Finite Group Databasebeta

← #190441

Subgroups — #190441

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2318C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic
3#21786C211yesmaximal, characteristic
4#8704C212yesmaximal, complemented
5#8702C212yesmaximal, complemented
6#8700C212yesmaximal, complemented
7#8698C212yesmaximal, complemented
8#7916C212yesmaximal, complemented
9#7914C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2318—24no—
14#2318—24no—
15C2×#388C411yescharacteristic
16C2×#388C411yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C411yescharacteristic
19C2×#386C411yescharacteristic
20C2×#386C411yescharacteristic
21C2×#385C411yescharacteristic
22C2×#385C411yescharacteristic
23C2×#383C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#381C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C22×#95C2211yescharacteristic
29#1955—24no—
30#1954—24no—
31#1779C2212yes—
32#1779C2212yes—
33#1779C2212yes—
34#1779C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C2×C4×D4—28no—
45C2×C4×D4D414yes—
46C2×C4×D4D414yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63#432—28no—
64#431—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78C2×C4×C8—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#387—24no—
83#386—24no—
84#386—24no—
85#385—24no—
86#385—24no—
87#383—22no—
88#383—22no—
89#382—22no—
90#382—22no—
91#381—22no—
92#381—22no—
93#380—22no—
94#380—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2311yescharacteristic
110C2×#95C2311yescharacteristic
111C2×#95C2311yescharacteristic
112C2×#95C2311yescharacteristic
113C25C22×C411yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—42no—
118C22×D4C22⋊C414yes—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—42no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
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×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
150C22×C8—24no—
151C22×C8—24no—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4C22⋊C414yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—81no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C22⋊C4—81no—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C22⋊C414yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42#4118yes—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—28no—
192C22⋊C8—28no—
193C22⋊C8—82no—
194C22⋊C8—82no—
195C22⋊C8—82no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198C22⋊C8—82no—
199C22⋊C8—82no—
200C22⋊C8—82no—
201#97—44no—
202C4×C8—44no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24C2×C22⋊C412yes—
235C24C2×C22⋊C412yes—
236C24#13114yes—
237C24C23×C411yescharacteristic
238C2×D4—28no—
239C2×D4—28no—
240C2×D4—44no—
241C2×D4—416no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
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—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
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×C4—44no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4#11714yes—
313C22×C4#11714yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4114yes—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C4⋊C4—28no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—82no—
337C42—82no—
338C42#12314yes—
339C42#12314yes—
340C42#12314yes—
341C42#12314yes—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—28no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23#40814yes—
374C23C22×C22⋊C411yescharacteristic
375C23C2×#11712yes—
376C23C2×#13112yes—
377C23#56812yes—
378D4—416no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—88no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#38114yes—
437C2×C4#38114yes—
438C2×C4C2×#12312yes—
439C2×C4C2×#12312yes—
440C8—84no—
441C8—84no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—416no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—88no—
464C22C2×#38111yescharacteristic
465C22C2×#40811yescharacteristic
466C22#287911yescharacteristic
467C22#288311yescharacteristic
468C22#288311yescharacteristic
469C22#288811yescharacteristic
470C22#288911yescharacteristic
471C22#292414yes—
472C4—24no—
473C4—22no—
474C4—22no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C2—24no—
487C2—44no—
488C2—82no—
489C2#1738611yescharacteristic
490C2#1738711yescharacteristic
491C2#1740511yescharacteristic
492C2#1740611yescharacteristic
493C2#2068511yescharacteristic
494C2#2069011yescharacteristic
495C2#2069511yescharacteristic
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
4. #8704C2semidirect
5. #8702C2semidirect
6. #8700C2semidirect
7. #8698C2semidirect
8. #7916C2semidirect
9. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23182
#217902
#217862
#87042
#87022
#87002
#86982
#79162
#79142