Finite Group Databasebeta

← #189751

Subgroups — #189751

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2#21740C212yesmaximal
3#8703C212yesmaximal, complemented
4#8702C212yesmaximal, complemented
5#8701C212yesmaximal, complemented
6#8700C212yesmaximal, complemented
7#7913C212yesmaximal, complemented
8#7911C211yesmaximal, characteristic, complemented
9#7907C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#407C2212yes—
12C22×C22⋊C8C2212yes—
13#2357—22no—
14#2357—22no—
15#2357—22no—
16#2357—22no—
17C2×#390C412yes—
18C2×#388C412yes—
19C2×#388C412yes—
20C2×#382C2211yescharacteristic
21C2×#381C2211yescharacteristic
22C2×#381C2211yescharacteristic
23C2×C4×C4⋊C4C412yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1870—22no—
28#1870—22no—
29#1869—22no—
30#1869—22no—
31#1817C2214yes—
32#1779C2214yes—
33#1280C2212yes—
34#1280C2212yes—
35#1280C2212yes—
36#1280C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C23×C8—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—22no—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C42C2×C412yes—
61C22×C42C2×C412yes—
62C22×C42C2×C412yes—
63C22×C42C2311yescharacteristic
64#407—24no—
65#407—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74#390—24no—
75#390—24no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#382—22no—
81#382—22no—
82#381—22no—
83#381—22no—
84#381—22no—
85#381—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2311yescharacteristic
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102C2×#95C2311yescharacteristic
103#336—24no—
104#336—24no—
105C25C22×C411yescharacteristic
106C25C22×C411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4C22⋊C412yes—
112C22×D4C22⋊C412yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—82no—
117C23×C4—82no—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4#4112yes—
131C23×C4#4114yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C2×M16—44no—
135C2×M16—84no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—44no—
139C22×C8—84no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—44no—
188C2×C42—82no—
189C2×C42—82no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—84no—
199C22⋊C8—84no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209C24—22no—
210C24—22no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—42no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24C2×C22⋊C412yes—
222C24C2×C22⋊C412yes—
223C24#11712yes—
224C24#11712yes—
225C24C23×C411yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—48no—
233C2×D4—48no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
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—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—84no—
279C22×C4—82no—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#11712yes—
288C22×C4#11712yes—
289C22×C4#12314yes—
290C22×C4#12314yes—
291C22×C4#12314yes—
292C22×C4#12314yes—
293C22×C4#13114yes—
294C22×C4C22×D412yes—
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298M16—88no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—88no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—48no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—44no—
327C42—82no—
328C42—82no—
329C42—82no—
330C42—82no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—24no—
335C23—24no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23#38114yes—
366C23C22×C22⋊C411yescharacteristic
367C23C2×#11711yescharacteristic
368C23C2×#11711yescharacteristic
369C23C2×#13112yes—
370C23#56812yes—
371D4—48no—
372D4—48no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#38112yes—
427C2×C4#38112yes—
428C2×C4#41312yes—
429C2×C4#41312yes—
430C2×C4C2×#12312yes—
431C2×C4C2×#12312yes—
432C8—84no—
433C8—84no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—82no—
453C22—82no—
454C22—84no—
455C22—82no—
456C22—82no—
457C22—84no—
458C22C2×#38111yescharacteristic
459C22#288012yes—
460C22#288312yes—
461C22C2×#41311yescharacteristic
462C22C2×#41311yescharacteristic
463C22#292211yescharacteristic
464C22#292311yescharacteristic
465C22#292512yes—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—44no—
473C4—42no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—82no—
478C4—82no—
479C2—21no—
480C2—21no—
481C2—22no—
482C2—44no—
483C2—82no—
484C2#1742412yes—
485C2#1742912yes—
486C2#2068411yescharacteristic
487C2#2068911yescharacteristic
488C2#2069411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8703C2semidirect
4. #8702C2semidirect
5. #8701C2semidirect
6. #8700C2semidirect
7. #7913C2semidirect
8. #7911C2semidirect
9. #7907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#217402
#87032
#87022
#87012
#87002
#79132
#79112
#79072