Finite Group Databasebeta

← #181982

Subgroups — #181982

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2334C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic, complemented
3#20533C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal
5#8155C212yesmaximal
6#8154C212yesmaximal, complemented
7C4×#324C212yesmaximal, complemented
8C4×#323C212yesmaximal
9#7914C212yesmaximal, complemented
10C22×#126C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2334—24no—
14#2334—24no—
15C2×#389C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#385C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1771—24no—
30C8×C4⋊C4—24no—
31#1749C2212yes—
32C4×C22⋊C8C2212yes—
33C4×C22⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1473C4116yescomplemented
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C24×C4C2×C411yescharacteristic
41C2×#126—22no—
42C2×#126—22no—
43C2×#126—24no—
44C2×#126C2×C418yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#432—24no—
68#432C2×C418yescomplemented
69#431—24no—
70#431C2×C418yescomplemented
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2314yes—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76C2×#97—22no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81#389—24no—
82#389—24no—
83#388—24no—
84#388—22no—
85#388—22no—
86#388—24no—
87#387—24no—
88#387—22no—
89#387—22no—
90#387—24no—
91#385—24no—
92#385—24no—
93#382—22no—
94#382—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98C4×C4⋊C4D414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
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×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2311yescharacteristic
112C2×#95C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#324C2×C4116yescomplemented
118#323C2×C4116yes—
119C25C22×C411yescharacteristic
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—81no—
123C23×C4—81no—
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×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—24no—
139C22×C8—24no—
140#126—28no—
141#126—28no—
142#126—28no—
143#126—28no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4—81no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4C4214yes—
164C2×C4⋊C4C4214yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22×C414yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—81no—
180C2×C22⋊C4—81no—
181C2×C22⋊C4C4214yes—
182C2×C22⋊C4C4214yes—
183C2×C22⋊C4C22×C414yes—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—81no—
189C2×C42—81no—
190C2×C42—81no—
191C2×C42—81no—
192C2×C42C22×C414yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42#4114yes—
200C4⋊C8—28no—
201C4⋊C8—28no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8C22×C418yes—
205C22⋊C8C22×C418yes—
206#97—28no—
207#97—82no—
208C4×C8—28no—
209C4×C8—82no—
210C4×C8—82no—
211C4×C8—82no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C22⋊C414yes—
225#95C22⋊C414yes—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24C2×C4212yes—
235C24C23×C411yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—44no—
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—82no—
282C22×C4—82no—
283C22×C4—88no—
284C22×C4C2×C4212yes—
285C22×C4C2×C4212yes—
286C22×C4C2×C4212yes—
287C22×C4C2×C4212yes—
288C22×C4C2×C4212yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4#11712yes—
296C22×C4#11712yes—
297C22×C4#13114yes—
298C22×C4#13114yes—
299C22×C4C23×C412yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4112yes—
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—44no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—82no—
333C42—82no—
334C42C4×D414yes—
335C42C4×D414yes—
336C42C4×D414yes—
337C42C4×D414yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—24no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23C22×C4211yescharacteristic
360C23C22×C22⋊C411yescharacteristic
361C23C2×#11711yescharacteristic
362C23C2×#13112yes—
363C23C2×#13112yes—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
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—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
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—88no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4C4×C22⋊C414yes—
425C2×C4C4×C22⋊C414yes—
426C2×C4#40814yes—
427C2×C4#40814yes—
428C2×C4C2×C4×D412yes—
429C2×C4C2×C4×D412yes—
430C8—48no—
431C8—48no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—84no—
444C22—84no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22C2×C4×C22⋊C411yescharacteristic
450C22C4×#13112yes—
451C22C2×#40811yescharacteristic
452C22C2×#40811yescharacteristic
453C22#288311yescharacteristic
454C22#288411yescharacteristic
455C22#288811yescharacteristic
456C22#288911yescharacteristic
457C22#292212yes—
458C4—24no—
459C4—22no—
460C4—22no—
461C4—24no—
462C4—24no—
463C4—24no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C2—22no—
474C2—22no—
475C2—82no—
476C2#1738611yescharacteristic
477C2#1738711yescharacteristic
478C2#1739011yescharacteristic
479C2#1739111yescharacteristic
480C2C4×#40811yescharacteristic
481C2#2039311yescharacteristic
482C2#2039711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ 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. #20534C2semidirect
3. #20533C2semidirect
6. #8154C2semidirect
7. C4×#324C2semidirect
9. #7914C2semidirect
35. #1473C4semidirect
68. #432C2×C4semidirect
70. #431C2×C4semidirect
117. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23342
#205342
#205332
C2×#14732
#81552
#81542
C4×#3242
C4×#3232
#79142