Finite Group Databasebeta

← #197869

Subgroups — #197869

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#2975C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1544C211yesmaximal, characteristic, complemented
6#9227C214yesmaximal, complemented
7#9225C214yesmaximal, complemented
8#8046C211yesmaximal, characteristic, complemented
9#7878C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3110—24no—
13#3110C418yescomplemented
14C2×#484—21no—
15C2×#474C2212yes—
16C2×#461C2212yes—
17C2×#450—21no—
18#2975—24no—
19#2975—24no—
20C2×#440—22no—
21C2×#436—22no—
22C2×#436C2214yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#407—22no—
28#2285C2211yescharacteristic
29#1990C2212yescomplemented
30#1956C2212yescomplemented
31#1873—22no—
32#1823—21no—
33#1805—22no—
34#1802C2214yes—
35#1786—21no—
36C4×#102—22no—
37#1544—24no—
38C2×#332C2212yescomplemented
39C2×#332C2212yescomplemented
40#1296C2214yescomplemented
41#1277C2214yescomplemented
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C22×SD16—22no—
45C22×M16—24no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533C2×C414yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×D4—21no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C42C2311yescharacteristic
70#484—28no—
71#474—24no—
72#474C2×C418yes—
73#461—24no—
74#461C2×C418yes—
75#450—28no—
76#440—216no—
77#436—216no—
78#436—28no—
79#436—28no—
80#432—22no—
81#432C2×C414yes—
82#431—22no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431D414yes—
88C2×#107—22no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#103—22no—
94C2×#103—24no—
95C2×#102—22no—
96C2×#102—24no—
97#407—216no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8D412yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2312yes—
107#384C2312yes—
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113#340—24no—
114#340—24no—
115#332—24no—
116#332—24no—
117C25—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—44no—
123C22×D4C22⋊C414yes—
124C22×D4C2×D412yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×SD16—28no—
136C2×SD16—48no—
137C2×M16—216no—
138C2×M16—84no—
139C2×M16—84no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D414yes—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C22×C412yes—
151C4⋊Q8C22×C412yes—
152#125—24no—
153#125—42no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4C22⋊C418yes—
170C4×D4C22×C414yes—
171C4×D4C2×D414yes—
172#117—24no—
173#117—42no—
174#117—82no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42#4111yescharacteristic
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204#107—28no—
205C4⋊C8—24no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8C22×C414yes—
211C4⋊C8C2×D414yes—
212#103—28no—
213#103—84no—
214#102—28no—
215#102—48no—
216C22⋊C8—216no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221#97—24no—
222C4×C8—24no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—42no—
226#95—41no—
227#95—41no—
228C24—22no—
229C24—24no—
230C24—44no—
231C24—81no—
232C24—81no—
233C24—82no—
234C24#12111yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—48no—
249C2×D4C2×C22⋊C414yes—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12012yes—
274C22×C4#12112yes—
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12312yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282SD16—416no—
283M16—816no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—44no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—44no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—82no—
309C4⋊C4C2×C22⋊C414yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C42—44no—
316C42—82no—
317C42C2×C22⋊C412yes—
318C42C4×D412yes—
319C42C4×D412yes—
320C42#12012yes—
321C42#12112yes—
322C42#12312yes—
323C42C23×C411yescharacteristic
324C42C22×D411yescharacteristic
325C42C22×D411yescharacteristic
326C42C2×#4111yescharacteristic
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—24no—
332C23—48no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—84no—
338C23—84no—
339C23—88no—
340C23—88no—
341C23#38612yes—
342C23C2×#12111yescharacteristic
343Q8—44no—
344Q8—44no—
345Q8—82no—
346Q8—82no—
347D4—24no—
348D4—24no—
349D4—48no—
350D4—44no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—88no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38612yes—
378C2×C4#38612yes—
379C2×C4#38612yes—
380C2×C4C22×C22⋊C411yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×#12011yescharacteristic
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4#57514yes—
387C2×C4#57612yes—
388C2×C4#57712yes—
389C8—44no—
390C8—44no—
391C8—44no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—816no—
402C22—84no—
403C22—84no—
404C22C2×#38611yescharacteristic
405C22#304111yescharacteristic
406C22#304211yescharacteristic
407C22#305412yes—
408C22#306912yes—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—44no—
415C4—82no—
416C4—82no—
417C4—162no—
418C4#289612yes—
419C4#294612yes—
420C4#301212yes—
421C4#305412yes—
422C2—42no—
423C2—84no—
424C2#1942412yes—
425C2#1943112yes—
426C2#2092911yescharacteristic
427C2#2097011yescharacteristic
428C2#2100611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21864C2semidirect
5. C2×#1544C2semidirect
6. #9227C2semidirect
7. #9225C2semidirect
8. #8046C2semidirect
9. #7878C2semidirect
13. #3110C4semidirect
29. #1990C22semidirect
30. #1956C22semidirect
38. C2×#332C22semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect
41. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#29752
#218642
#207322
C2×#15442
#92272
#92252
#80462
#78782