Finite Group Databasebeta

← #380056

Subgroups — #380056

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3C2×#1992C212yesmaximal, complemented
4#14686C211yesmaximal, characteristic
5#11228C212yesmaximal, complemented
6#11227C212yesmaximal, complemented
7#11200C212yesmaximal
8#11199C212yesmaximal
9#10843C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—24no—
13C2×#432—28no—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2212yes—
17#2851—24no—
18#2851C418yescomplemented
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#385—21no—
24C2×#385—21no—
25C2×#384C2212yes—
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32#1992—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125C2×C414yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2314yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42C2×C414yes—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D414yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#432—232no—
71C2×C4⋊C8—28no—
72C2×C4⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—28no—
75C2×#97—24no—
76C2×#97—24no—
77C2×#97D418yes—
78#389—22no—
79#389—22no—
80#389C2×C414yes—
81#389C2×C414yes—
82#387—28no—
83#387—28no—
84#385—28no—
85#385—28no—
86#384—24no—
87#384—24no—
88#382—22no—
89#382C2×C414yes—
90#380—22no—
91#380—22no—
92#380—44no—
93#380—44no—
94#380—44no—
95#380C2×C414yes—
96#380C2×C414yes—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340—28no—
107#340—28no—
108#340—44no—
109#340—44no—
110C25C2×D411yescharacteristic
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22⋊C414yes—
130C23×C4C22⋊C414yes—
131C23×C4C22×C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8C2×D418yes—
146#125—216no—
147#125—44no—
148#125—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—216no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42C22⋊C414yes—
202C2×C42C22⋊C414yes—
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D414yes—
207C2×C42C2×D412yes—
208C2×C42#4114yes—
209C4⋊C8—216no—
210C22⋊C8—216no—
211#97—216no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—22no—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24C2×C22⋊C412yes—
230C24AGL(1,ℤ/8)12yes—
231C24#13712yes—
232C24C22×D411yescharacteristic
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4#12114yes—
305C22×C4#12114yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C42−1+414yes—
314C22×C42−1+412yes—
315C2×C8—28no—
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—88no—
345C42—24no—
346C42—22no—
347C42—22no—
348C42—28no—
349C42—28no—
350C42C4×D418yes—
351C42#12014yes—
352C42#12314yes—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#41714yes—
376C23C22×C22⋊C411yescharacteristic
377C23C2×#12112yes—
378C23#53511yescharacteristic
379C23#53612yes—
380C23#53611yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×AGL(1,ℤ/8)11yescharacteristic
383C23C2×#13711yescharacteristic
384C23C2×#13711yescharacteristic
385C23#57512yes—
386C23#57512yes—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—416no—
406C2×C4—416no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4#38614yes—
444C2×C4#38614yes—
445C2×C4#41714yes—
446C2×C4C2×C4×D412yes—
447C2×C4#51912yes—
448C2×C4#52012yes—
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4#53614yes—
452C2×C4#53612yes—
453C2×C4#53714yes—
454C2×C4#53712yes—
455C2×C4#54114yes—
456C2×C4#54214yes—
457C8—48no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22C2×#38611yescharacteristic
470C22#231011yescharacteristic
471C22#231111yescharacteristic
472C22#262012yes—
473C22#262112yes—
474C22C2×#41711yescharacteristic
475C22C2×#41711yescharacteristic
476C22#289611yescharacteristic
477C22#289611yescharacteristic
478C22#305512yes—
479C22#305512yes—
480C22#305612yes—
481C22#305612yes—
482C22#321711yescharacteristic
483C22#321811yescharacteristic
484C22#321912yes—
485C22#323412yes—
486C22#323512yes—
487C22#323612yes—
488C22#323712yes—
489C22#323811yescharacteristic
490C22#323911yescharacteristic
491C22#324012yes—
492C4—24no—
493C4—24no—
494C4—24no—
495C4—42no—
496C4—42no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—164no—
502C4#232314yes—
503C4#261612yes—
504C4#262512yes—
505C2—44no—
506C2#1425811yescharacteristic
507C2#2097111yescharacteristic
508C2#2097111yescharacteristic
509C2#2183811yescharacteristic
510C2#2183911yescharacteristic
511C2#2186711yescharacteristic
512C2#2186811yescharacteristic
513C2#2985112yes—
514C2#2985212yes—
515C2#2988312yes—
516C2#2988412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1992C2semidirect
5. #11228C2semidirect
6. #11227C2semidirect
9. #10843C2semidirect
18. #2851C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×#28512
C2×#19922
#146862
#112282
#112272
#112002
#111992
#108432