Finite Group Databasebeta

← #350568

Subgroups — #350568

Order 512.

1161 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2#33146C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#27950C211yesmaximal, characteristic
5#8987C214yesmaximal, complemented
6#8884C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8706C211yesmaximal, characteristic
9#8311C211yesmaximal, characteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#462C2212yes—
13C2×#451—21no—
14#2857C2211yescharacteristic
15Q8×C4⋊C4—22no—
16Q8×C4⋊C4—22no—
17Q8×C4⋊C4—22no—
18Q8×C4⋊C4C2214yes—
19C2×#389—22no—
20C2×#389—21no—
21C2×#384—21no—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#2083—22no—
26#1990C2212yes—
27#1988C2212yes—
28#1947C2212yes—
29#1918—22no—
30#1878—22no—
31#1872—21no—
32#1852—22no—
33#1852—21no—
34#1845C2212yes—
35#1823—21no—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C2×Q8⋊C8C2214yes—
41#1296C2214yes—
42#1279C2214yes—
43C23×Q8D411yescharacteristic
44C22×Q16—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C22⋊Q8—21no—
47C2×C4×Q8—28no—
48C2×C4×Q8—24no—
49C2×C4×Q8—24no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—44no—
53C2×C4×Q8—44no—
54C2×C4×Q8D412yes—
55C2×C4×Q8C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C42—21no—
66C22×C42—22no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#462—28no—
70#451—44no—
71C2×#107—22no—
72C2×#107—24no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8C2312yes—
81#391—21no—
82#389—24no—
83#389—28no—
84#389—24no—
85#389—28no—
86#389—21no—
87#389—21no—
88#384—24no—
89#384—24no—
90#384—21no—
91#384—21no—
92#384D412yes—
93#384C2311yescharacteristic
94#384C2312yes—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4C2312yes—
103C2×#95—22no—
104C2×#95—21no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—22no—
110#336—21no—
111#336—21no—
112#336—21no—
113Q8⋊C8—28no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8D814yes—
120C22×Q8C2×D412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×Q16—44no—
130C2×Q16—44no—
131C2×Q16—44no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—24no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C4×Q8—28no—
139C4×Q8—28no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—48no—
144C4×Q8—416no—
145C4×Q8—48no—
146C4×Q8#4118yes—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4Q1614yescomplemented
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D414yes—
180C2×C22⋊C4—41no—
181C2×C42—28no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—48no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195#107—44no—
196#107—84no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199#103—44no—
200#103—44no—
201C22⋊C8—24no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204C4×C8—24no—
205#95—28no—
206#95—22no—
207#95—22no—
208#95—41no—
209#95—41no—
210C24#12011yescharacteristic
211C2×Q8—24no—
212C2×Q8—216no—
213C2×Q8—24no—
214C2×Q8—216no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8#12314yes—
224C2×Q8C2×D814yes—
225C2×Q8#13714yes—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4C2×D812yes—
259C22×C4C2×Q1612yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4#13712yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—88no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—28no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—416no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—82no—
298C4⋊C4—88no—
299C4⋊C4—82no—
300C4⋊C4#12314yes—
301C4⋊C4C2×Q1614yes—
302C22⋊C4—82no—
303C42—28no—
304C42—22no—
305C42—28no—
306C42—22no—
307C42—22no—
308C42—416no—
309C42—42no—
310C42—42no—
311C42—44no—
312C42C2×#4114yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23#44712yes—
318C23#45112yes—
319C23#45612yes—
320C23C2×#12011yescharacteristic
321Q8—216no—
322Q8—24no—
323Q8—24no—
324Q8—24no—
325Q8—82no—
326Q8—82no—
327Q8#48018yes—
328C2×C4—216no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4—88no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#44712yes—
369C2×C4#45112yes—
370C2×C4C2×#12312yes—
371C2×C4#52512yes—
372C2×C4#52512yes—
373C2×C4C22×D811yescharacteristic
374C2×C4C22×Q1611yescharacteristic
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C2×#13712yes—
377C2×C4C2×#13711yescharacteristic
378C2×C4#57512yes—
379C8—44no—
380C8—82no—
381C8—82no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22#164914yes—
389C22#284911yescharacteristic
390C22C2×#44711yescharacteristic
391C22C2×#45111yescharacteristic
392C22#300411yescharacteristic
393C22#300511yescharacteristic
394C22C2×#45611yescharacteristic
395C22#301911yescharacteristic
396C4—28no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—44no—
401C4—41no—
402C4—44no—
403C4—82no—
404C4—84no—
405C4—161no—
406C4—161no—
407C4C2×#48012yes—
408C4#309212yes—
409C4#310412yes—
410C4#311712yes—
411C2—24no—
412C2C2×#164911yescharacteristic
413C2#1854111yescharacteristic
414C2#1876611yescharacteristic
415C2#1876911yescharacteristic
416C2#3297811yescharacteristic
417C2#3300011yescharacteristic
418C2#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. #8987C2semidirect
6. #8884C2semidirect
175. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
#331462
#329062
#279502
#89872
#88842
#87242
#87062
#83112