Finite Group Databasebeta

← #196286

Subgroups — #196286

Order 512.

1271 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2C4×#536C211yesmaximal, characteristic
3#17449C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9103C214yesmaximal, complemented
6#9100C214yesmaximal, complemented
7#7856C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2991—24no—
12#2982C2212yes—
13#2975—22no—
14#2975C2212yes—
15#2971—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2327C2212yes—
20#2326C2212yes—
21#2323—22no—
22#2322—22no—
23#2317C411yescharacteristic
24#2311C411yescharacteristic
25#2310C411yescharacteristic
26C4×C4⋊Q8C412yes—
27C4×#124C412yes—
28C4×C22⋊Q8—22no—
29C4×#123—22no—
30C4×#117C2211yescharacteristic
31#1491—24no—
32#1487—24no—
33#1479C2218yes—
34#1465—22no—
35#1464—22no—
36#1461C2212yes—
37#1460C2212yes—
38#1274C2218yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41#536—28no—
42C2×C4⋊Q8C2×C412yes—
43C2×#124C2×C412yes—
44C2×#123—22no—
45C2×C22⋊Q8—22no—
46C4×#41—22no—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2312yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4C2311yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C42—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C8×Q8—28no—
74#436—216no—
75#436—28no—
76#432—24no—
77#432—22no—
78#432—22no—
79#432—22no—
80#432—22no—
81#431—24no—
82#431—24no—
83#431—22no—
84#431D414yes—
85#431C2314yes—
86#431C2314yes—
87#423—24no—
88#423—22no—
89#423—22no—
90#407—24no—
91#407—22no—
92#407—22no—
93C4×M16—24no—
94C4×M16—22no—
95C4×M16D414yes—
96#391—22no—
97#391C2×C412yes—
98#391C2×C412yes—
99#391C2×C412yes—
100#390C2×C412yes—
101#389—24no—
102#389—22no—
103#388—24no—
104#388—24no—
105#387—24no—
106#387—24no—
107#386—22no—
108#385—24no—
109#385—22no—
110#384C2×C412yes—
111#382C2×C412yes—
112#381—22no—
113#380—22no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C2312yes—
121C43C2×C412yes—
122Q8⋊C8—216no—
123#324—24no—
124#323—24no—
125C2×#41—24no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—42no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8C22×C412yes—
135C22×Q8C22×C411yescharacteristic
136C22×D4C22⋊C412yes—
137C22×D4C22×C411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—28no—
147C2×M16—24no—
148C2×M16—42no—
149C4⋊Q8—24no—
150C4⋊Q8M1618yescomplemented
151#124—24no—
152#124—24no—
153#123—216no—
154C22⋊Q8—216no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177#117—22no—
178#117—22no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—42no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C22⋊C412yes—
217C2×C42C22⋊C412yes—
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—28no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—42no—
234C22⋊C8—28no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—42no—
239#97—24no—
240#97—42no—
241C4×C8—28no—
242C4×C8—24no—
243C4×C8C2×D418yes—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C411yescharacteristic
252#41—48no—
253#41—48no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8C2×M1614yes—
266C2×Q8C2×M1614yes—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—42no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4#12012yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311M16—84no—
312M16—88no—
313M16—84no—
314C2×C8—28no—
315C2×C8—44no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—216no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—44no—
329C4⋊C4—84no—
330C22⋊C4—24no—
331C22⋊C4—216no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—84no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—24no—
345C42—44no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—82no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12011yescharacteristic
353C42#12011yescharacteristic
354C42#12112yes—
355C42#12112yes—
356C42#12311yescharacteristic
357C42#12311yescharacteristic
358C42C2×M1614yes—
359C42C22×D412yes—
360C42C2×#4112yes—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23#38612yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—28no—
373Q8—28no—
374Q8—44no—
375Q8—44no—
376Q8—44no—
377D4—48no—
378D4—44no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38612yes—
420C2×C4#38611yescharacteristic
421C2×C4#40714yes—
422C2×C4C2×C4×D412yes—
423C2×C4C2×#12011yescharacteristic
424C2×C4C2×#12112yes—
425C2×C4C2×#12311yescharacteristic
426C2×C4C22×M1612yes—
427C2×C4#56812yes—
428C8—48no—
429C8—84no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22C2×#38611yescharacteristic
443C22C2×#40711yescharacteristic
444C22#287911yescharacteristic
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—24no—
453C4—44no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—82no—
458C4—82no—
459C4#190912yes—
460C4D4×M1614yes—
461C2—21no—
462C2—21no—
463C2—42no—
464C2—82no—
465C2#2089811yescharacteristic
466C2C2×#190911yescharacteristic
467C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17449C2semidirect
5. #9103C2semidirect
6. #9100C2semidirect
7. #7856C2semidirect
150. C4⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
#602842
C4×#5362
#174492
#173662
#91032
#91002
#78562