Finite Group Databasebeta

← #197511

Subgroups — #197511

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#17575C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9199C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2848—22no—
12#2626C2214yes—
13#2580C414yescomplemented
14#2566—22no—
15#2563C412yescomplemented
16#2542C412yescomplemented
17#2538—22no—
18#2349—24no—
19D4×C4⋊C4—24no—
20#2317—21no—
21#2316—21no—
22#2315C2214yes—
23#2309—21no—
24#2285C2211yescharacteristic
25#2285C2212yes—
26C2×#380—22no—
27#1520C2218yes—
28#1518—24no—
29#1503—24no—
30#1460—22no—
31#1460—22no—
32C2×#323C2214yes—
33#1289C2218yes—
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36#533—22no—
37#533—22no—
38#533—22no—
39#533—22no—
40C2×C4⋊Q8C2×C411yescharacteristic, complemented
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8C2312yes—
46#518—44no—
47#518—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yescomplemented
50C2×C4×D4—28no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4—24no—
65C22×C22⋊C4C2312yes—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C4×C8—22no—
74#399C2×C418yescomplemented
75#396—28no—
76#393C2×C418yescomplemented
77#390—22no—
78#390—22no—
79#390—24no—
80#389—28no—
81#389—24no—
82#385—28no—
83#385—24no—
84#385—216no—
85#384—22no—
86#384—22no—
87#384—21no—
88#382—22no—
89#382C2×C412yescomplemented
90#382D414yes—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—24no—
95#380—21no—
96#380—22no—
97#380C2312yes—
98#380C2314yes—
99#379—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C412yescomplemented
103C4×C4⋊C4D414yes—
104C2×#95—24no—
105#328—216no—
106#323—28no—
107C25C2×D412yes—
108C22×Q8—21no—
109C22×Q8C22×C412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C22⋊C412yes—
115C22×D4C22⋊C412yes—
116C22×D4SD1614yes—
117C22×D4Q1614yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—28no—
123C23×C4—84no—
124C23×C4C22⋊C414yes—
125C23×C4C22⋊C414yes—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—24no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135#125—42no—
136#125—42no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—88no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—432no—
157#117—42no—
158#117—42no—
159#117—44no—
160#117—42no—
161#117—44no—
162#117—42no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—88no—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4C22×C414yes—
193C2×C22⋊C4C2×D414yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C4⋊C8—42no—
206C4⋊C8—42no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C4×C8—42no—
210C4×C8—42no—
211#95—28no—
212#95—22no—
213#95—28no—
214#95—24no—
215#95C22×C414yes—
216#95C2×D418yes—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—82no—
222C24—82no—
223C24C2×C22⋊C414yes—
224C24C2×SD1614yes—
225C24C2×Q1614yes—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—28no—
240C2×D4#10318yes—
241C2×D4#10318yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—416no—
256C22×C4—84no—
257C22×C4—81no—
258C22×C4—84no—
259C22×C4—82no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12012yes—
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12312yes—
275C22×C4C2×SD1612yes—
276C22×C4C2×Q1612yes—
277C22×C4AGL(1,ℤ/8)12yes—
278C22×C4#13712yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4112yes—
283C2×C8—44no—
284C2×C8—84no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—44no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—416no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—88no—
313C22⋊C4—88no—
314C42—44no—
315C42—44no—
316C42—48no—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C4×D412yes—
320C42C4×D412yes—
321C42#12112yes—
322C42#12112yes—
323C42#12112yes—
324C42#12112yes—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—41no—
332C23—41no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23C2×#10318yes—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12011yescharacteristic
346C23C2×#12311yescharacteristic
347C23C22×SD1611yescharacteristic
348C23C22×Q1611yescharacteristic
349C23C2×AGL(1,ℤ/8)11yescharacteristic
350C23C2×#13711yescharacteristic
351Q8—84no—
352Q8—82no—
353Q8—82no—
354D4—216no—
355D4—216no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—1616no—
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38611yescharacteristic
388C2×C4#38612yes—
389C2×C4#38612yes—
390C2×C4#38611yescharacteristic
391C2×C4C2×#10312yes—
392C2×C4C2×#10312yes—
393C2×C4#41712yes—
394C2×C4#41712yes—
395C2×C4#45014yes—
396C2×C4#45114yes—
397C2×C4#48314yes—
398C2×C4#48414yes—
399C2×C4C2×C4×D412yes—
400C2×C4C2×#12112yes—
401C8—84no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—28no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—48no—
414C22—48no—
415C22—48no—
416C22—44no—
417C22—44no—
418C22—84no—
419C22—84no—
420C22—84no—
421C22C2×#38611yescharacteristic
422C22C22×#10311yescharacteristic
423C22C2×#41711yescharacteristic
424C22C2×#45011yescharacteristic
425C22C2×#45111yescharacteristic
426C22C2×#48311yescharacteristic
427C22C2×#48411yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—84no—
435C4—88no—
436C4—82no—
437C4—82no—
438C4#187214yes—
439C4#187214yes—
440C4#191112yes—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—48no—
448C2C2×#187211yescharacteristic
449C2C2×#191111yescharacteristic
450C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. #17575C2semidirect
5. #9199C2semidirect
6. #7981C2semidirect
13. #2580C4semidirect
15. #2563C4semidirect
16. #2542C4semidirect
40. C2×C4⋊Q8C2×C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
49. C2×C4×Q8C2×C4semidirect
74. #399C2×C4semidirect
76. #393C2×C4semidirect
89. #382C2×C4semidirect
102. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351722
#341812
#175752
C2×#14602
#91992
#79812