Finite Group Databasebeta

← #196820

Subgroups — #196820

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4C2×#1518C212yesmaximal, complemented
5#9152C214yesmaximal, complemented
6#9143C214yesmaximal
7#8026C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C4⋊C4—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12#2857C412yescomplemented
13#2857C412yescomplemented
14#2855—21no—
15#2855C2212yes—
16#2851—22no—
17#2848—21no—
18#2848C2212yes—
19C4×C22⋊Q8—22no—
20C4×C22⋊Q8C414yescomplemented
21#2296—21no—
22#2285C2211yescharacteristic
23C2×#382—21no—
24C2×#382—21no—
25C2×#382—28no—
26C2×#382C2212yes—
27C2×#382C2212yes—
28#2280—21no—
29C22×#95—21no—
30C22×#95C2212yes—
31#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34C4×C22⋊C8—22no—
35#1518—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—28no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4—44no—
60C22×C42—24no—
61C22×C42—24no—
62C22×C42C2×C412yes—
63C22×C42C2×C412yes—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#431—24no—
71#431—24no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8C2314yes—
75C2×C4×C8—22no—
76#391—22no—
77#391C2×C414yescomplemented
78#389—22no—
79#389—22no—
80#389—22no—
81#389C2×C414yescomplemented
82#387—24no—
83#387—24no—
84#385—24no—
85#385—22no—
86#385—24no—
87#385—22no—
88#384C2×C412yescomplemented
89#382—28no—
90#382—28no—
91#382—21no—
92#382—24no—
93#382—28no—
94#382D418yes—
95#381—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—21no—
100#380C2×C414yes—
101#380C2312yes—
102#379—22no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yescomplemented
106C4×C22⋊C4—24no—
107C2×#95—28no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—28no—
112C2×#95C2314yes—
113#336—24no—
114#328—216no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8C22×C412yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—44no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—48no—
132C23×C4—48no—
133C23×C4—44no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4SD1614yes—
139C23×C4Q1614yes—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151#125—44no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—48no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—48no—
158C4×Q8—28no—
159C4×Q8—44no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—432no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C414yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D414yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4114yes—
210C4⋊C8—44no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—48no—
214C4×C8—44no—
215#95—28no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—28no—
223#95—24no—
224#95—22no—
225#95C2×D418yes—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—24no—
231C24—24no—
232C24C2×C22⋊C412yes—
233C24C2×SD1612yes—
234C24C2×Q1612yes—
235C24C22×D411yescharacteristic
236C2×Q8—24no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—416no—
281C22×C4—416no—
282C22×C4—416no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—816no—
288C22×C4—816no—
289C22×C4#10318yes—
290C22×C4#10318yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#12012yes—
299C22×C4#12012yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4C2×SD1612yes—
303C22×C4C2×SD1612yes—
304C22×C4C2×Q1612yes—
305C22×C4C2×Q1612yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C2×#4112yes—
312C2×C8—48no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—44no—
316C4⋊C4—416no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—416no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—44no—
340C42—44no—
341C42C4×D414yes—
342C42C4×D414yes—
343C42#12114yes—
344C42#12114yes—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—28no—
350C23—28no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—28no—
361C23—28no—
362C23—22no—
363C23#41014yes—
364C23C2×#10314yes—
365C23#42114yes—
366C23C22×C22⋊C411yescharacteristic
367C23C2×#12011yescharacteristic
368C23C2×#12311yescharacteristic
369C23C22×SD1611yescharacteristic
370C23C22×Q1611yescharacteristic
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372C23C2×#13711yescharacteristic
373Q8—84no—
374Q8—84no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—216no—
386C2×C4—216no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—832no—
416C2×C4—88no—
417C2×C4—88no—
418C2×C4#38612yes—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4C2×#10314yes—
423C2×C4C2×#10312yes—
424C2×C4C2×#10312yes—
425C2×C4#41712yes—
426C2×C4#41712yes—
427C2×C4#45014yes—
428C2×C4#45114yes—
429C2×C4#48314yes—
430C2×C4#48414yes—
431C2×C4C2×C4×D412yes—
432C2×C4C2×#12112yes—
433C8—84no—
434C22—28no—
435C22—28no—
436C22—28no—
437C22—28no—
438C22—22no—
439C22—22no—
440C22—24no—
441C22—24no—
442C22—22no—
443C22—22no—
444C22—24no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22#151818yescomplemented
453C22C2×#38611yescharacteristic
454C22C2×#41012yes—
455C22C22×#10311yescharacteristic
456C22C2×#41711yescharacteristic
457C22C2×#42112yes—
458C22C2×#45011yescharacteristic
459C22C2×#45111yescharacteristic
460C22C2×#48311yescharacteristic
461C22C2×#48411yescharacteristic
462C4—24no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—88no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4#187214yes—
474C4#187214yes—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—28no—
480C2C2×#151812yes—
481C2#1758812yes—
482C2C2×#187211yescharacteristic
483C2#2097911yescharacteristic
484C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1518C2semidirect
5. #9152C2semidirect
7. #8026C2semidirect
12. #2857C4semidirect
13. #2857C4semidirect
20. C4×C22⋊Q8C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
52. C2×C4×Q8C2×C4semidirect
77. #391C2×C4semidirect
81. #389C2×C4semidirect
88. #384C2×C4semidirect
105. C4×C4⋊C4C2×C4semidirect
452. C22#1518semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×#3822
#207142
C2×#15182
#91522
#91432
#80262