Finite Group Databasebeta

← #480283

Subgroups — #480283

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#17357C212yesmaximal
3#11513C214yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11511C212yesmaximal, complemented
6#11447C214yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9C23×C42C411yescharacteristic
10#2920—22no—
11#2285—22no—
12#2285—22no—
13#2285C412yes—
14#2285C412yes—
15#2285C2212yes—
16#2285C2212yes—
17C2×#384C411yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C411yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380—22no—
22C2×C4×C4⋊C4—22no—
23#1464—22no—
24#1464C2214yes—
25#1463C2214yes—
26#1462—22no—
27#1458—22no—
28#1458C2214yes—
29#1325C2218yescomplemented
30#1324C2218yescomplemented
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C2×#117—21no—
34C2×#117—21no—
35C2×#117—41no—
36C2×#117—41no—
37C2×#117—41no—
38C2×#117—41no—
39C2×#117—41no—
40C2×#117—41no—
41C2×#117—41no—
42C2×#117—41no—
43C2×#117C2×C412yes—
44C2×#117C2×C412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117D412yes—
48C2×#117D412yes—
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4D412yes—
59C22×C42—21no—
60C22×C42—21no—
61C22×C42—21no—
62C22×C42—28no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42D412yes—
69#432—24no—
70#432—42no—
71#431—24no—
72#431—42no—
73#423—42no—
74#407—24no—
75C4×M16—42no—
76#384—22no—
77#384—22no—
78#384C2×C412yes—
79#384C2×C412yes—
80#384C2×C412yes—
81#384C2312yes—
82#382—22no—
83#382—22no—
84#382C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2312yes—
88#380—24no—
89#380—24no—
90#380—24no—
91#380—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C2×#95—22no—
97C2×#95—22no—
98#324—24no—
99#324C2318yes—
100#323—24no—
101#323—24no—
102C25C22⋊C411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—28no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D412yes—
126C2×M16—44no—
127#117—41no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—42no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—41no—
139#117—42no—
140#117—42no—
141#117—42no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146#117—42no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2411yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42M1618yescomplemented
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C4⋊C8—44no—
211C4⋊C8—44no—
212C22⋊C8—28no—
213C22⋊C8—44no—
214#97—44no—
215C4×C8—44no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—432no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4#9911yescharacteristic
283C22×C4#9911yescharacteristic
284C22×C4#9911yescharacteristic
285C22×C4#9911yescharacteristic
286C22×C4#10114yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×M1614yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D412yes—
296M16—88no—
297C2×C8—48no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C42—24no—
319C42—28no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—28no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42C2×M1614yes—
337C42C2×M1614yes—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—48no—
353C23—48no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23#40712yes—
359C23C2×#9911yescharacteristic
360C23C2×#9911yescharacteristic
361C23C2×#10112yes—
362C23C22×C22⋊C411yescharacteristic
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—216no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—416no—
393C2×C4—416no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#40712yes—
414C2×C4C2×#9911yescharacteristic
415C2×C4C2×#9911yescharacteristic
416C2×C4C2×#9911yescharacteristic
417C2×C4C2×#9911yescharacteristic
418C2×C4C2×#10112yes—
419C2×C4C2×#10112yes—
420C2×C4C22×M1612yes—
421C8—88no—
422C22—24no—
423C22—24no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—48no—
437C22—48no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#212612yes—
443C22#213212yes—
444C22C2×#40711yescharacteristic
445C22C22×#9911yescharacteristic
446C22C22×#10111yescharacteristic
447C4—28no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—22no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—82no—
458C4—82no—
459C4#146312yes—
460C4#146312yes—
461C2—21no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—42no—
466C2—42no—
467C2C2×#146311yescharacteristic
468C2C2×#212611yescharacteristic
469C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11513C2semidirect
4. #11512C2semidirect
5. #11511C2semidirect
6. #11447C2semidirect
29. #1325C22semidirect
30. #1324C22semidirect
205. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#173572
#115132
#115122
#115112
#114472