Finite Group Databasebeta

← #161190

Subgroups — #161190

Order 512.

2694 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#7640C214yesmaximal, complemented
4#7626C214yesmaximal, complemented
5#7592C214yesmaximal, complemented
6C24×D4C411yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8C23×C22⋊C4—22no—
9C23×C42C2211yescharacteristic
10C2×#432—22no—
11C2×#390—22no—
12C2×#390—22no—
13C2×#390—22no—
14C2×#390C412yes—
15C2×#390C412yes—
16C2×#390C412yes—
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#390C2212yes—
20#1817C2214yescomplemented
21#1797—22no—
22#1797C2214yescomplemented
23#1779—22no—
24#1749—22no—
25#1749C2214yescomplemented
26#1270C22116yescomplemented
27C26—21no—
28C23×D4—28no—
29C23×D4—21no—
30C23×D4—21no—
31C23×D4—21no—
32C23×D4—44no—
33C23×D4—44no—
34C23×D4C2×C412yes—
35C23×D4C2×C412yes—
36C23×D4C2×C412yes—
37C24×C4—21no—
38C24×C4C2×C411yescharacteristic
39C24×C4D412yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—24no—
51C22×C22⋊C4—24no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—416no—
54C22×C42—44no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#432—48no—
65C2×C4⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×#97—24no—
70#390—24no—
71#390—24no—
72#390—24no—
73#390—24no—
74#390—24no—
75#390—24no—
76#390C2×C414yes—
77#390C2×C414yes—
78#390C2×C414yes—
79#390C2314yes—
80#336—24no—
81#336C2318yescomplemented
82C25—28no—
83C25—22no—
84C25—22no—
85C25—22no—
86C25—48no—
87C25—44no—
88C25—48no—
89C25—44no—
90C25C22⋊C411yescharacteristic
91C22×D4—28no—
92C22×D4—28no—
93C22×D4—28no—
94C22×D4—28no—
95C22×D4—28no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—24no—
100C22×D4—44no—
101C22×D4—44no—
102C22×D4—44no—
103C22×D4—44no—
104C22×D4—464no—
105C22×D4—44no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—44no—
111C22×D4—44no—
112C22×D4—44no—
113C22×D4M1618yes—
114C22×D4C22×C414yes—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—28no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—48no—
125C23×C4—44no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C22×C8—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C22×C414yes—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—416no—
154C2×C22⋊C4—416no—
155C2×C22⋊C4—416no—
156C2×C22⋊C4—416no—
157C2×C22⋊C4—416no—
158C2×C22⋊C4—416no—
159C2×C42—28no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2411yescharacteristic
178C4⋊C8—48no—
179C22⋊C8—48no—
180C22⋊C8—48no—
181C22⋊C8—48no—
182#97—48no—
183C24—22no—
184C24—22no—
185C24—24no—
186C24—24no—
187C24—28no—
188C24—28no—
189C24—24no—
190C24—28no—
191C24—48no—
192C24—48no—
193C24—44no—
194C24—44no—
195C24—432no—
196C24—464no—
197C24—48no—
198C24—48no—
199C24—44no—
200C24—44no—
201C24—44no—
202C24—44no—
203C24—48no—
204C24—48no—
205C24—832no—
206C24—816no—
207C24—816no—
208C24#10014yes—
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—24no—
215C2×D4—28no—
216C2×D4—416no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—432no—
220C2×D4—432no—
221C2×D4—432no—
222C2×D4—432no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—432no—
228C2×D4—416no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4C2×M1618yes—
232C22×C4—232no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—432no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—816no—
275C22×C4#9914yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×M1614yes—
283C22×C4AGL(1,ℤ/8)18yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286C2×C8—48no—
287C2×C8—48no—
288C2×C8—48no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C22⋊C4—416no—
294C22⋊C4—416no—
295C22⋊C4—416no—
296C22⋊C4—416no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—24no—
301C42—42no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—42no—
306C42—41no—
307C23—216no—
308C23—216no—
309C23—28no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—28no—
319C23—28no—
320C23—28no—
321C23—416no—
322C23—416no—
323C23—432no—
324C23—432no—
325C23—432no—
326C23—44no—
327C23—44no—
328C23—48no—
329C23—48no—
330C23—416no—
331C23—8128no—
332C23—88no—
333C23—88no—
334C23—816no—
335C23—816no—
336C23—816no—
337C23—88no—
338C23—88no—
339C23—88no—
340C23—88no—
341C23#40712yes—
342C23C2×#9912yes—
343C23C2×#10012yes—
344C23C2×#10012yes—
345C23C2×#10012yes—
346C23#41814yes—
347C23#42114yes—
348C23C22×C22⋊C411yescharacteristic
349D4—48no—
350D4—48no—
351D4—416no—
352D4—416no—
353D4—416no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—88no—
387C2×C4—88no—
388C2×C4—88no—
389C2×C4#40712yes—
390C2×C4C2×#9912yes—
391C2×C4C2×#9912yes—
392C2×C4#41814yes—
393C2×C4C22×M1612yes—
394C2×C4C2×AGL(1,ℤ/8)14yes—
395C8—88no—
396C22—232no—
397C22—24no—
398C22—24no—
399C22—24no—
400C22—22no—
401C22—22no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—416no—
410C22—48no—
411C22—832no—
412C22—832no—
413C22—832no—
414C22—84no—
415C22—88no—
416C22—84no—
417C22#146214yes—
418C22#153314yes—
419C22#155514yes—
420C22C2×#40711yescharacteristic
421C22C22×#9911yescharacteristic
422C22C22×#10011yescharacteristic
423C22C2×#41812yes—
424C22C2×#42112yes—
425C4—24no—
426C4—24no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—84no—
434C2—24no—
435C2—24no—
436C2—24no—
437C2—88no—
438C2C2×#146211yescharacteristic
439C2#1744812yes—
440C2#1757712yes—
441C2C2×#153311yescharacteristic
442C2C2×#155511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24×D4C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #7640C2semidirect
4. #7626C2semidirect
5. #7592C2semidirect
20. #1817C22semidirect
22. #1797C22semidirect
25. #1749C22semidirect
26. #1270C22semidirect
81. #336C23semidirect

Maximal subgroups

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

subgroupindex
C22×#3902
#207152
#76402
#76262
#75922