Finite Group Databasebeta

← #182013

Subgroups — #182013

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3C2×#1465C212yesmaximal
4#8155C214yesmaximal, complemented
5#8154C214yesmaximal, complemented
6#7911C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#117C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#407C2214yes—
11C2×C4×M16—22no—
12#2290—22no—
13#2290—22no—
14#2290—22no—
15#2290—22no—
16C2×#388C2212yes—
17C2×#387C2212yes—
18C2×#386C412yes—
19C2×#385C412yes—
20C2×C4×C4⋊C4C411yescharacteristic
21C2×C4×C22⋊C4C411yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×#376C411yescharacteristic
24#1749—22no—
25#1749C2214yes—
26C4×C22⋊C8—22no—
27C4×C22⋊C8C2214yes—
28#1728—22no—
29#1465C4116yescomplemented
30C2×#324C2214yes—
31C2×#323C2214yes—
32#1280C2218yes—
33C23×D4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C412yes—
36C24×C4C2311yescharacteristic
37C22×M16—24no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—24no—
41C2×C4×D4C2×C418yes—
42C2×#117—22no—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117—44no—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C42—44no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2312yes—
65#407—28no—
66#407C2×C4116yes—
67C2×C22⋊C8—24no—
68C2×C22⋊C8C2318yes—
69C4×M16—48no—
70C2×#97—24no—
71C2×C4×C8—24no—
72#388—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#386—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96#376—21no—
97#376—22no—
98#376—21no—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2312yes—
104#336—24no—
105#336—24no—
106#324C2×C4116yescomplemented
107#323C2×C4116yescomplemented
108C25C22×C411yescharacteristic
109C25C22×C411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—44no—
113C22×D4C4214yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—44no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4C4214yes—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C414yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4#4114yes—
141C23×C4C2411yescharacteristic
142C2×M16—216no—
143C2×M16—48no—
144C2×M16—48no—
145C22×C8—28no—
146C4×D4—216no—
147C4×D4—216no—
148#117—48no—
149#117—44no—
150#117—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4C22×C414yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4C4214yes—
174C2×C22⋊C4C22×C414yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—28no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—44no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C4214yes—
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C22⋊C8—216no—
203C22⋊C8—28no—
204C22⋊C8C22×C4116yes—
205#97—48no—
206C4×C8—48no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217C24—24no—
218C24—21no—
219C24—21no—
220C24—24no—
221C24—41no—
222C24—41no—
223C24—42no—
224C24—41no—
225C24—42no—
226C24—41no—
227C24C2×C4212yes—
228C24C2×C4212yes—
229C24C2×C22⋊C412yes—
230C24#11712yes—
231C24C23×C411yescharacteristic
232C2×D4—28no—
233C2×D4—48no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—48no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
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—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—432no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C2×C4212yes—
293C22×C4C2×C4212yes—
294C22×C4C2×C4212yes—
295C22×C4C2×C4212yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4#11712yes—
303C22×C4C4×D418yes—
304C22×C4C4×D418yes—
305C22×C4C23×C412yes—
306C22×C4C22×D412yes—
307C22×C4C2×#4112yes—
308M16—432no—
309C2×C8—216no—
310C2×C8—48no—
311C2×C8—48no—
312C4⋊C4—28no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42#9914yes—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—24no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—48no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—84no—
370C23—84no—
371C23C4×C22⋊C414yes—
372C23C22×C4211yescharacteristic
373C23C22×C22⋊C411yescharacteristic
374C23C2×#11711yescharacteristic
375C23#56814yes—
376D4—416no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—216no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4C4×C22⋊C412yes—
427C2×C4C4×C22⋊C412yes—
428C2×C4C2×#9914yes—
429C2×C4C2×#9912yes—
430C2×C4#41012yes—
431C2×C4#41314yes—
432C2×C4C2×C4×D414yes—
433C8—416no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—48no—
444C22—48no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—84no—
453C22—84no—
454C22—82no—
455C22—82no—
456C22C2×C4×C22⋊C411yescharacteristic
457C22#287312yes—
458C22#287912yes—
459C22C22×#9911yescharacteristic
460C22C2×#41011yescharacteristic
461C22C2×#41312yes—
462C22#292112yes—
463C4—24no—
464C4—22no—
465C4—24no—
466C4—24no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—82no—
474C4C4×#9914yes—
475C2—22no—
476C2—22no—
477C2—44no—
478C2—82no—
479C2#1740012yes—
480C2#1740112yes—
481C2#2037711yescharacteristic
482C2C2×C4×#9911yescharacteristic
483C2C4×#41311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
4. #8155C2semidirect
5. #8154C2semidirect
6. #7911C2semidirect
29. #1465C4semidirect
106. #324C2×C4semidirect
107. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C4×#4072
C2×#14652
#81552
#81542
#79112