Finite Group Databasebeta

← #204221

Subgroups — #204221

Order 512.

1043 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2664C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#20534C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal
5C2×#1470C212yesmaximal, complemented
6#9654C212yesmaximal, complemented
7#9652C212yesmaximal
8#9651C212yesmaximal
9#9650C212yesmaximal, complemented
10C22×#126C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#432C2212yes—
13C2×#431C2212yes—
14#2664—24no—
15#2664—24no—
16C2×#400C411yescharacteristic
17C2×#399C411yescharacteristic
18C2×#398C411yescharacteristic
19C2×#398C411yescharacteristic
20C2×#398C411yescharacteristic
21C2×#397C411yescharacteristic
22C2×#396C411yescharacteristic
23C2×#396C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2211yescharacteristic
27C2×#388C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1771—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1473—28no—
36#1470—28no—
37C2×#324C2214yes—
38C2×#324C2214yes—
39C2×#324C2214yes—
40C2×#323C2214yes—
41C2×#126—22no—
42C2×#126—22no—
43C2×#126—24no—
44C2×#126—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—24no—
48C2×#125—24no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#432—24no—
75#432D418yes—
76#431—24no—
77#431—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8C2314yes—
83C2×C22⋊C8C2314yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#400—24no—
89#399—24no—
90#398—24no—
91#398—24no—
92#398—24no—
93#397—24no—
94#396—24no—
95#396—24no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4D414yes—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—28no—
121#324—28no—
122#324—28no—
123#323—28no—
124C25C22×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×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141#126—44no—
142#126—44no—
143#126—44no—
144#126—44no—
145#125—44no—
146#125—44no—
147#125—44no—
148#125—44no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—81no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22⋊C414yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D414yes—
209C2×C42C2×D412yes—
210C2×C42#4114yes—
211C4⋊C8—28no—
212C4⋊C8—44no—
213C4⋊C8—82no—
214C4⋊C8—82no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8C2×D418yes—
219#97—28no—
220#97—82no—
221C4×C8—44no—
222C4×C8—82no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95C22⋊C414yes—
232#95C22⋊C414yes—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24C23×C411yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
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—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
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—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×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C4×D414yes—
294C22×C4C4×D414yes—
295C22×C4#13114yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D412yes—
299C22×C4C2×#4112yes—
300C2×C8—28no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C4⋊C4—28no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42#12114yes—
341C42#12114yes—
342C42#12414yes—
343C42#12714yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
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—
362C23C22×C22⋊C411yescharacteristic
363C23C2×C4×D412yes—
364C23C2×#13112yes—
365C23#56812yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—28no—
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—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
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—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4#39014yes—
422C2×C4#39014yes—
423C2×C4#40814yes—
424C2×C4C2×#12112yes—
425C2×C4C2×#12411yescharacteristic
426C2×C4C2×#12711yescharacteristic
427C8—48no—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—164no—
443C22C2×#39011yescharacteristic
444C22C2×#40811yescharacteristic
445C22#287911yescharacteristic
446C22#288311yescharacteristic
447C22#288411yescharacteristic
448C22#288911yescharacteristic
449C22#288911yescharacteristic
450C22#293514yes—
451C4—24no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—24no—
456C4—24no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C2—24no—
468C2—161no—
469C2#1738711yescharacteristic
470C2#1739111yescharacteristic
471C2#1740611yescharacteristic
472C2#1741011yescharacteristic
473C2#2118911yescharacteristic
474C2#2121411yescharacteristic
475C2#2122011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. #20534C2semidirect
5. C2×#1470C2semidirect
6. #9654C2semidirect
9. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26642
#219512
#205342
C2×#14732
C2×#14702
#96542
#96522
#96512
#96502