Finite Group Databasebeta

← #189949

Subgroups — #189949

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#384C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#8725C214yesmaximal, complemented
5#8724C214yesmaximal, complemented
6#8043C214yesmaximal, complemented
7C23×C4⋊C4—21no—
8C23×C4⋊C4—21no—
9C23×C4⋊C4C412yes—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12C2×#423C2212yes—
13C2×#407—21no—
14C2×#384—21no—
15C2×#384—21no—
16C2×#384—21no—
17C2×#384—21no—
18C2×#384—21no—
19C2×#384—21no—
20C2×#384C412yes—
21C2×#384C412yes—
22C2×#384C412yes—
23C2×#384C2212yes—
24C2×#384C2212yes—
25C2×#384C2212yes—
26#1817—21no—
27#1797—21no—
28#1797—21no—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1779—21no—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36#1728—21no—
37#1296C22116yescomplemented
38C24×C4—22no—
39C24×C4C2×C411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C22×M16—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—48no—
59C22×C4⋊C4—48no—
60C22×C4⋊C4—48no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4C2×C414yes—
64C22×C42—28no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#431—28no—
78#423—28no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2314yes—
81C2×C4⋊C8C2314yes—
82#407—44no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—22no—
101#384C2×C414yes—
102#384C2×C414yes—
103#384C2×C414yes—
104#384C2314yes—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109C25C22⋊C411yescharacteristic
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—48no—
122C23×C4—48no—
123C23×C4—48no—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4SD1614yes—
134C23×C4Q1614yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—24no—
148C22×C8—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4C22×C418yes—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4118yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—28no—
208C4⋊C8—28no—
209C4⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—44no—
212C22⋊C8—44no—
213C22⋊C8—44no—
214C22⋊C8—44no—
215C4×C8—44no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24#10314yes—
223C24C2×C22⋊C411yescharacteristic
224C24C2×C22⋊C411yescharacteristic
225C24C2×C22⋊C411yescharacteristic
226C22×C4—232no—
227C22×C4—232no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
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—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—216no—
269C22×C4—216no—
270C22×C4—28no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—832no—
281C22×C4#10014yes—
282C22×C4#10114yes—
283C22×C4#10314yes—
284C22×C4#10314yes—
285C22×C4#10314yes—
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×C4C2×C22⋊C411yescharacteristic
295C22×C4#11714yes—
296C22×C4#12314yes—
297C22×C4#12314yes—
298C22×C4C2×SD1612yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×SD1612yes—
301C22×C4C2×Q1612yes—
302C22×C4C2×Q1612yes—
303C22×C4C2×Q1612yes—
304C22×C4AGL(1,ℤ/8)14yes—
305C22×C4AGL(1,ℤ/8)14yes—
306C22×C4#13714yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310M16—88no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—48no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42#11714yes—
342C42#12314yes—
343C42#12314yes—
344C42C2×#4114yes—
345C23—28no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—28no—
351C23—28no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23#38112yes—
365C23C2×#10012yes—
366C23C2×#10112yes—
367C23C2×#10312yes—
368C23C2×#10312yes—
369C23C2×#10312yes—
370C23#41812yes—
371C23#41812yes—
372C23#41912yes—
373C23C22×C22⋊C411yescharacteristic
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
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—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—816no—
413C2×C4—816no—
414C2×C4—816no—
415C2×C4#38112yes—
416C2×C4#38112yes—
417C2×C4#38112yes—
418C2×C4C2×#10012yes—
419C2×C4C2×#10012yes—
420C2×C4C2×#10112yes—
421C2×C4C2×#10112yes—
422C2×C4C2×#10312yes—
423C2×C4C2×#10312yes—
424C2×C4C2×#10312yes—
425C2×C4#41812yes—
426C2×C4#41812yes—
427C2×C4#41912yes—
428C2×C4#48118yes—
429C2×C4#48418yes—
430C2×C4C2×#11712yes—
431C2×C4C2×#12312yes—
432C2×C4C2×#12312yes—
433C2×C4C22×SD1611yescharacteristic
434C2×C4C22×Q1611yescharacteristic
435C2×C4C2×AGL(1,ℤ/8)12yes—
436C2×C4C2×AGL(1,ℤ/8)12yes—
437C2×C4C2×#13712yes—
438C8—84no—
439C8—84no—
440C22—232no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22#154814yes—
460C22#154914yes—
461C22#182414yes—
462C22C2×#38111yescharacteristic
463C22C22×#10011yescharacteristic
464C22C22×#10111yescharacteristic
465C22C22×#10311yescharacteristic
466C22C2×#41811yescharacteristic
467C22C2×#41811yescharacteristic
468C22C2×#41911yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—88no—
478C4#182114yes—
479C4#182414yes—
480C4C2×#48112yes—
481C4C2×#48412yes—
482C4#309214yes—
483C2—24no—
484C2—24no—
485C2—24no—
486C2C2×#154811yescharacteristic
487C2C2×#154911yescharacteristic
488C2#1774111yescharacteristic
489C2#1774211yescharacteristic
490C2C2×#182111yescharacteristic
491C2C2×#182411yescharacteristic
492C2#2070011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8725C2semidirect
5. #8724C2semidirect
6. #8043C2semidirect
37. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3842
#207142
#206262
#87252
#87242
#80432