Finite Group Databasebeta

← #189833

Subgroups — #189833

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#14684C211yesmaximal, characteristic
4#8717C214yesmaximal, complemented
5#8716C214yesmaximal, complemented
6#8015C214yesmaximal
7C22×#117—21no—
8C23×C42C2211yescharacteristic
9C2×#407C2212yes—
10#2285—21no—
11#2285—21no—
12#2285—21no—
13#2285—21no—
14#2285C412yescomplemented
15#2285C412yescomplemented
16#2285C2212yes—
17#2285C2212yes—
18C2×#384C411yescharacteristic, complemented
19C2×#384C411yescharacteristic, complemented
20C2×#382—21no—
21C2×#381C2212yes—
22C2×#380—21no—
23C2×#380C2212yes—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C412yescomplemented
26#1817—21no—
27#1817—21no—
28#1817C2212yes—
29#1817C2212yes—
30#1728—21no—
31#1728—21no—
32#1434—21no—
33#1434—21no—
34#1434C2212yes—
35#1434C2212yes—
36C22⋊#95—21no—
37C22⋊#95—21no—
38#1294C22116yes—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—48no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C42—28no—
57C22×C42—22no—
58C22×C42C2×C411yescharacteristic
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
70#407—28no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2314yes—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384C2×C412yes—
78#384C2×C412yes—
79#384C2×C412yes—
80#384C2×C412yes—
81#382—22no—
82#382—22no—
83#382—22no—
84#382—22no—
85#381—28no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380C2×C414yes—
93#380C2314yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4C2×C414yes—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95C2314yes—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114C25C2×D411yescharacteristic
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4118yes—
144C23×C4C2411yescharacteristic
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—44no—
148C22×C8—24no—
149#117—48no—
150#117—48no—
151#117—48no—
152#117—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
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—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22×C414yes—
170C2×C4⋊C4C22×C414yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C42—24no—
178C2×C42—24no—
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—48no—
192C2×C42C22⋊C411yescharacteristic
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×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C22⋊C8—28no—
209C22⋊C8—44no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—28no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—41no—
225C24—41no—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24#12314yes—
229C24C22×D411yescharacteristic
230C22×C4—232no—
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—24no—
264C22×C4—24no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—416no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—416no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4C2×C22⋊C411yescharacteristic
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#11714yes—
306C22×C4#12314yes—
307C22×C4#12314yes—
308C22×C4#12314yes—
309C22×C4AGL(1,ℤ/8)14yes—
310C22×C4#13714yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4114yes—
314M16—88no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—88no—
333C42—22no—
334C42—22no—
335C42—22no—
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—22no—
349C42—48no—
350C42—48no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—28no—
360C23—28no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23#38112yes—
373C23#38112yes—
374C23#41012yes—
375C23#41012yes—
376C23C22×C22⋊C411yescharacteristic
377C23C2×#12312yes—
378C23C2×AGL(1,ℤ/8)12yes—
379C23C2×#13712yes—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—28no—
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—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—44no—
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—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—88no—
431C2×C4—88no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#38112yes—
440C2×C4#38112yes—
441C2×C4#41812yes—
442C2×C4#41812yes—
443C2×C4#41912yes—
444C2×C4#41912yes—
445C2×C4#42112yes—
446C2×C4#42112yes—
447C2×C4C2×#11712yes—
448C2×C4C2×#12312yes—
449C8—84no—
450C22—22no—
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—22no—
464C22—22no—
465C22—416no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22#182014yes—
471C22C2×#38111yescharacteristic
472C22C2×#41011yescharacteristic
473C22C2×#41011yescharacteristic
474C22C2×#41811yescharacteristic
475C22C2×#41911yescharacteristic
476C22C2×#42111yescharacteristic
477C22C2×#42111yescharacteristic
478C22#309214yes—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—84no—
492C4—84no—
493C4—164no—
494C2—24no—
495C2—24no—
496C2—42no—
497C2#1756111yescharacteristic
498C2#1756211yescharacteristic
499C2#1758911yescharacteristic
500C2#1759011yescharacteristic
501C2C2×#182011yescharacteristic
502C2#2070011yescharacteristic
503C2#2070511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
4. #8717C2semidirect
5. #8716C2semidirect
14. #2285C4semidirect
15. #2285C4semidirect
18. C2×#384C4semidirect
19. C2×#384C4semidirect
25. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#206822
#146842
#87172
#87162
#80152