Finite Group Databasebeta

← #379877

Subgroups — #379877

Order 512.

2225 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1896C212yesmaximal
4#20682C211yesmaximal, characteristic, complemented
5#11226C214yesmaximal, complemented
6#11075C214yesmaximal, complemented
7#10719C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C4⋊C4—21no—
10C23×C42C2212yes—
11C23×C42C2211yescharacteristic
12C22×#104C2214yes—
13C2×#407C2212yes—
14#2855C412yescomplemented
15#2855C412yescomplemented
16#2854C412yescomplemented
17#2854C412yescomplemented
18#2853—22no—
19#2849—21no—
20#2849—21no—
21C4×#123—22no—
22C4×#123—22no—
23#2297C2212yes—
24#2296C2212yes—
25#2280C2211yescharacteristic
26C2×C4×C4⋊C4—28no—
27C2×C4×C4⋊C4—22no—
28C2×C4×C4⋊C4C2214yes—
29C22×#95—21no—
30#1896—28no—
31#1817—22no—
32#1817—22no—
33C2×#339C2214yes—
34C2×#339C2218yes—
35#1728C2214yes—
36C22×#41C2311yescharacteristic
37C24×C4—22no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16C2314yes—
42#525—24no—
43#525—24no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×C22⋊Q8C2×C411yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#121C2×C411yescharacteristic
51C2×#120—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×#117C2311yescharacteristic
55C2×#117C2312yes—
56C22×C4⋊C4—216no—
57C22×C4⋊C4—28no—
58C22×C4⋊C4—24no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42—22no—
62C22×C42—24no—
63C22×C42—48no—
64C22×C42C2×C412yes—
65C22×C42C2×C412yes—
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42D418yes—
70C22×C42C2314yes—
71C22×C42C2312yes—
72C2×#104—28no—
73C2×#104—48no—
74#407—24no—
75#407—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#391C2×C414yescomplemented
79#390C2×C414yescomplemented
80#389C2×C414yescomplemented
81#389C2×C414yescomplemented
82#388—24no—
83#388—24no—
84#387—24no—
85#387—24no—
86#386—22no—
87#386—22no—
88#386—22no—
89#386—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#381—22no—
95#381—22no—
96#381—21no—
97#380C2×C412yes—
98#380C2×C412yes—
99#379—21no—
100C4×C4⋊C4—216no—
101C4×C4⋊C4—216no—
102C4×C4⋊C4—24no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—24no—
106C2×#95—28no—
107C2×#95—24no—
108#339—28no—
109#339—48no—
110#336—28no—
111C25C2×D411yescharacteristic
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—24no—
115C2×#41—24no—
116C22×Q8C22×C411yescharacteristic
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C22×C411yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—28no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—216no—
127C23×C4—28no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—21no—
132C23×C4—44no—
133C23×C4—48no—
134C23×C4—48no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D414yes—
146C23×C4C2×D412yes—
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—28no—
152C2×M16—28no—
153C22×C8—24no—
154#123—48no—
155#123—48no—
156#123—42no—
157#123—42no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160#121—24no—
161#121—24no—
162#120—42no—
163#120—42no—
164C4×D4—44no—
165C4×D4—44no—
166#117—22no—
167#117—22no—
168#117—24no—
169#117—24no—
170C2×C4⋊C4—264no—
171C2×C4⋊C4—232no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C414yes—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C42—28no—
191C2×C42—28no—
192C2×C42—24no—
193C2×C42—28no—
194C2×C42—28no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—416no—
205C2×C42—48no—
206C2×C42—416no—
207C2×C42C22⋊C414yes—
208C2×C42C22⋊C414yes—
209C2×C42C22×C414yes—
210C2×C42C2×D418yes—
211C2×C42C2×D414yes—
212#104—416no—
213C22⋊C8—44no—
214C22⋊C8—44no—
215#95—216no—
216#95—22no—
217#95—22no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222C24—22no—
223C24—22no—
224C24—24no—
225C24—24no—
226C24—21no—
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C24#12012yes—
230C24#12312yes—
231C24C22×D411yescharacteristic
232#41—44no—
233#41—44no—
234#41—48no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C22×C4—216no—
248C22×C4—216no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—216no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—216no—
266C22×C4—232no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—416no—
281C22×C4—464no—
282C22×C4—48no—
283C22×C4—416no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4#10418yescomplemented
290C22×C4C2×C22⋊C414yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C414yes—
294C22×C4C4×D414yes—
295C22×C4C4×D414yes—
296C22×C4#12012yes—
297C22×C4#12114yes—
298C22×C4#12114yes—
299C22×C4#12312yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4H(5,𝔽2)14yes—
305C22×C42−1+414yes—
306M16—44no—
307M16—44no—
308M16—48no—
309C2×C8—44no—
310C2×C8—48no—
311C4⋊C4—264no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—28no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—48no—
331C42—28no—
332C42—28no—
333C42—28no—
334C42—28no—
335C42—28no—
336C42—28no—
337C42—416no—
338C42—42no—
339C42—48no—
340C42—44no—
341C23—216no—
342C23—216no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23#38612yes—
361C23#38612yes—
362C23C2×#10414yes—
363C23#42114yes—
364C23C22×C22⋊C411yescharacteristic
365C23C2×#12011yescharacteristic
366C23C2×#12311yescharacteristic
367C23#53512yes—
368C23#53612yes—
369Q8—44no—
370D4—44no—
371D4—81no—
372D4—81no—
373D4—82no—
374D4—84no—
375D4—84no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—216no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—216no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—216no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—464no—
396C2×C4—432no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—84no—
413C2×C4#38612yes—
414C2×C4#38612yes—
415C2×C4C2×#10418yes—
416C2×C4C2×C4×D412yes—
417C2×C4#52014yes—
418C2×C4C2×#12112yes—
419C2×C4#53414yes—
420C2×C4#53614yes—
421C2×C4#53714yes—
422C2×C4#53918yes—
423C8—84no—
424C22—28no—
425C22—28no—
426C22—216no—
427C22—24no—
428C22—24no—
429C22—28no—
430C22—28no—
431C22—22no—
432C22—22no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—84no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22#189614yes—
444C22#189714yes—
445C22C2×#38611yescharacteristic
446C22#231112yes—
447C22#261311yescharacteristic
448C22#261411yescharacteristic
449C22#262312yes—
450C22C22×#10412yes—
451C22C2×#42112yes—
452C4—22no—
453C4—22no—
454C4—28no—
455C4—416no—
456C4—44no—
457C4—84no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4#232214yes—
462C4#261814yes—
463C2—21no—
464C2—21no—
465C2—22no—
466C2—24no—
467C2—24no—
468C2—161no—
469C2#1425611yescharacteristic
470C2C2×#189611yescharacteristic
471C2C2×#189711yescharacteristic
472C2#2189914yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
4. #20682C2semidirect
5. #11226C2semidirect
6. #11075C2semidirect
7. #10719C2semidirect
14. #2855C4semidirect
15. #2855C4semidirect
16. #2854C4semidirect
17. #2854C4semidirect
78. #391C2×C4semidirect
79. #390C2×C4semidirect
80. #389C2×C4semidirect
81. #389C2×C4semidirect
289. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
#366262
C22×C4×C4⋊C42
C2×#18962
#206822
#112262
#110752
#107192