Finite Group Databasebeta

← #201487

Subgroups — #201487

Order 512.

1815 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35237C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic, complemented
3#21815C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic, complemented
5#9475C218yesmaximal, complemented
6#7941C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10#2920C2212yes—
11#2847—22no—
12#2631C414yescomplemented
13#2563C412yescomplemented
14#2541C412yescomplemented
15#2535—22no—
16#2504—21no—
17#2433—21no—
18#2404C2214yes—
19#2328—22no—
20#2283C2211yescharacteristic
21#2275—24no—
22#2274—21no—
23#2273C2214yes—
24C42×D4—24no—
25C4×#117C2212yes—
26C2×C4×C22⋊C4—22no—
27#1980—24no—
28#1962—24no—
29#1789C2218yes—
30#1466—22no—
31#1462C2214yes—
32#1460—22no—
33#1284C2218yes—
34C22×#41C2312yes—
35C23×D4D411yescharacteristic
36C24×C4D412yes—
37#532—22no—
38#532—22no—
39#532—22no—
40#532—22no—
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#127—22no—
43C2×#127C2×C411yescharacteristic, complemented
44C2×#124C2×C412yescomplemented
45C2×#121—21no—
46C2×#121—21no—
47C2×#121—24no—
48C2×#120—24no—
49#518—216no—
50C2×C4×D4—28no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C412yes—
59C2×C4×D4D412yes—
60C2×C4×D4D412yes—
61C2×#117—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C22⋊C4—24no—
64C22×C22⋊C4C2312yes—
65C22×C42—24no—
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#431—22no—
70#431—22no—
71#423—22no—
72C2×#104—216no—
73#407C2314yes—
74C4×M16—22no—
75#397—24no—
76#393C2×C418yescomplemented
77#392—24no—
78#392C2×C418yescomplemented
79#390—22no—
80#390—22no—
81#386—28no—
82#386—24no—
83#386—28no—
84#383C2×C412yescomplemented
85#382—22no—
86#382—21no—
87#381C2312yes—
88#380—24no—
89#380—21no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C412yescomplemented
93C4×C4⋊C4D414yescomplemented
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—216no—
100C4×C22⋊C4C2314yes—
101#376—21no—
102#376—22no—
103C2×#95—24no—
104C43D414yescomplemented
105#323—28no—
106C25C2×D412yes—
107C2×#41—24no—
108C2×#41—24no—
109C2×#41—84no—
110C22×Q8C22×C412yes—
111C22×D4—28no—
112C22×D4—22no—
113C22×D4—28no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22⋊C412yes—
119C22×D4C22⋊C412yes—
120C22×D4C22×C412yes—
121C23×C4—28no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—84no—
128C23×C4C22⋊C414yes—
129C23×C4C22⋊C414yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C2×M16—24no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137#127—22no—
138#127—22no—
139#127—48no—
140#124—24no—
141#124—24no—
142#121—41no—
143#121—42no—
144#121—41no—
145#121—42no—
146#121—41no—
147#121—41no—
148#121—88no—
149#120—88no—
150C4×D4—264no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—42no—
162C4×D4—42no—
163#117—216no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—232no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4C22×C414yes—
183C2×C22⋊C4C2×D414yes—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—28no—
188C2×C42—88no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D418yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42#4112yes—
197C2×C42#4112yes—
198C4⋊C8—42no—
199C4⋊C8—42no—
200#104—816no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203#97—42no—
204C4×C8—42no—
205#95—28no—
206#95—22no—
207#95—22no—
208#95—28no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95C22×C414yes—
213C24—22no—
214C24—22no—
215C24—28no—
216C24—22no—
217C24—44no—
218C24—82no—
219C24C2×C22⋊C414yes—
220C24C22×D411yescharacteristic
221#41—416no—
222#41—88no—
223C2×Q8—24no—
224C2×Q8—82no—
225C2×D4—216no—
226C2×D4—216no—
227C2×D4—28no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—44no—
233C2×D4—48no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—416no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—44no—
241C2×D4—82no—
242C2×D4—88no—
243C2×D4—88no—
244C2×D4#10418yescomplemented
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—232no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—44no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4—84no—
265C22×C4—88no—
266C22×C4—88no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12012yes—
276C22×C4#12011yescharacteristic
277C22×C4#12312yes—
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4112yes—
284M16—48no—
285C2×C8—44no—
286C4⋊C4—216no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C22⋊C4—216no—
294C22⋊C4—232no—
295C22⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—88no—
302C22⋊C4—88no—
303C42—216no—
304C42—48no—
305C42—48no—
306C42—44no—
307C42—88no—
308C42C4×D412yes—
309C42C4×D412yes—
310C42C4×D412yes—
311C42C4×D412yes—
312C42#12112yes—
313C42#12112yes—
314C42#12112yes—
315C42#12112yes—
316C23—28no—
317C23—28no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—24no—
322C23—41no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—42no—
331C23—82no—
332C23—82no—
333C23—816no—
334C23C2×#10418yes—
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12011yescharacteristic
337C23C2×#12311yescharacteristic
338Q8—48no—
339D4—216no—
340D4—216no—
341D4—48no—
342D4—82no—
343D4—84no—
344D4—84no—
345D4—84no—
346D4—82no—
347D4—1616no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—88no—
377C2×C4—82no—
378C2×C4—832no—
379C2×C4#38612yes—
380C2×C4#38611yescharacteristic
381C2×C4#38611yescharacteristic
382C2×C4#38611yescharacteristic
383C2×C4#38612yes—
384C2×C4#38611yescharacteristic
385C2×C4C2×#10414yes—
386C2×C4C2×C4×D412yes—
387C2×C4C2×#12112yes—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—28no—
397C22—44no—
398C22—48no—
399C22—416no—
400C22—44no—
401C22—44no—
402C22—84no—
403C22—88no—
404C22—88no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22C2×#38611yescharacteristic
409C22C22×#10412yes—
410C22#302712yes—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—416no—
418C4—44no—
419C4—84no—
420C4—82no—
421C4#189614yes—
422C4#191112yes—
423C2—22no—
424C2—22no—
425C2—41no—
426C2—48no—
427C2—162no—
428C2#2096111yescharacteristic
429C2C2×#189611yescharacteristic
430C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35237C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#518C2semidirect
3. #21815C2semidirect
4. #17364C2semidirect
5. #9475C2semidirect
6. #7941C2semidirect
12. #2631C4semidirect
13. #2563C4semidirect
14. #2541C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
43. C2×#127C2×C4semidirect
44. C2×#124C2×C4semidirect
76. #393C2×C4semidirect
78. #392C2×C4semidirect
84. #383C2×C4semidirect
92. C4×C4⋊C4C2×C4semidirect
93. C4×C4⋊C4D4semidirect
104. C43D4semidirect
244. C2×D4#104semidirect

Maximal subgroups

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

subgroupindex
#352372
C4×#5182
#218152
#173642
#94752
#79412