Finite Group Databasebeta

← #379866

Subgroups — #379866

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#20911C212yesmaximal
4C2×#1822C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11210C212yesmaximal, complemented
7#11055C212yesmaximal, complemented
8#11055C212yesmaximal, complemented
9#10733C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13C2×#419C2214yes—
14C2×#407C2212yes—
15#2499—24no—
16#2499C418yescomplemented
17C2×#397—22no—
18#2317—24no—
19#2316C2212yes—
20#2316C2212yes—
21#2315—22no—
22#2296—24no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#387—22no—
26#2285—22no—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1873—24no—
31#1872—24no—
32#1822—24no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16C2314yes—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8C2×C414yes—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—44no—
52C2×#117—44no—
53C2×#117C2×C414yes—
54C2×#117D414yes—
55C2×#117D414yes—
56C2×#117C2314yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D412yes—
67C22×C42—22no—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#419—44no—
72#419—44no—
73C2×#103—28no—
74#407—24no—
75#407—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#397—216no—
79#391—22no—
80#391—22no—
81#391C2×C414yescomplemented
82#391C2×C414yescomplemented
83#389—28no—
84#389—28no—
85#387—216no—
86#385—216no—
87#384—24no—
88#384—22no—
89#384C2×C414yescomplemented
90#382—24no—
91#382—22no—
92#382C2×C414yescomplemented
93#380—24no—
94#380—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4C2×C414yes—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103#343—24no—
104#343—24no—
105#340—28no—
106#339—44no—
107#339—44no—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8C22×C412yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4C22×C412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—24no—
141C22×C8—24no—
142C22⋊Q8—28no—
143C22⋊Q8—28no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146#117—28no—
147#117—28no—
148#117—28no—
149#117—28no—
150#117—24no—
151#117—24no—
152#117—24no—
153#117—24no—
154#117—48no—
155#117—44no—
156#117—44no—
157#117—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D418yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4C22⋊C418yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—48no—
205C2×C42C22⋊C418yes—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208#103—416no—
209C22⋊C8—44no—
210C22⋊C8—44no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—28no—
215#95—28no—
216#95—28no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—42no—
222C24—42no—
223C24C2×C22⋊C412yes—
224C24#12012yes—
225C24#12312yes—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—44no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C4×D414yes—
286C22×C4C4×D414yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12114yes—
290C22×C4#12114yes—
291C22×C4#12312yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C42−1+412yes—
299C22×C42−1+412yes—
300M16—44no—
301M16—44no—
302M16—44no—
303M16—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—416no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—88no—
320C4⋊C4—88no—
321C22⋊C4—216no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—48no—
327C42—28no—
328C42—28no—
329C42—24no—
330C42—24no—
331C42—48no—
332C42—44no—
333C42—44no—
334C42—42no—
335C42—42no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23#38614yes—
355C23C22×C22⋊C411yescharacteristic
356C23C2×#12011yescharacteristic
357C23C2×#12311yescharacteristic
358C23#53511yescharacteristic
359C23#53511yescharacteristic
360C23#53611yescharacteristic
361C23#53611yescharacteristic
362Q8—48no—
363Q8—84no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—816no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4#38612yes—
410C2×C4#38612yes—
411C2×C4C2×C4×D412yes—
412C2×C4#52012yes—
413C2×C4#52012yes—
414C2×C4C2×#12112yes—
415C2×C4#53414yes—
416C2×C4#53614yes—
417C2×C4#53712yes—
418C2×C4#53712yes—
419C2×C4#53914yes—
420C2×C4#53914yes—
421C8—84no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22C2×#38611yescharacteristic
440C22#231111yescharacteristic
441C22#231111yescharacteristic
442C22#261311yescharacteristic
443C22#261411yescharacteristic
444C22#262311yescharacteristic
445C22#262311yescharacteristic
446C22#301812yes—
447C22#302212yes—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—82no—
455C4—88no—
456C4—82no—
457C4#232214yes—
458C4#261814yes—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—44no—
464C2#1425611yescharacteristic
465C2#2095211yescharacteristic
466C2#2095511yescharacteristic
467C2#2962112yes—
468C2#2965512yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11211C2semidirect
6. #11210C2semidirect
7. #11055C2semidirect
8. #11055C2semidirect
16. #2499C4semidirect
81. #391C2×C4semidirect
82. #391C2×C4semidirect
89. #384C2×C4semidirect
92. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#340522
#209112
C2×#18222
#112112
#112102
#110552
#107332