Finite Group Databasebeta

← #182715

Subgroups — #182715

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic
3#20364C211yesmaximal, characteristic
4C2×#1528C212yesmaximal
5#8227C214yesmaximal, complemented
6#8225C214yesmaximal, complemented
7#8033C212yesmaximal
8C22×#125C2211yescharacteristic
9C2×#432C2212yes—
10C2×C4×M16C2212yes—
11C4×#125—21no—
12C4×#125—21no—
13C4×#125—21no—
14C4×#125—21no—
15C4×#125C412yes—
16C4×#125C412yes—
17C4×#125C2212yes—
18C4×#125C2212yes—
19C2×#389C411yescharacteristic
20C2×#389C411yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C43C2211yescharacteristic
29#1767—21no—
30C4×C4⋊C8—21no—
31#1749—21no—
32C4×C22⋊C8—21no—
33#1745C2212yes—
34#1745C2212yes—
35#1745C2212yes—
36#1744—21no—
37#1744—21no—
38#1728—21no—
39#1728—21no—
40C4×#97C2212yes—
41#1528C4116yescomplemented
42C2×#330C2218yes—
43#1295C2218yes—
44C22×M16—22no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—24no—
48C2×#125C2×C418yes—
49C2×#125C2×C412yes—
50C2×#125C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#432—24no—
69#432C2×C418yes—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—22no—
72C4×M16—24no—
73C4×M16C2×C418yes—
74C2×#97—22no—
75C2×#97C2314yes—
76C2×#97C2314yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#382—21no—
96#382—21no—
97#382C2×C412yes—
98#382C2312yes—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43C2311yescharacteristic
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#330C2×C4132yescomplemented
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—28no—
148C2×M16—44no—
149C2×M16—44no—
150C22×C8—24no—
151C22×C8—24no—
152#125—24no—
153#125—28no—
154#125—28no—
155#125—24no—
156#125C4218yes—
157#125C22×C418yes—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C4214yes—
179C2×C4⋊C4C4214yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C414yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4114yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—28no—
227C22⋊C8—28no—
228#97—24no—
229#97—24no—
230#97—44no—
231#97C22×C418yes—
232#97C22×C418yes—
233C4×C8—28no—
234C4×C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247C24C2×C22⋊C411yescharacteristic
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C2×C4212yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4#11712yes—
295C22×C4C4×D414yes—
296C22×C4C4×D414yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—416no—
301C2×C8—28no—
302C2×C8—28no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4C2×C4214yes—
324C4⋊C4C2×C4214yes—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—44no—
352C42#10114yes—
353C42C2×C4212yes—
354C42#11712yes—
355C42C4×D414yes—
356C42C4×D414yes—
357C42C23×C412yes—
358C42C2×#4112yes—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23C4×C22⋊C412yes—
364C23#42112yes—
365C23#42112yes—
366C23C22×C22⋊C411yescharacteristic
367C2×C4—28no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4C4×C22⋊C412yes—
422C2×C4C4×C22⋊C412yes—
423C2×C4C4×C22⋊C412yes—
424C2×C4C2×#10112yes—
425C2×C4C2×#10112yes—
426C2×C4C2×#10112yes—
427C2×C4#41312yes—
428C2×C4#42112yes—
429C2×C4#42112yes—
430C2×C4C22×C4211yescharacteristic
431C2×C4C2×#11711yescharacteristic
432C2×C4C2×C4×D412yes—
433C2×C4C2×C4×D412yes—
434C8—48no—
435C8—48no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22#175914yes—
443C22C2×C4×C22⋊C411yescharacteristic
444C22C22×#10111yescharacteristic
445C22C2×#41311yescharacteristic
446C22C2×#42111yescharacteristic
447C22C2×#42111yescharacteristic
448C22C2×#42111yescharacteristic
449C22C2×#42111yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—21no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—44no—
465C4—44no—
466C4—82no—
467C4—82no—
468C4C4×#10114yes—
469C2—24no—
470C2#1766711yescharacteristic
471C2#1766811yescharacteristic
472C2#1767011yescharacteristic
473C2#1767211yescharacteristic
474C2C2×C4×#10111yescharacteristic
475C2C2×#175911yescharacteristic
476C2C4×#42111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8227C2semidirect
6. #8225C2semidirect
41. #1528C4semidirect
135. #330C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C4×#4322
#203642
C2×#15282
#82272
#82252
#80332