Finite Group Databasebeta

← #196859

Subgroups — #196859

Order 512.

2133 conjugacy classes of subgroups.

The subgroup lattice has 460 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
3#20682C211yesmaximal, characteristic, complemented
4C2×#1504C212yesmaximal
5#9148C214yesmaximal, complemented
6#9144C214yesmaximal, complemented
7#8020C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C4⋊C4—21no—
10C23×C42—21no—
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13#2855C412yescomplemented
14#2855C2212yes—
15#2854C412yescomplemented
16#2854C2212yes—
17#2853—22no—
18#2849—21no—
19#2849—21no—
20C4×#123—22no—
21C4×#123—22no—
22#2297C412yescomplemented
23#2296C412yescomplemented
24#2280C2211yescharacteristic
25C2×C4×C4⋊C4—22no—
26C2×C4×C4⋊C4—28no—
27C2×C4×C4⋊C4C2214yes—
28C22×#95C2212yes—
29#1817—22no—
30#1817C2214yes—
31#1728—22no—
32#1504—28no—
33C2×#328C2214yes—
34C2×#327C2214yes—
35#1294C2218yes—
36C22×#41C2×C411yescharacteristic
37C24×C4—22no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42#525—24no—
43#525—24no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×C22⋊Q8C2311yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#121C2311yescharacteristic
51C2×#120—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C412yes—
56C22×C4⋊C4—216no—
57C22×C4⋊C4—28no—
58C22×C4⋊C4—24no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42—28no—
62C22×C42—24no—
63C22×C42—48no—
64C22×C42C2×C412yes—
65C22×C42C2×C412yes—
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42C2312yes—
70#407—24no—
71#407—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8C2314yes—
74#391C2×C414yescomplemented
75#390C2×C414yescomplemented
76#389C2×C414yescomplemented
77#389C2×C414yescomplemented
78#388—24no—
79#388—24no—
80#387—24no—
81#387—24no—
82#386—22no—
83#386—22no—
84#386—22no—
85#386—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#381—22no—
91#381—22no—
92#381—21no—
93#380C2×C412yes—
94#380C2312yes—
95#379—21no—
96C4×C4⋊C4—216no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—216no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—24no—
102C2×#95—22no—
103C2×#95—24no—
104C2×#95D418yes—
105C2×#95C2314yes—
106#336—28no—
107#328—28no—
108#327—28no—
109C25C2×D411yescharacteristic
110C2×#41—24no—
111C2×#41—44no—
112C22×Q8C22×C411yescharacteristic
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4C22×C411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—216no—
125C23×C4—28no—
126C23×C4—24no—
127C23×C4—21no—
128C23×C4—44no—
129C23×C4—48no—
130C23×C4—48no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D414yes—
143C23×C4#4114yes—
144C23×C4C2411yescharacteristic
145C2×M16—48no—
146C2×M16—44no—
147C22×C8—24no—
148#123—48no—
149#123—44no—
150#123—44no—
151#123—44no—
152C22⋊Q8—24no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155#121—24no—
156#121—22no—
157#121—22no—
158#120—42no—
159#120—42no—
160C4×D4—42no—
161C4×D4—44no—
162C4×D4—42no—
163#117—24no—
164#117—44no—
165C2×C4⋊C4—264no—
166C2×C4⋊C4—232no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C414yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42—24no—
188C2×C42—28no—
189C2×C42—28no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—216no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—432no—
197C2×C42—48no—
198C2×C42—48no—
199C2×C42C22⋊C414yes—
200C2×C42C22⋊C414yes—
201C2×C42C22×C414yes—
202C2×C42C2×D414yes—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—44no—
206#95—22no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—28no—
212#95—24no—
213#95—22no—
214#95C2×D418yes—
215C24—22no—
216C24—22no—
217C24—24no—
218C24—24no—
219C24—21no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24#12012yes—
223C24#12312yes—
224C24C22×D411yescharacteristic
225#41—88no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—44no—
238C22×C4—28no—
239C22×C4—216no—
240C22×C4—216no—
241C22×C4—216no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—216no—
261C22×C4—232no—
262C22×C4—28no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—416no—
275C22×C4—464no—
276C22×C4—48no—
277C22×C4—416no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4#10418yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C4×D414yes—
286C22×C4C4×D414yes—
287C22×C4#12012yes—
288C22×C4#12114yes—
289C22×C4#12114yes—
290C22×C4#12312yes—
291C22×C4AGL(1,ℤ/8)14yes—
292C22×C4#13714yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C2×#4112yes—
297M16—88no—
298C2×C8—48no—
299C2×C8—44no—
300C4⋊C4—264no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C42—28no—
321C42—28no—
322C42—28no—
323C42—28no—
324C42—432no—
325C42—44no—
326C42—42no—
327C23—216no—
328C23—216no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—24no—
339C23—24no—
340C23—22no—
341C23—22no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23#38612yes—
347C23#38612yes—
348C23#41014yes—
349C23C2×#10414yes—
350C23C22×C22⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23C2×AGL(1,ℤ/8)12yes—
354C23C2×#13712yes—
355Q8—82no—
356D4—82no—
357D4—82no—
358D4—84no—
359D4—84no—
360D4—81no—
361D4—81no—
362C2×C4—28no—
363C2×C4—216no—
364C2×C4—216no—
365C2×C4—28no—
366C2×C4—216no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—216no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—216no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—464no—
390C2×C4—432no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—84no—
403C2×C4#38612yes—
404C2×C4#38612yes—
405C2×C4#41718yes—
406C2×C4C2×#10418yes—
407C2×C4C2×C4×D412yes—
408C2×C4C2×#12112yes—
409C8—84no—
410C22—28no—
411C22—216no—
412C22—28no—
413C22—24no—
414C22—24no—
415C22—28no—
416C22—28no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—84no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#189614yes—
430C22#189914yes—
431C22C2×#38611yescharacteristic
432C22C2×#41012yes—
433C22C2×#41712yes—
434C22C22×#10412yes—
435C22#300311yescharacteristic
436C22#300411yescharacteristic
437C22#309212yes—
438C4—28no—
439C4—28no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—416no—
445C4—84no—
446C4—81no—
447C4—81no—
448C4—81no—
449C2—21no—
450C2—22no—
451C2—24no—
452C2—24no—
453C2—21no—
454C2—161no—
455C2#1754614yes—
456C2#2090911yescharacteristic
457C2C2×#189611yescharacteristic
458C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
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
2. C22×C4×C4⋊C4C2semidirect
3. #20682C2semidirect
5. #9148C2semidirect
6. #9144C2semidirect
7. #8020C2semidirect
13. #2855C4semidirect
15. #2854C4semidirect
22. #2297C4semidirect
23. #2296C4semidirect
74. #391C2×C4semidirect
75. #390C2×C4semidirect
76. #389C2×C4semidirect
77. #389C2×C4semidirect

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
#206822
C2×#15042
#91482
#91442
#80202