Finite Group Databasebeta

← #358122

Subgroups — #358122

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×C4×#99C212yesmaximal
4#13776C211yesmaximal, characteristic, complemented
5#11203C212yesmaximal, complemented
6#11197C212yesmaximal, complemented
7#11043C212yesmaximal
8#10966C212yesmaximal
9C4×#342C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14#2322—24no—
15#2322—24no—
16#2313—28no—
17C2×#389C412yescomplemented
18C2×#388C412yescomplemented
19C2×#387—24no—
20C2×#387C412yescomplemented
21C2×#386C2211yescharacteristic
22C2×#385C412yescomplemented
23C2×#385C2211yescharacteristic
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2211yescharacteristic
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31C4×#99—28no—
32C2×#342C2214yes—
33C2×#342C2214yes—
34C2×#342C2214yes—
35#1436—24no—
36C22⋊#95C2212yes—
37C23×D4C2311yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×#123—24no—
43C2×#123—22no—
44C2×#123—22no—
45C2×#123—24no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4D414yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42—22no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C412yes—
72C2×#99—28no—
73C2×#99—28no—
74C2×#99—28no—
75#389—28no—
76#388—28no—
77#387—232no—
78#387—28no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—28no—
84#381—22no—
85#381—22no—
86#381—28no—
87#381—28no—
88#380—28no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4D418yes—
95C2×#95—24no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—24no—
99C2×#95—22no—
100C2×#95—24no—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106#342—28no—
107#342—28no—
108#342—28no—
109C25C22×C411yescharacteristic
110C25C2411yescharacteristic
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—24no—
117C22×D4#4114yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—82no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D414yes—
148C23×C4#4114yes—
149#123—216no—
150#123—216no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C418yes—
171C2×C4⋊C4C22⋊C418yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4C2×D418yes—
200C2×C22⋊C4#4118yes—
201C2×C22⋊C4#4114yes—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—28no—
206C2×C42—44no—
207C2×C42—84no—
208C2×C42—84no—
209C2×C42—82no—
210#99—232no—
211#95—28no—
212#95—216no—
213#95—216no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95—44no—
221#95—84no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24#11712yes—
231C24C23×C411yescharacteristic
232C24C22×D411yescharacteristic
233C24C22×D411yescharacteristic
234C24C22×D412yes—
235C24C2×#4112yes—
236C24C2×#4112yes—
237C24C2×#4112yes—
238C24C2×#4114yes—
239C2×D4—28no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—28no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—42no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4C2×C22⋊C414yes—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4#11712yes—
317C22×C4#11714yes—
318C22×C4C4×D414yes—
319C22×C4C4×D414yes—
320C22×C4#12014yes—
321C22×C4#12114yes—
322C22×C4#12114yes—
323C22×C4#12314yes—
324C4⋊C4—216no—
325C4⋊C4—216no—
326C4⋊C4—216no—
327C4⋊C4—44no—
328C4⋊C4—84no—
329C4⋊C4—88no—
330C22⋊C4—216no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—41no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—84no—
348C42—44no—
349C42—84no—
350C42—88no—
351C42—88no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—24no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—48no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—82no—
375C23—84no—
376C23—82no—
377C23C22×C22⋊C411yescharacteristic
378C23C2×#11711yescharacteristic
379C23C2×#11711yescharacteristic
380C23C2×C4×D412yes—
381C23C4×#4112yes—
382C23C2×#12011yescharacteristic
383C23C2×#12112yes—
384C23C2×#12311yescharacteristic
385C23#52512yes—
386C23#52512yes—
387C23#52512yes—
388C23#52514yes—
389C23#52914yes—
390C23#52914yes—
391C23#53212yes—
392C23#53214yes—
393D4—44no—
394D4—48no—
395D4—44no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—416no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—88no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#38614yes—
451C2×C4#38614yes—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—22no—
457C22—24no—
458C22—24no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—21no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—42no—
472C22—44no—
473C22—84no—
474C22—84no—
475C22—82no—
476C22—81no—
477C22—81no—
478C22#228011yescharacteristic
479C22C2×#38611yescharacteristic
480C22#229012yes—
481C22#229312yes—
482C22#229711yescharacteristic
483C22#238812yes—
484C22#239214yes—
485C22#239512yes—
486C22#288312yes—
487C22#288412yes—
488C4—24no—
489C4—24no—
490C4—44no—
491C4—42no—
492C4—44no—
493C4—44no—
494C4—88no—
495C4—88no—
496C4—82no—
497C4—81no—
498C4—81no—
499C2—22no—
500C2—24no—
501C2—22no—
502C2—22no—
503C2—22no—
504C2—82no—
505C2#1383911yescharacteristic
506C2#2072512yes—
507C2#2098411yescharacteristic
508C2#2098611yescharacteristic
509C2#2142312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
4. #13776C2semidirect
5. #11203C2semidirect
6. #11197C2semidirect
9. C4×#342C2semidirect
17. C2×#389C4semidirect
18. C2×#388C4semidirect
20. C2×#387C4semidirect
22. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#23132
C2×C4×#992
#137762
#112032
#111972
#110432
#109662
C4×#3422