Finite Group Databasebeta

← #196788

Subgroups — #196788

Order 512.

2102 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4C2×#1503C212yesmaximal, complemented
5#9144C214yesmaximal, complemented
6#9143C214yesmaximal
7#8014C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C4⋊C4—21no—
10C23×C42—21no—
11C23×C42C2211yescharacteristic
12C2×#431C2212yes—
13#2857C412yescomplemented
14#2857C412yescomplemented
15#2855—21no—
16#2855C2212yes—
17#2851—22no—
18#2848—21no—
19#2848C2212yes—
20C4×C22⋊Q8—22no—
21C4×C22⋊Q8C414yescomplemented
22#2296—21no—
23#2285C2211yescharacteristic
24#2280—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—28no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C22×#95C2212yes—
31#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34C4×C22⋊C8—22no—
35#1503—28no—
36C2×#328C2218yes—
37#1294C2218yes—
38C24×C4—22no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42#533—24no—
43#533—24no—
44C2×C4⋊Q8C2×C411yescharacteristic, complemented
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C414yes—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yescomplemented
53C2×#117—21no—
54C2×#117—21no—
55C22×C4⋊C4—216no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42—28no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—44no—
65C22×C42—44no—
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#431—24no—
75#431—24no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8C2314yes—
79C2×C4×C8—22no—
80#391—22no—
81#391C2×C414yescomplemented
82#389—22no—
83#389—22no—
84#389—22no—
85#389C2×C414yescomplemented
86#387—24no—
87#387—24no—
88#385—24no—
89#385—22no—
90#385—24no—
91#385—22no—
92#384C2×C412yescomplemented
93#382—21no—
94#381—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—21no—
99#380C2×C414yes—
100#380C2312yes—
101#379—22no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4C2×C412yescomplemented
109C4×C4⋊C4D418yes—
110C4×C22⋊C4—24no—
111C2×#95—22no—
112C2×#95—28no—
113C2×#95C2314yes—
114#336—24no—
115#328—216no—
116C25C2×D411yescharacteristic
117C22×Q8—21no—
118C22×Q8C22×C412yes—
119C23×C4—28no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—216no—
127C23×C4—28no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—44no—
132C23×C4—48no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4SD1614yes—
140C23×C4Q1614yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152#125—44no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—48no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C22⋊Q8—48no—
159C4×Q8—28no—
160C4×Q8—44no—
161#117—42no—
162#117—42no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—264no—
166C2×C4⋊C4—216no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C414yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—28no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—44no—
200C2×C42—432no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C412yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D414yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4114yes—
216C4⋊C8—44no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—48no—
220C4×C8—44no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—24no—
228#95—22no—
229#95C2×D418yes—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—24no—
235C24—24no—
236C24C2×C22⋊C412yes—
237C24C2×SD1612yes—
238C24C2×Q1612yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—44no—
244C22×C4—216no—
245C22×C4—216no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—232no—
268C22×C4—216no—
269C22×C4—21no—
270C22×C4—28no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—416no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—416no—
287C22×C4—432no—
288C22×C4—48no—
289C22×C4—432no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4#10318yes—
293C22×C4#10318yes—
294C22×C4#10418yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4#12012yes—
303C22×C4#12012yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4C2×SD1612yes—
307C22×C4C2×SD1612yes—
308C22×C4C2×Q1612yes—
309C22×C4C2×Q1612yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4#13712yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C2×#4112yes—
316C2×C8—48no—
317C2×C8—44no—
318C4⋊C4—232no—
319C4⋊C4—232no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—48no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—416no—
344C42—44no—
345C42—44no—
346C42—416no—
347C42C4×D414yes—
348C42C4×D414yes—
349C42#12114yes—
350C42#12114yes—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—28no—
355C23—24no—
356C23—28no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—28no—
367C23—28no—
368C23—22no—
369C23#41014yes—
370C23C2×#10314yes—
371C23C2×#10414yes—
372C23C22×C22⋊C411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×#12311yescharacteristic
375C23C22×SD1611yescharacteristic
376C23C22×Q1611yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13711yescharacteristic
379Q8—84no—
380Q8—84no—
381C2×C4—24no—
382C2×C4—216no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—216no—
396C2×C4—216no—
397C2×C4—24no—
398C2×C4—216no—
399C2×C4—24no—
400C2×C4—216no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—464no—
409C2×C4—416no—
410C2×C4—416no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4#38612yes—
423C2×C4#38612yes—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4C2×#10314yes—
427C2×C4C2×#10312yes—
428C2×C4C2×#10312yes—
429C2×C4#41712yes—
430C2×C4#41712yes—
431C2×C4C2×#10414yes—
432C2×C4C2×#10414yes—
433C2×C4#45014yes—
434C2×C4#45114yes—
435C2×C4#48314yes—
436C2×C4#48414yes—
437C2×C4C2×C4×D412yes—
438C2×C4C2×#12112yes—
439C8—84no—
440C22—28no—
441C22—28no—
442C22—28no—
443C22—28no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—21no—
453C22—21no—
454C22—24no—
455C22—21no—
456C22—21no—
457C22—24no—
458C22#150318yescomplemented
459C22C2×#38611yescharacteristic
460C22C2×#41012yes—
461C22C22×#10311yescharacteristic
462C22C2×#41711yescharacteristic
463C22C22×#10412yes—
464C22C2×#45011yescharacteristic
465C22C2×#45111yescharacteristic
466C22C2×#48311yescharacteristic
467C22C2×#48411yescharacteristic
468C4—24no—
469C4—22no—
470C4—24no—
471C4—22no—
472C4—22no—
473C4—416no—
474C4—42no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4#187214yes—
479C4#187214yes—
480C4#189614yes—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—28no—
486C2C2×#150312yes—
487C2#1754612yes—
488C2C2×#187211yescharacteristic
489C2C2×#189611yescharacteristic
490C2#2098311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1503C2semidirect
5. #9144C2semidirect
7. #8014C2semidirect
13. #2857C4semidirect
14. #2857C4semidirect
21. C4×C22⋊Q8C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
52. C2×C4×Q8C2×C4semidirect
81. #391C2×C4semidirect
85. #389C2×C4semidirect
92. #384C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect
458. C22#1503semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×C4×C4⋊C42
#207142
C2×#15032
#91442
#91432
#80142