Finite Group Databasebeta

← #161158

Subgroups — #161158

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3#7617C214yesmaximal, complemented
4#7616C214yesmaximal, complemented
5#7598C212yesmaximal, complemented
6#7597C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C2×C4×#41C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423—22no—
11C2×#407—22no—
12C4×C22⋊Q8C412yes—
13C4×C22⋊Q8C2212yes—
14C4×#123—22no—
15C4×#121C412yes—
16C4×#121C2212yes—
17C4×#120—22no—
18#2297C2211yescharacteristic
19#2296C2211yescharacteristic
20#2290—22no—
21#2280C411yescharacteristic
22C42×D4—22no—
23C4×#117C411yescharacteristic
24#1797—22no—
25#1797—22no—
26#1797C2214yes—
27#1797C2214yes—
28#1728C2214yes—
29#1272C2214yescomplemented
30#1271C2218yescomplemented
31#1270C2214yescomplemented
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36#525—24no—
37#525—24no—
38C2×#123—21no—
39C2×C22⋊Q8C2×C412yes—
40C2×#121C2×C412yes—
41C2×#120—21no—
42C4×#41—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C412yes—
49C2×#117—21no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C22×C42—21no—
54C22×C42—44no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2312yes—
62#423—48no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—24no—
65#407—48no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68#391C2×C411yescharacteristic
69#391C2311yescharacteristic
70#390C2×C411yescharacteristic
71#390C2311yescharacteristic
72#389C2×C412yes—
73#389C2312yes—
74#388—22no—
75#388—22no—
76#387—22no—
77#387—22no—
78#386—22no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—22no—
84#381—21no—
85#380C2×C412yes—
86#380C2×C412yes—
87#379—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100#376—21no—
101C43—21no—
102#336—24no—
103#336C2318yes—
104C25C22⋊C411yescharacteristic
105C2×#41—22no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41M1614yescomplemented
109C22×Q8C22×C411yescharacteristic
110C22×D4—22no—
111C22×D4—21no—
112C22×D4C22×C411yescharacteristic
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C2×M16—48no—
134C2×M16—48no—
135C2×M16—48no—
136C22×C8—28no—
137#123—28no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140#121—24no—
141#121—24no—
142#120—28no—
143C4×Q8—44no—
144C4×D4—216no—
145C4×D4—44no—
146C4×D4—48no—
147C4×D4—48no—
148C4×D4—44no—
149C4×D4—44no—
150#117—22no—
151#117—44no—
152#117—42no—
153#117—42no—
154#117M1614yescomplemented
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C42—21no—
174C2×C42—24no—
175C2×C42—28no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22×C412yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2411yescharacteristic
197C4⋊C8—48no—
198C4⋊C8—48no—
199C22⋊C8—48no—
200C22⋊C8—48no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209C24—21no—
210C24—22no—
211C24—24no—
212C24—41no—
213C24—41no—
214C24—42no—
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C412yes—
217#41—48no—
218#41—48no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8C2×M1612yes—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4C2×M1612yes—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
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—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×M1612yes—
285C22×C4C2×M1612yes—
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4#13714yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D412yes—
290M16—816no—
291C2×C8—48no—
292C2×C8—48no—
293C2×C8—48no—
294C4⋊C4—28no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4C2×M1614yes—
304C22⋊C4—28no—
305C22⋊C4—28no—
306C22⋊C4—24no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—41no—
323C42—42no—
324C42—41no—
325C42—42no—
326C42—44no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42#9914yes—
331C23—22no—
332C23—21no—
333C23—22no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—48no—
343C23—44no—
344C23—48no—
345C23—44no—
346C23—44no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23#40712yes—
352C23#41714yes—
353C23C22×C22⋊C411yescharacteristic
354Q8—42no—
355Q8—42no—
356D4—42no—
357D4—42no—
358D4—416no—
359D4—48no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—88no—
403C2×C4#40712yes—
404C2×C4C2×#9912yes—
405C2×C4C2×#9914yes—
406C2×C4#41312yes—
407C2×C4#41714yes—
408C2×C4#42114yes—
409C2×C4C22×M1611yescharacteristic
410C2×C4C22×M1611yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)12yes—
412C2×C4C2×#13712yes—
413C8—88no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—22no—
419C22—416no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—44no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22C2×#40711yescharacteristic
433C22C22×#9911yescharacteristic
434C22C2×#41311yescharacteristic
435C22C2×#41712yes—
436C22C2×#42112yes—
437C4—22no—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—44no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4#146514yes—
450C4#151318yes—
451C2—44no—
452C2—42no—
453C2—82no—
454C2C2×#146511yescharacteristic
455C2#1745112yes—
456C2C2×#151312yes—
457C2#1767311yescharacteristic
458C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
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. #20626C2semidirect
3. #7617C2semidirect
4. #7616C2semidirect
5. #7598C2semidirect
6. #7597C2semidirect
29. #1272C22semidirect
30. #1271C22semidirect
31. #1270C22semidirect
108. C2×#41M16semidirect
154. #117M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#206262
#76172
#76162
#75982
#75972