Finite Group Databasebeta

← #200617

Subgroups — #200617

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34936C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic, complemented
3#17544C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic, complemented
5#9409C218yesmaximal, complemented
6#8048C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#525—22no—
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11#2920C2212yes—
12#2542C412yescomplemented
13#2541C412yescomplemented
14#2540—22no—
15#2539C2214yes—
16#2501—21no—
17#2433—21no—
18#2397C414yescomplemented
19C4×#127—22no—
20#2275—24no—
21#2274—21no—
22#2273C2214yes—
23C42×D4—24no—
24C4×#117C2211yescharacteristic
25C4×#117C2212yes—
26C2×C4×C22⋊C4—22no—
27#1507C2218yes—
28#1505—24no—
29#1502—24no—
30#1466—22no—
31#1462C2214yes—
32#1460—22no—
33#1297C2218yes—
34C22×#41—21no—
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×#124—21no—
45#525—416no—
46C2×#123—24no—
47C2×C22⋊Q8—24no—
48C2×#121—24no—
49C2×#121—21no—
50C2×#121C2312yes—
51C2×#120—24no—
52#518—216no—
53C2×C4×D4—28no—
54C2×C4×D4—28no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yescomplemented
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×#117—22no—
66C2×#117—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4C2312yes—
70C22×C42—24no—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#431—22no—
75#431—22no—
76#423—22no—
77#407C2314yes—
78C4×M16—22no—
79#399—24no—
80#399C2×C418yescomplemented
81#392—24no—
82#392C2×C418yescomplemented
83#390—22no—
84#390—22no—
85#390—28no—
86#384—22no—
87C4×C4⋊C4—28no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yescomplemented
91C4×C4⋊C4D414yescomplemented
92C4×C22⋊C4—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—216no—
98C4×C22⋊C4C2312yes—
99C4×C22⋊C4C2314yes—
100#376C2×C412yescomplemented
101#376D414yescomplemented
102C2×#95—24no—
103C43—21no—
104C43—22no—
105#327—216no—
106#323—28no—
107C25C2×D412yes—
108C2×#41—48no—
109C2×#41—41no—
110C2×#41—41no—
111C2×#41—41no—
112C2×#41—41no—
113C22×Q8—21no—
114C22×D4—28no—
115C22×D4—22no—
116C22×D4—28no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4C22⋊C412yes—
122C22×D4C22⋊C412yes—
123C22×D4C22×C412yes—
124C23×C4—28no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—48no—
131C23×C4C22⋊C414yes—
132C23×C4C22⋊C414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
137C2×M16—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140#127—22no—
141#127—22no—
142#127—48no—
143#124—42no—
144#124—42no—
145#123—416no—
146C22⋊Q8—416no—
147#121—24no—
148#121—24no—
149#121—416no—
150#121—42no—
151#121—42no—
152#120—416no—
153C4×D4—264no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—432no—
165C4×D4—42no—
166C4×D4—42no—
167#117—216no—
168#117—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—232no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22×C414yes—
189C2×C22⋊C4C2×D414yes—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—28no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—88no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C4⋊C8—42no—
207C4⋊C8—42no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210#97—42no—
211C4×C8—42no—
212#95—28no—
213#95—22no—
214#95—24no—
215#95C22×C414yes—
216#95C2×D418yes—
217C24—22no—
218C24—22no—
219C24—28no—
220C24—22no—
221C24—44no—
222C24—82no—
223C24C2×C22⋊C414yes—
224C24C22×D411yescharacteristic
225#41—84no—
226#41—84no—
227#41—84no—
228#41—84no—
229C2×Q8—44no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×D4—216no—
233C2×D4—216no—
234C2×D4—28no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—416no—
240C2×D4—44no—
241C2×D4—48no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—416no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—44no—
249C2×D4—88no—
250C2×D4—82no—
251C2×D4#10418yes—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—232no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—81no—
275C22×C4—88no—
276C22×C4—88no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12012yes—
284C22×C4#12011yescharacteristic
285C22×C4#12312yes—
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4112yes—
292M16—84no—
293C2×C8—44no—
294C4⋊C4—216no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—416no—
300C4⋊C4—42no—
301C22⋊C4—216no—
302C22⋊C4—232no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—28no—
306C22⋊C4—416no—
307C22⋊C4—416no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C42—216no—
311C42—28no—
312C42—44no—
313C42—48no—
314C42—84no—
315C42—82no—
316C42—88no—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C4×D412yes—
320C42C4×D412yes—
321C42#12112yes—
322C42#12112yes—
323C42#12112yes—
324C42#12112yes—
325C23—28no—
326C23—28no—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23—41no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—42no—
340C23—82no—
341C23—82no—
342C23—816no—
343C23C2×#10418yes—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12011yescharacteristic
346C23C2×#12311yescharacteristic
347Q8—82no—
348Q8—82no—
349D4—216no—
350D4—216no—
351D4—84no—
352D4—82no—
353D4—832no—
354D4—84no—
355D4—84no—
356D4—82no—
357D4—84no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—216no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—82no—
387C2×C4—832no—
388C2×C4—84no—
389C2×C4#38612yes—
390C2×C4#38611yescharacteristic
391C2×C4#38611yescharacteristic
392C2×C4#38611yescharacteristic
393C2×C4#38612yes—
394C2×C4#38611yescharacteristic
395C2×C4C2×#10414yes—
396C2×C4#45314yes—
397C2×C4C2×C4×D412yes—
398C2×C4C2×#12112yes—
399C8—84no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—28no—
408C22—44no—
409C22—48no—
410C22—416no—
411C22—44no—
412C22—44no—
413C22—84no—
414C22—88no—
415C22—88no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22C2×#38611yescharacteristic
420C22C22×#10412yes—
421C22C2×#45312yes—
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—416no—
428C4—48no—
429C4—44no—
430C4—82no—
431C4#188712yes—
432C4#188712yes—
433C4#189614yes—
434C2—22no—
435C2—22no—
436C2—41no—
437C2—48no—
438C2—162no—
439C2C2×#188711yescharacteristic
440C2C2×#189611yescharacteristic
441C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34936C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #17544C2semidirect
4. #17364C2semidirect
5. #9409C2semidirect
6. #8048C2semidirect
12. #2542C4semidirect
13. #2541C4semidirect
18. #2397C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
43. C2×#127C2×C4semidirect
62. C2×C4×D4C2×C4semidirect
80. #399C2×C4semidirect
82. #392C2×C4semidirect
90. C4×C4⋊C4C2×C4semidirect
91. C4×C4⋊C4D4semidirect
100. #376C2×C4semidirect
101. #376D4semidirect

Maximal subgroups

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

subgroupindex
#349362
C4×#5182
#175442
#173642
#94092
#80482