Finite Group Databasebeta

← #189968

Subgroups — #189968

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#8726C214yesmaximal, complemented
5#8725C212yesmaximal, complemented
6#8724C212yesmaximal, complemented
7#8042C214yesmaximal, complemented
8C22×#117—21no—
9C23×C42C2211yescharacteristic
10C2×#432C2212yes—
11C2×#423C2212yes—
12C2×#407—21no—
13#2285—21no—
14#2285—21no—
15#2285—21no—
16#2285—21no—
17#2285C412yes—
18#2285C412yes—
19#2285C2212yes—
20#2285C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×#380—21no—
25C2×#380C412yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C412yes—
28#1817—21no—
29#1797—21no—
30#1797—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1779—21no—
36#1749—21no—
37#1749—21no—
38#1728—21no—
39#1296C2218yescomplemented
40#1296C2218yescomplemented
41C24×C4C2×C411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C22×M16—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—48no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C22⋊C4—22no—
57C22×C42—28no—
58C22×C42—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#432—28no—
72#423—28no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2314yes—
75C2×C4⋊C8C2314yes—
76#407—44no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2312yes—
90#384C2312yes—
91#382—22no—
92#382—22no—
93#382—22no—
94#382—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380C2×C414yes—
102#380C2×C414yes—
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⋊C4C2×C414yes—
110C4×C4⋊C4C2×C414yes—
111C2×#95—22no—
112C2×#95—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117C25C22⋊C411yescharacteristic
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—44no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—24no—
152C22×C8—24no—
153#117—48no—
154#117—48no—
155#117—48no—
156#117—48no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C414yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—48no—
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×C42—24no—
191C2×C42—24no—
192C2×C42—48no—
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4118yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—28no—
211C4⋊C8—28no—
212C4⋊C8—44no—
213C22⋊C8—44no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218#97—44no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C22×C4—232no—
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—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
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—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—416no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4#11714yes—
305C22×C4#12314yes—
306C22×C4#12314yes—
307C22×C4AGL(1,ℤ/8)12yes—
308C22×C4AGL(1,ℤ/8)14yes—
309C22×C4#13712yes—
310C22×C4#13714yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—88no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—48no—
349C42—48no—
350C42#11714yes—
351C42#12314yes—
352C42#12314yes—
353C42C2×#4114yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—28no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23#38112yes—
378C23#41312yes—
379C23#41712yes—
380C23#41812yes—
381C23#41912yes—
382C23C22×C22⋊C411yescharacteristic
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—28no—
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—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—88no—
433C2×C4—88no—
434C2×C4#38112yes—
435C2×C4#38112yes—
436C2×C4#38112yes—
437C2×C4#41312yes—
438C2×C4#41712yes—
439C2×C4#41812yes—
440C2×C4#41912yes—
441C2×C4C2×#11712yes—
442C2×C4C2×#12312yes—
443C2×C4C2×#12312yes—
444C2×C4C2×AGL(1,ℤ/8)12yes—
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13712yes—
447C2×C4C2×#13711yescharacteristic
448C2×C4#57512yes—
449C8—84no—
450C8—84no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—416no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#155914yes—
473C22C2×#38111yescharacteristic
474C22C2×#41311yescharacteristic
475C22C2×#41311yescharacteristic
476C22C2×#41811yescharacteristic
477C22C2×#41911yescharacteristic
478C22C2×#41711yescharacteristic
479C22#289611yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—84no—
491C4—84no—
492C4#309214yes—
493C4#309412yes—
494C4#309512yes—
495C2—24no—
496C2—42no—
497C2—42no—
498C2#1773111yescharacteristic
499C2C2×#155911yescharacteristic
500C2#1774511yescharacteristic
501C2#1774711yescharacteristic
502C2#2069411yescharacteristic
503C2#2070011yescharacteristic
504C2#2070211yescharacteristic
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. #8726C2semidirect
5. #8725C2semidirect
6. #8724C2semidirect
7. #8042C2semidirect
39. #1296C22semidirect
40. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#207152
#206262
#87262
#87252
#87242
#80422