Finite Group Databasebeta

← #216202

Subgroups — #216202

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2649C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3#21902C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9651C212yesmaximal, complemented
6#9081C212yesmaximal, complemented
7#9079C212yesmaximal
8#8562C212yesmaximal, complemented
9#8561C212yesmaximal, complemented
10C22×C22⋊C8C2212yes—
11#2649—24no—
12#2649—24no—
13C2×#400C411yescharacteristic
14C2×#400C411yescharacteristic
15C2×#399C411yescharacteristic
16C2×#397C411yescharacteristic
17C2×#397C411yescharacteristic
18C2×#396C411yescharacteristic
19C2×#394C411yescharacteristic
20C2×#394C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1923—24no—
29#1850—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1779C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C23×C8—22no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊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×C22⋊C4C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—44no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×C4×C8—22no—
73#400—24no—
74#400—24no—
75#399—24no—
76#397—24no—
77#397—24no—
78#396—24no—
79#394—24no—
80#394—24no—
81#389—22no—
82#389—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#386—22no—
90#386—22no—
91#385—22no—
92#385—22no—
93C4×C22⋊C4D414yes—
94C4×C22⋊C4D414yes—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#324—44no—
112#323—44no—
113C25C22×C411yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—22no—
116C22×D4—24no—
117C22×D4—81no—
118C22×D4—81no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—81no—
122C23×C4—81no—
123C23×C4C22×C411yescharacteristic
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×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2411yescharacteristic
138C22×C8—24no—
139C22×C8—24no—
140C22×C8—44no—
141C22×C8—82no—
142C22×C8—82no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—81no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C22⋊C4—81no—
180C2×C22⋊C4—81no—
181C2×C22⋊C4—81no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4C22⋊C414yes—
184C2×C22⋊C4C22⋊C414yes—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C4⋊C8—82no—
192C4⋊C8—82no—
193C4⋊C8—82no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C22⋊C8—82no—
200C22⋊C8—82no—
201C4×C8—82no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C22⋊C414yes—
215#95C22⋊C414yes—
216C24—22no—
217C24—22no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24C2×C22⋊C412yes—
231C24#13114yes—
232C24C23×C411yescharacteristic
233C2×D4—44no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—88no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
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—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
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—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—41no—
287C22×C4—41no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4#13114yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+412yes—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—88no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—44no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—28no—
345C23—44no—
346C23—44no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—44no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23#40814yes—
366C23C22×C22⋊C411yescharacteristic
367C23#52012yes—
368C23C2×#13112yes—
369C23C2×#13112yes—
370D4—88no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
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—44no—
416C2×C4—84no—
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—82no—
427C2×C4#40814yes—
428C2×C4#53511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53712yes—
431C8—84no—
432C8—84no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—88no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—164no—
455C22#231111yescharacteristic
456C22C2×#40811yescharacteristic
457C22C2×#40811yescharacteristic
458C22#288311yescharacteristic
459C22#288311yescharacteristic
460C22#288811yescharacteristic
461C22#288911yescharacteristic
462C22#299414yes—
463C4—21no—
464C4—22no—
465C4—21no—
466C4—24no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C2—24no—
479C2—44no—
480C2—161no—
481C2#1738611yescharacteristic
482C2#1738711yescharacteristic
483C2#1738711yescharacteristic
484C2#1738911yescharacteristic
485C2#2184811yescharacteristic
486C2#2187611yescharacteristic
487C2#2188111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21930C2semidirect
3. #21902C2semidirect
4. #9879C2semidirect
5. #9651C2semidirect
6. #9081C2semidirect
8. #8562C2semidirect
9. #8561C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26492
#219302
#219022
#98792
#96512
#90812
#90792
#85622
#85612