Finite Group Databasebeta

← #194709

Subgroups — #194709

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#20699C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9026C212yesmaximal, complemented
6#9025C212yesmaximal, complemented
7#9022C212yesmaximal, complemented
8#9021C212yesmaximal, complemented
9#8012C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2395—24no—
14#2395C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#399C2211yescharacteristic
17C2×#398C2211yescharacteristic
18C2×#397—21no—
19C2×#397C2211yescharacteristic
20C2×#394—21no—
21#2318—24no—
22#2315C2212yes—
23#2313C2212yes—
24#2312—21no—
25#2308—21no—
26#2290—24no—
27#2283—21no—
28#2280C2212yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1825—24no—
34#1824C2214yes—
35#1823C2214yes—
36#1735—24no—
37#1459—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×D4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—24no—
49C2×C4×D4C2×C414yes—
50C2×C4×D4D414yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117C2×C414yescomplemented
56C2×#117D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42D411yescharacteristic
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2314yes—
77#400—22no—
78#400C2×C414yescomplemented
79#399—22no—
80#399C2×C414yes—
81#398—22no—
82#398C2×C414yescomplemented
83#397—28no—
84#397—22no—
85#397C2×C414yes—
86#394—28no—
87#388—28no—
88#387—28no—
89#386—28no—
90#385—28no—
91#383—22no—
92#382—22no—
93#381—22no—
94#381—22no—
95#380—22no—
96#380D414yes—
97#380C2314yes—
98#379—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—28no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4C2×C414yescomplemented
109#376—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117#340—28no—
118#340—28no—
119#324—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—42no—
127C22×D4C22⋊C414yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4C22⋊C414yes—
139C23×C4C22×C412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C22×C8—48no—
151C22×C8—44no—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156#117—28no—
157#117—28no—
158#117—44no—
159#117—44no—
160#117—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D418yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4C22⋊C414yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—44no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—42no—
212C2×C42—84no—
213C2×C42C22⋊C414yes—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—44no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—21no—
227#95—21no—
228#95—24no—
229#95C22×C412yes—
230#95C22×C412yes—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C412yes—
242C24#12012yes—
243C24#12312yes—
244C24#13514yes—
245C24C22×D411yescharacteristic
246C2×D4—28no—
247C2×D4—28no—
248C2×D4—44no—
249C2×D4—48no—
250C2×D4—48no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C4×D414yes—
301C22×C4C4×D414yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12114yes—
305C22×C4#12114yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C2×C8—48no—
312C2×C8—44no—
313C2×C8—88no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—84no—
324C22⋊C4—28no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C42—24no—
337C42—24no—
338C42—48no—
339C42—44no—
340C42—44no—
341C42—48no—
342C42—84no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—48no—
362C23—42no—
363C23—42no—
364C23—81no—
365C23—84no—
366C23—84no—
367C23—84no—
368C23—81no—
369C23#38614yes—
370C23#41614yes—
371C23C22×C22⋊C411yescharacteristic
372C23C2×#12011yescharacteristic
373C23C2×#12311yescharacteristic
374C23C2×#13512yes—
375C23#57512yes—
376D4—416no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—816no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4#45814yes—
428C2×C4C2×C4×D412yes—
429C2×C4C2×#12112yes—
430C8—84no—
431C22—24no—
432C22—24no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—44no—
442C22—48no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—84no—
450C22—84no—
451C22C2×#38611yescharacteristic
452C22C2×#41611yescharacteristic
453C22#289611yescharacteristic
454C22#300212yes—
455C22#301812yes—
456C22C2×#45812yes—
457C22#309112yes—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—82no—
466C4—82no—
467C4—88no—
468C4—82no—
469C4#189014yes—
470C2—21no—
471C2—21no—
472C2—22no—
473C2—44no—
474C2—44no—
475C2#1810714yes—
476C2#2090811yescharacteristic
477C2#2095211yescharacteristic
478C2C2×#189011yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20699C2semidirect
5. #9026C2semidirect
6. #9025C2semidirect
7. #9022C2semidirect
8. #9021C2semidirect
9. #8012C2semidirect
14. #2395C4semidirect
55. C2×#117C2×C4semidirect
78. #400C2×C4semidirect
82. #398C2×C4semidirect
108. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
#340572
#206992
C2×#14592
#90262
#90252
#90222
#90212
#80122