Finite Group Databasebeta

← #369254

Subgroups — #369254

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3138C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3C2×#1994C211yesmaximal, characteristic, complemented
4#21260C211yesmaximal, characteristic
5#14713C211yesmaximal, characteristic
6#11063C214yesmaximal, complemented
7#10958C214yesmaximal, complemented
8#10857C211yesmaximal, characteristic, complemented
9C4×#339C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3138—24no—
13#3138C418yescomplemented
14C2×#490—22no—
15C2×#486—22no—
16C2×#423C2211yescharacteristic
17C2×#418C2212yes—
18C2×#418C2214yes—
19C2×#418C2212yes—
20C2×C4×M16C2211yescharacteristic
21#2360—24no—
22#2355—22no—
23Q8×C4⋊C4—22no—
24#2330—22no—
25#2317C2212yes—
26C4×C4⋊Q8—22no—
27C2×#390—24no—
28C2×#390C2214yes—
29C2×#384C2211yescharacteristic
30#2274C2212yes—
31C4×#117C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1994—24no—
34#1982—22no—
35#1982—22no—
36C2×#343C2214yes—
37C2×#339C2212yes—
38C2×#339C2214yes—
39C2×#339C2212yes—
40C23×D4—22no—
41C23×D4C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44#533—22no—
45#533—22no—
46#533C2×C414yes—
47#533D414yes—
48C2×C4⋊Q8D412yes—
49C2×#127—21no—
50C2×#127—21no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127C2×C414yes—
56C2×#125D412yes—
57C2×C22⋊Q8—22no—
58C2×C4×Q8—24no—
59C2×C4×Q8—22no—
60C2×#117C2311yescharacteristic
61C2×#117C2312yes—
62C2×#117C2311yescharacteristic
63C22×C4⋊C4D414yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—28no—
66C22×C42—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#490—216no—
70#486—216no—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—22no—
74#418—28no—
75#418—28no—
76#418—44no—
77#418C2×C4116yes—
78C2×#102—28no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—24no—
82C4×M16—22no—
83C4×M16C2×C414yes—
84C2×#97—21no—
85C2×C4×C8—21no—
86#391—24no—
87#390—28no—
88#390—44no—
89#390—416no—
90#389—24no—
91#389—28no—
92#389—24no—
93#387—28no—
94#385—24no—
95#384—22no—
96#384—21no—
97#384—21no—
98#384—22no—
99#382—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107#376—21no—
108C43—21no—
109C43—22no—
110C43—22no—
111#343—44no—
112#339—28no—
113#339—24no—
114#339—24no—
115C25—22no—
116C22×Q8—22no—
117C22×D4—28no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—48no—
128C22×D4—48no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—44no—
132C22×D4C22×C414yes—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D414yes—
136C23×C4C2411yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—24no—
144C2×M16—22no—
145C2×M16—24no—
146C2×M16C22×C414yes—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—24no—
150C4⋊Q8C22⋊C418yescomplemented
151#127—24no—
152#127—24no—
153#127—28no—
154#127—44no—
155#127—42no—
156#127—42no—
157#125—24no—
158#125C22⋊C418yescomplemented
159C22⋊Q8—28no—
160C22⋊Q8—28no—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—88no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117C22×C414yes—
170#117C2×D414yes—
171#117#4114yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—416no—
190C2×C22⋊C4—432no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—28no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—28no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42C22×C412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42#4114yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C4⋊C8—44no—
214#102—232no—
215#102—44no—
216#102—44no—
217#102—48no—
218#97—24no—
219C4×C8—24no—
220#95—28no—
221#95—28no—
222#95—22no—
223#95—22no—
224C24—24no—
225C24—48no—
226C24—48no—
227C24—42no—
228C24—42no—
229C24—44no—
230C24—44no—
231C24C22×D411yescharacteristic
232C2×Q8—28no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—432no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—48no—
255C2×D4—48no—
256C2×D4—84no—
257C2×D4—84no—
258C2×D4—816no—
259C2×D4—84no—
260C2×D4—84no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—28no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4C4×D414yes—
276C22×C4#12012yes—
277C22×C4#12012yes—
278C22×C4#12114yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4111yescharacteristic
287C22×C4H(5,𝔽2)12yes—
288C22×C42−1+412yes—
289M16—24no—
290M16—24no—
291M16—42no—
292M16—42no—
293M16—44no—
294M16—44no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—216no—
304C4⋊C4—28no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—84no—
314C4⋊C4—88no—
315C4⋊C4C2×C22⋊C418yes—
316C4⋊C4#12414yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—42no—
321C22⋊C4—832no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—416no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—44no—
333C42—88no—
334C42C2×C22⋊C412yes—
335C42C2×C22⋊C412yes—
336C42C4×D414yes—
337C42#12114yes—
338C42#12312yes—
339C42#12312yes—
340C23—22no—
341C23—22no—
342C23—416no—
343C23—48no—
344C23—44no—
345C23—88no—
346C23—88no—
347C23—816no—
348C23—88no—
349C23—84no—
350C23—82no—
351C23—82no—
352C23—84no—
353C23—84no—
354C23C2×C4×D411yescharacteristic
355C23C2×#12011yescharacteristic
356C23C2×#12111yescharacteristic
357C23#54612yes—
358C23#54812yes—
359Q8—416no—
360D4—44no—
361D4—48no—
362D4—44no—
363D4—816no—
364D4—816no—
365D4—84no—
366D4—82no—
367D4—82no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—24no—
374C2×C4—28no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—88no—
390C2×C4#38614yes—
391C2×C4#38614yes—
392C2×C4#45314yes—
393C2×C4#45414yes—
394C2×C4C22×C22⋊C411yescharacteristic
395C2×C4C2×C4×D411yescharacteristic
396C2×C4C4×#4112yes—
397C2×C4#52012yes—
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12311yescharacteristic
400C2×C4C2×#12411yescharacteristic
401C2×C4#52912yes—
402C2×C4#53211yescharacteristic
403C2×C4#53511yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#54212yes—
406C2×C4#54812yes—
407C2×C4#54812yes—
408C2×C4#55012yes—
409C2×C4#55012yes—
410C2×C4#55612yes—
411C2×C4#55912yes—
412C8—44no—
413C8—82no—
414C22—24no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—816no—
420C22—84no—
421C22—88no—
422C22—168no—
423C22—164no—
424C22—162no—
425C22—162no—
426C22C2×#38611yescharacteristic
427C22#236712yes—
428C22#242612yes—
429C22#243111yescharacteristic
430C22#243611yescharacteristic
431C22C2×#45312yes—
432C22C2×#45412yes—
433C4—24no—
434C4—48no—
435C4—44no—
436C4—48no—
437C4—81no—
438C4—84no—
439C4—81no—
440C4#188318yes—
441C4C4×#12411yescharacteristic
442C4#232611yescharacteristic
443C4#234112yes—
444C4#244812yes—
445C4#254511yescharacteristic
446C4#254811yescharacteristic
447C2—22no—
448C2—22no—
449C2—164no—
450C2#1402611yescharacteristic
451C2C2×#188312yes—
452C2#2558811yescharacteristic
453C2#2559111yescharacteristic
454C2#2592412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. C2×#1994C2semidirect
6. #11063C2semidirect
7. #10958C2semidirect
8. #10857C2semidirect
9. C4×#339C2semidirect
13. #3138C4semidirect
150. C4⋊Q8C22⋊C4semidirect
158. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31382
#342142
C2×#19942
#212602
#147132
#110632
#109582
#108572
C4×#3392