Finite Group Databasebeta

← #147321

Subgroups — #147321

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#101C211yesmaximal, characteristic
2#13976C211yesmaximal, characteristic
3C2×#1394C212yesmaximal
4C2×#1394C212yesmaximal
5#10763C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C22×#101C2212yes—
8C22×#101C2212yes—
9C2×C4×M16—21no—
10C2×#391C2211yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#389C411yescharacteristic
13C2×#384C411yescharacteristic
14C4×#101—24no—
15C2×#343—21no—
16C2×#343—21no—
17#1728—21no—
18#1444—21no—
19#1444—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic
23C4×#95C411yescharacteristic
24#1394—24no—
25#1394C418yescomplemented
26C23×Q8C2311yescharacteristic
27C22×M16—22no—
28C22×M16—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8—42no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C2×#101—22no—
48C2×#101—22no—
49C2×#101—22no—
50C2×#101—22no—
51C2×#101—22no—
52C2×#101—22no—
53C2×#101—22no—
54C2×#101—22no—
55C2×#101C2×C414yes—
56C2×#101C2×C414yes—
57C2×#101D414yes—
58C2×#101Q814yes—
59C4×M16—24no—
60C4×M16—42no—
61C2×#97—41no—
62C2×C4×C8—41no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391C2×C414yes—
67#389—41no—
68#389—41no—
69#384—42no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79#343—24no—
80#343—42no—
81#336—41no—
82#336—41no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8C22×C412yes—
88C22×Q8C2×D412yes—
89C22×Q8C2×Q812yes—
90C22×Q8#4112yes—
91C22×Q8#4112yes—
92C22×Q8#4112yes—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C22×C8—42no—
117C22×C8—42no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—81no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—81no—
147C2×C4⋊C4C4212yes—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42C4212yes—
168#101—24no—
169#101—24no—
170#101—24no—
171#101—24no—
172#101—24no—
173#101—42no—
174#101—42no—
175#101C4⋊C418yescomplemented
176#97—44no—
177C4×C8—44no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—42no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—42no—
186#95—41no—
187#95—81no—
188#95—81no—
189C24C2×C22⋊C411yescharacteristic
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8C2×C4⋊C412yes—
203C2×Q8#11712yes—
204C2×Q8C4×D412yes—
205C2×Q8C4×Q812yes—
206C2×Q8C22⋊Q812yes—
207C2×Q8#12312yes—
208C2×Q8#12512yes—
209C2×Q8#12612yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
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#9911yescharacteristic
269C22×C4#9911yescharacteristic
270C22×C4#9912yes—
271C22×C4C2×C4211yescharacteristic
272C22×C4#11711yescharacteristic
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4C4×D412yes—
276C22×C4C4×Q812yes—
277C22×C4#12011yescharacteristic
278C22×C4#12112yes—
279C22×C4C22⋊Q812yes—
280C22×C4#12311yescharacteristic
281C22×C4#12411yescharacteristic
282C22×C4C4⋊Q811yescharacteristic
283M16—28no—
284M16—44no—
285M16—44no—
286M16—44no—
287M16—44no—
288M16—44no—
289M16—82no—
290M16—82no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—82no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C4×C22⋊C411yescharacteristic
323C23#38611yescharacteristic
324C23#39111yescharacteristic
325C23C2×#9911yescharacteristic
326C23C2×#9911yescharacteristic
327C23#41011yescharacteristic
328C23#41011yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—42no—
336Q8—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4#35612yes—
412C2×C4C4×C4⋊C412yes—
413C2×C4#38512yes—
414C2×C4#38712yes—
415C2×C4#38712yes—
416C2×C4#38812yes—
417C2×C4#38912yes—
418C8—48no—
419C8—84no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22#143611yescharacteristic
433C22C4×#9911yescharacteristic
434C22C4×#9911yescharacteristic
435C22#189811yescharacteristic
436C22#189911yescharacteristic
437C22#199611yescharacteristic
438C22#199611yescharacteristic
439C22C2×#35611yescharacteristic
440C22#213211yescharacteristic
441C22#213311yescharacteristic
442C22#213411yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—42no—
452C4—41no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—82no—
457C4—161no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2#1091211yescharacteristic
463C2C4×#35611yescharacteristic
464C2#1263211yescharacteristic
465C2#1309311yescharacteristic
466C2#1309411yescharacteristic
467C2#1330211yescharacteristic
468C2#1330311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1394C4semidirect
175. #101C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1012
#139762
C2×#13942
#107632