Finite Group Databasebeta

← #350540

Subgroups — #350540

Order 512.

1135 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2#33148C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic
5#8975C214yesmaximal, complemented
6#8896C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8706C211yesmaximal, characteristic
9#8311C211yesmaximal, characteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#477C2212yes—
13#2857C2211yescharacteristic
14Q8×C4⋊C4—22no—
15Q8×C4⋊C4—22no—
16Q8×C4⋊C4—22no—
17Q8×C4⋊C4C2214yes—
18C2×#389—22no—
19C2×#389—21no—
20C2×#384—21no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#2072—22no—
25#1987C2212yes—
26#1956C2212yes—
27#1948C2212yes—
28#1920—22no—
29#1872—22no—
30#1863—22no—
31#1853—21no—
32#1852—21no—
33#1834C2212yes—
34#1823—21no—
35#1803C2212yes—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1787—21no—
39C4×C22⋊C8C2211yescharacteristic
40C2×Q8⋊C8C2214yes—
41#1296C2214yes—
42#1279C2214yes—
43C23×Q8D411yescharacteristic
44C2×C4⋊Q8C2311yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C4×Q8—28no—
47C2×C4×Q8—24no—
48C2×C4×Q8—24no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—44no—
52C2×C4×Q8—44no—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C42—21no—
65C22×C42—22no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#477—28no—
69C2×#107—22no—
70C2×#106—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8C2312yes—
80#391—21no—
81#389—24no—
82#389—28no—
83#389—24no—
84#389—28no—
85#389—21no—
86#389—21no—
87#384—24no—
88#384—24no—
89#384—21no—
90#384—21no—
91#384D412yes—
92#384C2311yescharacteristic
93#384C2312yes—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4C2312yes—
102C2×#95—22no—
103C2×#95—21no—
104#340—22no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112Q8⋊C8—28no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8SD1614yes—
119C22×Q8C2×D412yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—24no—
132C22⋊Q8—44no—
133C4×Q8—28no—
134C4×Q8—28no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—48no—
139C4×Q8—416no—
140C4×Q8—48no—
141C4×Q8#4118yes—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4Q1614yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D414yes—
175C2×C22⋊C4—41no—
176C2×C42—28no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—28no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—48no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2411yescharacteristic
190#107—44no—
191#106—44no—
192C4⋊C8—24no—
193C4⋊C8—24no—
194#103—44no—
195#103—44no—
196#103—84no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200C4×C8—24no—
201#95—28no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206C24#12011yescharacteristic
207C2×Q8—24no—
208C2×Q8—216no—
209C2×Q8—24no—
210C2×Q8—216no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8#12314yes—
219C2×Q8C2×SD1614yes—
220C2×Q8#13714yes—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4C2×SD1612yes—
254C22×C4C2×Q1612yes—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4#13712yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—28no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—416no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—88no—
293C4⋊C4#12314yes—
294C4⋊C4C2×Q1614yes—
295C22⋊C4—82no—
296C42—28no—
297C42—22no—
298C42—28no—
299C42—22no—
300C42—22no—
301C42—416no—
302C42—42no—
303C42—42no—
304C42—44no—
305C42C2×#4114yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#45012yes—
311C23#45112yes—
312C23#45412yes—
313C23C2×#12011yescharacteristic
314Q8—216no—
315Q8—24no—
316Q8—24no—
317Q8—24no—
318Q8—84no—
319Q8#48318yes—
320C2×C4—216no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4#45012yes—
361C2×C4#45112yes—
362C2×C4C2×#12312yes—
363C2×C4#52512yes—
364C2×C4#52512yes—
365C2×C4C22×SD1611yescharacteristic
366C2×C4C22×Q1611yescharacteristic
367C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C2×#13712yes—
369C2×C4C2×#13711yescharacteristic
370C2×C4#57512yes—
371C8—82no—
372C8—82no—
373C8—82no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22#163714yes—
381C22#284911yescharacteristic
382C22C2×#45011yescharacteristic
383C22C2×#45111yescharacteristic
384C22#300411yescharacteristic
385C22#300511yescharacteristic
386C22C2×#45411yescharacteristic
387C22#301911yescharacteristic
388C4—28no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—44no—
393C4—41no—
394C4—44no—
395C4—82no—
396C4—84no—
397C4—161no—
398C4—161no—
399C4C2×#48312yes—
400C4#309212yes—
401C4#310412yes—
402C4#311712yes—
403C2—24no—
404C2C2×#163711yescharacteristic
405C2#1853511yescharacteristic
406C2#1876711yescharacteristic
407C2#1876911yescharacteristic
408C2#3298211yescharacteristic
409C2#3299711yescharacteristic
410C2#3301011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8975C2semidirect
6. #8896C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
#331482
#329572
#279722
#89752
#88962
#87242
#87062
#83112