Finite Group Databasebeta

← #190697

Subgroups — #190697

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#8725C212yesmaximal
5#8593C214yesmaximal, complemented
6#8340C212yesmaximal
7#8236C214yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8—21no—
10C4×C4⋊Q8—21no—
11C4×C4⋊Q8—21no—
12C4×C4⋊Q8—21no—
13C4×C4⋊Q8—21no—
14C4×C4⋊Q8C412yes—
15C4×C4⋊Q8C412yes—
16C4×C4⋊Q8C2212yes—
17C4×C4⋊Q8C2212yes—
18C2×#391—21no—
19C2×#389—21no—
20C2×#389C412yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4C412yes—
24C2×C43C2211yescharacteristic
25#1839—21no—
26#1839—21no—
27#1831C2212yes—
28#1831C2212yes—
29#1817—21no—
30#1817—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1767C2212yes—
34C4×C4⋊C8C2212yes—
35#1749—21no—
36C4×C22⋊C8—21no—
37#1745—21no—
38#1744—21no—
39C2×#332C2218yes—
40#1296C2218yes—
41C23×Q8—21no—
42C2×C4⋊Q8—28no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—48no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C42—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8C2314yes—
67C2×C4⋊C8C2314yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#389C2×C414yes—
84#384—21no—
85#384—21no—
86#384C2×C412yes—
87#384C2312yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C414yes—
96C2×#95—22no—
97C2×#95—22no—
98C43C2×C411yescharacteristic
99C43C2×C411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43D411yescharacteristic
103C43D411yescharacteristic
104C43C2311yescharacteristic
105C43C2311yescharacteristic
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#332—216no—
115C22×Q8—28no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—44no—
119C22×Q8—44no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—24no—
130C22×C8—24no—
131C4⋊Q8—28no—
132C4⋊Q8—28no—
133C4⋊Q8—28no—
134C4⋊Q8—24no—
135C4⋊Q8#4118yes—
136C4×Q8—48no—
137C4×Q8—48no—
138C4×Q8—48no—
139C4×Q8—48no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—48no—
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2411yescharacteristic
196C4⋊C8—28no—
197C4⋊C8—28no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200#97—44no—
201C4×C8—44no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206C24C2×C22⋊C411yescharacteristic
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—432no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#13514yes—
253C22×C4#13714yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—24no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4#11718yes—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—48no—
302C42—48no—
303C42#10114yes—
304C42AGL(1,ℤ/8)14yes—
305C42C2×#4114yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#41312yes—
311C23#41612yes—
312C23#41912yes—
313C23C22×C22⋊C411yescharacteristic
314Q8—416no—
315Q8—416no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—28no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—88no—
359C2×C4C2×#10112yes—
360C2×C4C2×#10112yes—
361C2×C4C2×#10112yes—
362C2×C4#41612yes—
363C2×C4#41812yes—
364C2×C4#41812yes—
365C2×C4#41912yes—
366C2×C4C2×#11712yes—
367C2×C4#52512yes—
368C2×C4#52512yes—
369C2×C4C2×#13512yes—
370C2×C4C2×AGL(1,ℤ/8)12yes—
371C2×C4C2×#13712yes—
372C2×C4#57512yes—
373C8—84no—
374C8—84no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22#155414yes—
382C22#228011yescharacteristic
383C22C22×#10111yescharacteristic
384C22C2×#41311yescharacteristic
385C22C2×#41611yescharacteristic
386C22C2×#41811yescharacteristic
387C22C2×#41911yescharacteristic
388C22#289611yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—48no—
402C4—84no—
403C4#310514yes—
404C4#311714yes—
405C2—24no—
406C2C2×#155411yescharacteristic
407C2#1771511yescharacteristic
408C2#1774211yescharacteristic
409C2#1774411yescharacteristic
410C2#2072711yescharacteristic
411C2#2073311yescharacteristic
412C2#2074111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8593C2semidirect
7. #8236C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#207782
#207772
#87252
#85932
#83402
#82362