Finite Group Databasebeta

← #359914

Subgroups — #359914

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3C2×#1820C212yesmaximal
4#14684C211yesmaximal, characteristic
5#11203C214yesmaximal
6#10912C214yesmaximal
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2296—22no—
12#2296—22no—
13#2296—21no—
14#2296—21no—
15#2296—21no—
16#2296—21no—
17C2×#389C412yescomplemented
18C2×#389C412yescomplemented
19C2×#385C2212yes—
20C2×#385C2212yes—
21#2283—28no—
22C2×#383—21no—
23C2×#382—21no—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C414yescomplemented
29C2×C4×C22⋊C4C2212yes—
30C2×#376—21no—
31#1820—28no—
32C2×#342C2218yes—
33C2×#342C2218yes—
34#1434—22no—
35#1434—22no—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C2×#117—21no—
42C2×#117—21no—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117—24no—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—24no—
51C2×#117—28no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C414yes—
57C22×C4⋊C4C2×C414yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2312yes—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42—22no—
65C22×C42C2×C412yes—
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42D412yes—
70C22×C42D412yes—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#385—22no—
84#385—22no—
85#385—24no—
86#385—24no—
87#385—22no—
88#385—22no—
89#383—28no—
90#382—28no—
91#381—24no—
92#381—24no—
93#381—24no—
94#381D418yes—
95#380—28no—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C22⋊C4—28no—
102#376—28no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95C2×C414yes—
112C2×#95C2×C414yes—
113#342—216no—
114#342—216no—
115C25C2411yescharacteristic
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C414yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4114yes—
145#117—28no—
146#117—28no—
147#117—28no—
148#117—28no—
149#117—42no—
150#117—42no—
151#117—44no—
152#117—44no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4C2×D418yes—
187C2×C22⋊C4#4118yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—28no—
191C2×C42—28no—
192C2×C42—28no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—28no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—48no—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22⋊C412yes—
211C2×C42C22⋊C412yes—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22⋊C412yes—
216#95—28no—
217#95—28no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—28no—
223#95—28no—
224#95—48no—
225#95—48no—
226#95—44no—
227#95—44no—
228C24—42no—
229C24—42no—
230C24C23×C411yescharacteristic
231C24C22×D411yescharacteristic
232C24C22×D411yescharacteristic
233C24C22×D412yes—
234C24C2×#4114yes—
235C24C2×#4112yes—
236C242−1+414yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—216no—
254C22×C4—216no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4#11718yes—
308C22×C4C4×D414yes—
309C22×C4C4×D414yes—
310C22×C4#12012yes—
311C22×C4#12012yes—
312C22×C4#12114yes—
313C22×C4#12114yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C4⋊C4—28no—
317C4⋊C4—28no—
318C4⋊C4—416no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—216no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C42—28no—
337C42—28no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—416no—
347C42—44no—
348C42—44no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—48no—
360C23—48no—
361C23C22×C22⋊C411yescharacteristic
362C23C2×#11712yes—
363C23C2×C4×D412yes—
364C23#51912yes—
365C23C2×#12011yescharacteristic
366C23C2×#12112yes—
367C23C2×#12311yescharacteristic
368C23#52512yes—
369C23#52512yes—
370C23#53612yes—
371C23#53612yes—
372C23#54718yes—
373C23#55218yes—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—88no—
425C2×C4—88no—
426C2×C4—84no—
427C2×C4—84no—
428C2×C4—88no—
429C2×C4—88no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#41014yes—
438C2×C4#41014yes—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—48no—
454C22—48no—
455C22#228011yescharacteristic
456C22C2×#38611yescharacteristic
457C22#231011yescharacteristic
458C22#234414yes—
459C22#240612yes—
460C22#241314yes—
461C22#248412yes—
462C22C2×#41012yes—
463C22C2×#41012yes—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—24no—
469C4—24no—
470C4—48no—
471C4—42no—
472C4—42no—
473C4—88no—
474C4—82no—
475C4—82no—
476C4—164no—
477C2—22no—
478C2—24no—
479C2—24no—
480C2—22no—
481C2#1387011yescharacteristic
482C2#2072312yes—
483C2#2098311yescharacteristic
484C2#2098311yescharacteristic
485C2#2182612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#389C4semidirect
18. C2×#389C4semidirect
28. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#22832
C2×#18202
#146842
#112032
#109122
#107262