Finite Group Databasebeta

← #201296

Subgroups — #201296

Order 512.

1412 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
1C2×#2502C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4#17578C212yesmaximal
5#9461C218yesmaximal, complemented
6#7942C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2502—21no—
11#2502—21no—
12#2502—22no—
13#2502C414yescomplemented
14#2502C2214yes—
15C2×#398—22no—
16#2345—24no—
17#2344—24no—
18#2316—21no—
19#2313C2214yes—
20C2×#391C412yescomplemented
21C2×#385—22no—
22C2×#384C412yescomplemented
23#2280—22no—
24#2280C2212yes—
25#2274—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1821—22no—
28#1821C2214yes—
29C2×#343—22no—
30#1520—24no—
31#1511C2218yes—
32#1508—24no—
33#1284C2218yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4D412yes—
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8C2312yes—
46C2×#117—22no—
47C2×#117—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42C2×C412yes—
65#407—22no—
66#407—22no—
67#407—22no—
68#407C2314yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71#398—24no—
72#398—24no—
73#398—28no—
74#391—22no—
75#391—22no—
76#391C2×C418yes—
77#389—28no—
78#387—216no—
79#385—28no—
80#385—24no—
81#385—24no—
82#384—22no—
83#384—22no—
84#384C2×C418yes—
85#381—24no—
86#381C2314yes—
87#380—24no—
88#380—24no—
89#380—22no—
90#380D414yes—
91#379—22no—
92C4×C4⋊C4—28no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D414yes—
100C4×C22⋊C4C2312yes—
101C2×#95—24no—
102C2×#95—21no—
103#343—28no—
104#328—216no—
105C25C2×D411yescharacteristic
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—44no—
110C22×Q8C22×C412yes—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—44no—
119C23×C4—84no—
120C23×C4C22⋊C414yes—
121C23×C4C22⋊C414yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C2×M16—24no—
135C2×M16—42no—
136C2×M16—48no—
137C22×C8—42no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—42no—
143C22⋊Q8—42no—
144C22⋊Q8—42no—
145C22⋊Q8—42no—
146#117—48no—
147#117—88no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—88no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—28no—
196C2×C42—44no—
197C2×C42C22×C414yes—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202#95—28no—
203#95—28no—
204#95—22no—
205#95—28no—
206#95—28no—
207#95—22no—
208#95—24no—
209#95C2×D418yes—
210C24—24no—
211C24—42no—
212C24—42no—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C411yescharacteristic
215C24#12011yescharacteristic
216C24#12011yescharacteristic
217C24#12311yescharacteristic
218C24#12311yescharacteristic
219C24C22×D411yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—44no—
224C2×Q8—84no—
225C2×Q8—84no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12012yes—
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12312yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C2×#4112yes—
283M16—48no—
284M16—84no—
285C2×C8—44no—
286C2×C8—84no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C4⋊C4—88no—
301C22⋊C4—28no—
302C22⋊C4—28no—
303C22⋊C4—48no—
304C22⋊C4—48no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—48no—
310C42—44no—
311C42—48no—
312C42—44no—
313C42—88no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—416no—
335C23#38611yescharacteristic
336C23#38611yescharacteristic
337C23#38611yescharacteristic
338C23#38611yescharacteristic
339C23#45614yes—
340C23C22×C22⋊C411yescharacteristic
341C23C2×#12011yescharacteristic
342C23C2×#12311yescharacteristic
343Q8—84no—
344Q8—84no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—416no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#38612yes—
388C2×C4#38612yes—
389C2×C4#41014yes—
390C2×C4C2×C4×D412yes—
391C2×C4C2×#12112yes—
392C8—84no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—44no—
411C22—44no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22#188612yes—
417C22#188612yes—
418C22#191112yes—
419C22C2×#38611yescharacteristic
420C22C2×#41012yes—
421C22C2×#45612yes—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—48no—
426C4—44no—
427C4—44no—
428C4—88no—
429C4—82no—
430C4—82no—
431C4#189914yes—
432C2—22no—
433C2—22no—
434C2—22no—
435C2—22no—
436C2—22no—
437C2—41no—
438C2—41no—
439C2C2×#188611yescharacteristic
440C2C2×#189911yescharacteristic
441C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9461C2semidirect
13. #2502C4semidirect
20. C2×#391C4semidirect
22. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#341782
C2×#18212
#175782
#94612
#79422