
36#537—44no—
37#537—44no—
38#532—24no—
39#532—24no—
40#532—24no—
41#532—24no—
42C2×C4⋊Q8D412yes—
43C2×#127—22no—
44C2×#127D412yes—
45C2×#124—22no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121C2314yes—
54C2×C4×D4—22no—
55C2×C4×D4D414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#431—24no—
66#431—24no—
67C2×C4⋊C8—22no—
68C2×#102—24no—
69C2×#102—24no—
70C2×C22⋊C8C2314yes—
71C2×C4×C8—22no—
72#399—28no—
73#392—28no—
74#390—24no—
75#390—24no—
76#390—24no—
77#389—44no—
78#384—22no—
79#382—22no—
80#382D414yescomplemented
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2312yes—
91C2×#95C2312yes—
92#340—24no—
93#340—24no—
94#327—216no—
95C2×#41—44no—
96C2×#41—44no—
97C2×#41—44no—
98C2×#41—44no—
99C2×#41—44no—
100C2×#41—44no—
101C22×Q8—22no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4C2×D414yes—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—24no—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C22×C8—24no—
118C4⋊Q8—24no—
119C4⋊Q8—24no—
120#127—24no—
121#127—24no—
122#127—84no—
123#124—44no—
124#124—44no—
125C22⋊Q8—416no—
126C22⋊Q8—416no—
127#121—24no—
128#121—24no—
129#121—24no—
130#121—24no—
131#121—416no—
132#121—416no—
133#121—48no—
134#121—48no—
135#121—44no—
136#121—44no—
137#121—48no—
138#121—48no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—44no—
142C4×D4—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4D814yes—
161C2×C4⋊C4SD1614yes—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—82no—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42#4114yes—
177C2×C42#4112yes—
178C4⋊C8—44no—
179#102—48no—
180#102—48no—
181C22⋊C8—24no—
182C22⋊C8—24no—
183C4×C8—44no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95—44no—
188#95—41no—
189#95—41no—
190#95—44no—
191#95—82no—
192#95C2×D414yes—
193C24—24no—
194C24C22×D411yescharacteristic
195#41—88no—
196#41—88no—
197#41—88no—
198#41—88no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—84no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12012yes—
264C22×C4#12112yes—
265C22×C4#12112yes—
266C22×C4#12412yes—
267C22×C4#12412yes—
268C22×C4C2×D814yes—
269C22×C4C2×D812yes—
270C22×C4C2×SD1614yes—
271C22×C4C2×SD1612yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4C22×D412yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—88no—
294C22⋊C4—416no—
295C22⋊C4—416no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—48no—
299C42—44no—
300C42—82no—
301C42—84no—
302C42#12114yes—
303C42#12114yes—
304C42#12314yes—
305C42#12314yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23C2×#12011yescharacteristic
318C23C2×#12111yescharacteristic
319C23C2×#12411yescharacteristic
320C23C22×D811yescharacteristic
321C23C22×SD1611yescharacteristic
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×AGL(1,ℤ/8)11yescharacteristic
324Q8—88no—
325D4—48no—
326D4—84no—
327D4—88no—
328D4—84no—
329D4—84no—
330D4—84no—
331D4—84no—
332D4—88no—
333D4—168no—
334D4—168no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—816no—
381C2×C4#39412yes—
382C2×C4#39412yes—
383C2×C4#39414yes—
384C2×C4#44714yes—
385C2×C4#45014yes—
386C2×C4#45914yes—
387C2×C4#46014yes—
388C2×C4#48614yes—
389C2×C4#49014yes—
390C2×C4C2×#12112yes—
391C2×C4C2×#12312yes—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—84no—
406C22—84no—
407C22—84no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22—84no—
412C22C2×#39411yescharacteristic
413C22C2×#44711yescharacteristic
414C22C2×#45011yescharacteristic
415C22#301911yescharacteristic
416C22#302011yescharacteristic
417C22#302511yescharacteristic
418C22#302511yescharacteristic
419C22C2×#45911yescharacteristic
420C22C2×#46011yescharacteristic
421C22C2×#48611yescharacteristic
422C22C2×#49011yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—44no—
428C4—44no—
429C4—84no—
430C4—82no—
431C4—82no—
432C4—84no—
433C4#204118yes—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—164no—
439C2#1821011yescharacteristic
440C2#1821111yescharacteristic
441C2#1821811yescharacteristic
442C2#1821911yescharacteristic
443C2#2349911yescharacteristic
444C2C2×#204111yescharacteristic
445C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35166C2
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
2. #21386C2semidirect
3. C2×#1517C2semidirect
4. #14150C2semidirect
5. #10403C2semidirect
6. #10402C2semidirect
7. #7908C2semidirect
80. #382D4semidirect

Maximal subgroups

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

subgroupindex
#351662
#213862
C2×#15172
#141502
#104032
#104022
#79082