Finite Group Databasebeta

← #216203

Subgroups — #216203

Order 512.

975 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2653C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic, complemented
3#21902C211yesmaximal, characteristic, complemented
4#9878C212yesmaximal, complemented
5#9651C212yesmaximal, complemented
6#9081C212yesmaximal, complemented
7#9079C212yesmaximal
8#8562C212yesmaximal, complemented
9#8561C212yesmaximal, complemented
10C22×C22⋊C8C2212yes—
11#2653—24no—
12#2653—24no—
13C2×#401C411yescharacteristic
14C2×#398C411yescharacteristic
15C2×#397C411yescharacteristic
16C2×#397C411yescharacteristic
17C2×#394C411yescharacteristic
18C2×#394C411yescharacteristic
19C2×#393C411yescharacteristic
20C2×#393C411yescharacteristic
21C2×#391C2211yescharacteristic
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×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
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⋊C4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—44no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×C4×C8—22no—
71#401—24no—
72#398—24no—
73#397—24no—
74#397—24no—
75#394—24no—
76#394—24no—
77#393—24no—
78#393—24no—
79#391—22no—
80#391—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#386—22no—
88#386—22no—
89#385—22no—
90#385—22no—
91C4×C22⋊C4D414yes—
92C4×C22⋊C4D414yes—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C411yescharacteristic
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×Q8—22no—
116C22×Q8—24no—
117C22×Q8—81no—
118C22×Q8—81no—
119C22×D4—22no—
120C22×D4—24no—
121C22×D4—81no—
122C22×D4—81no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—81no—
126C23×C4—81no—
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×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—44no—
145C22×C8—82no—
146C22×C8—82no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
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×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
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—81no—
176C2×C22⋊C4—81no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C22⋊C4—81no—
180C2×C22⋊C4—81no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C22⋊C4C22⋊C414yes—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—82no—
190C4⋊C8—82no—
191C4⋊C8—82no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—82no—
198C22⋊C8—82no—
199C4×C8—82no—
200#95—22no—
201#95—22no—
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×Q8—44no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—88no—
237C2×D4—44no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—88no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
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—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
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—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4#13114yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+412yes—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—88no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—44no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—28no—
346C23—44no—
347C23—44no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—44no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23#40814yes—
367C23C22×C22⋊C411yescharacteristic
368C23#52012yes—
369C23C2×#13112yes—
370C23C2×#13112yes—
371Q8—88no—
372D4—88no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
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—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
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—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—84no—
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—82no—
428C2×C4—82no—
429C2×C4#40814yes—
430C2×C4#53511yescharacteristic
431C2×C4#53611yescharacteristic
432C2×C4#53712yes—
433C8—84no—
434C8—84no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—88no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—164no—
457C22#231111yescharacteristic
458C22C2×#40811yescharacteristic
459C22C2×#40811yescharacteristic
460C22#288311yescharacteristic
461C22#288311yescharacteristic
462C22#288811yescharacteristic
463C22#288911yescharacteristic
464C22#299414yes—
465C4—21no—
466C4—22no—
467C4—21no—
468C4—24no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C2—24no—
481C2—44no—
482C2—161no—
483C2#1738611yescharacteristic
484C2#1738611yescharacteristic
485C2#1738711yescharacteristic
486C2#1738811yescharacteristic
487C2#2184811yescharacteristic
488C2#2187611yescharacteristic
489C2#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. #9878C2semidirect
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×#26532
#219302
#219022
#98782
#96512
#90812
#90792
#85622
#85612