Finite Group Databasebeta

← #374562

Subgroups — #374562

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#21325C212yesmaximal, complemented
4C2×C4×#100C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11210C212yesmaximal, complemented
7#10851C212yesmaximal, complemented
8#10749C212yesmaximal
9#10749C212yesmaximal
10C22×#125C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×#100C2212yes—
14C2×C4×M16C2212yes—
15#2566—24no—
16#2566C418yescomplemented
17C2×#396—22no—
18#2349—24no—
19D4×C4⋊C4—24no—
20#2316C2212yes—
21#2315—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#385—22no—
25#2285C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380—22no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1992—24no—
31#1991—24no—
32C4×#100—24no—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×D4C2311yescharacteristic
39C24×C4D412yes—
40C22×M16C2314yes—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—24no—
45C2×#125C2×C414yescomplemented
46#518—24no—
47#518—24no—
48#518—44no—
49C2×C4×D4—28no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4C2×C414yes—
54C2×C4×D4D414yes—
55C2×#117C2314yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—24no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#100—24no—
72C2×#100—24no—
73C2×#100Q818yes—
74C4×M16—24no—
75C4×M16—24no—
76C2×#97—22no—
77C2×C4×C8—22no—
78#396—216no—
79#390—22no—
80#390—22no—
81#390C2×C414yes—
82#390C2×C414yes—
83#389—28no—
84#385—216no—
85#385—216no—
86#384—22no—
87#384C2×C414yescomplemented
88#384D414yes—
89#382—22no—
90#382C2×C414yescomplemented
91#382D414yes—
92#380—24no—
93#380—24no—
94#380—22no—
95#380—24no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C414yescomplemented
101C2×#95—24no—
102C2×#95—22no—
103C2×#95—21no—
104#343—24no—
105#340—28no—
106#340—28no—
107#339—44no—
108#339—44no—
109C25C2×D412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C22×C412yes—
115C22×D4C2×D412yes—
116C22×D4C2×Q814yes—
117C22×D4#4114yes—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—28no—
126C23×C4—84no—
127C23×C4C22⋊C418yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C2×M16C2×Q818yes—
138C22×C8—24no—
139#125—28no—
140#125—28no—
141#125—44no—
142#125—44no—
143C4×D4—28no—
144C4×D4—28no—
145C4×D4—28no—
146C4×D4—28no—
147C4×D4—416no—
148C4×D4—416no—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—88no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C2×D418yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—84no—
191C2×C22⋊C4C22⋊C418yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42#4114yes—
206#100—216no—
207#97—44no—
208C4×C8—44no—
209#95—28no—
210#95—24no—
211#95—28no—
212#95—24no—
213#95—24no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—44no—
219C24—44no—
220C24C2×C22⋊C414yes—
221C24#12414yes—
222C24C4⋊Q814yes—
223C24C22×D411yescharacteristic
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—28no—
229C2×D4—28no—
230C2×D4—28no—
231C2×D4—28no—
232C2×D4C4×Q818yes—
233C2×D4C22⋊Q818yes—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—416no—
262C22×C4—84no—
263C22×C4—88no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4—88no—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4#12011yescharacteristic
271C22×C4#12012yes—
272C22×C4#12011yescharacteristic
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D412yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q812yes—
279C22×C4C2×#4112yes—
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283M16—28no—
284M16—28no—
285M16—44no—
286M16—44no—
287C2×C8—28no—
288C2×C8—44no—
289C2×C8—44no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—84no—
304C4⋊C4—88no—
305C4⋊C4—88no—
306C22⋊C4—216no—
307C22⋊C4—416no—
308C22⋊C4—416no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—88no—
312C42—24no—
313C42—24no—
314C42—42no—
315C42—42no—
316C42—44no—
317C42—44no—
318C42C4×D414yes—
319C42C4×D414yes—
320C42#12114yes—
321C42#12114yes—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—44no—
331C23—44no—
332C23—48no—
333C23—48no—
334C23—44no—
335C23—44no—
336C23—82no—
337C23—84no—
338C23—82no—
339C23—84no—
340C23#39118yes—
341C23C22×C22⋊C411yescharacteristic
342C23C2×#12011yescharacteristic
343C23C2×#12311yescharacteristic
344C23C2×#12411yescharacteristic
345C23C2×C4⋊Q811yescharacteristic
346C23#53511yescharacteristic
347C23#53611yescharacteristic
348D4—216no—
349D4—216no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—416no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—88no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—1616no—
385C2×C4#38614yes—
386C2×C4#38612yes—
387C2×C4#38612yes—
388C2×C4C2×C4×D412yes—
389C2×C4C2×C4×Q812yes—
390C2×C4#52012yes—
391C2×C4C2×#12112yes—
392C2×C4C2×C22⋊Q812yes—
393C2×C4#53712yes—
394C2×C4#54614yes—
395C2×C4D4×Q814yes—
396C2×C4#55214yes—
397C2×C4#55414yes—
398C8—48no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—28no—
408C22—48no—
409C22—48no—
410C22—44no—
411C22—44no—
412C22—48no—
413C22—84no—
414C22—84no—
415C22—84no—
416C22C2×#38611yescharacteristic
417C22C2×#39111yescharacteristic
418C22#231111yescharacteristic
419C22#243411yescharacteristic
420C22#243511yescharacteristic
421C22#250211yescharacteristic
422C22#250311yescharacteristic
423C22#301812yes—
424C22#301912yes—
425C4—24no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—44no—
430C4—82no—
431C4—88no—
432C4—82no—
433C4—82no—
434C4#236014yes—
435C4#241714yes—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—48no—
441C2#1413911yescharacteristic
442C2#2095211yescharacteristic
443C2#2095311yescharacteristic
444C2#2582812yes—
445C2#2587412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. #21325C2semidirect
5. #11211C2semidirect
6. #11210C2semidirect
7. #10851C2semidirect
16. #2566C4semidirect
45. C2×#125C2×C4semidirect
87. #384C2×C4semidirect
90. #382C2×C4semidirect
100. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
#341812
#213252
C2×C4×#1002
#112112
#112102
#108512
#107492