Finite Group Databasebeta

← #190781

Subgroups — #190781

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8726C212yesmaximal
5#8591C212yesmaximal, complemented
6#8586C212yesmaximal, complemented
7#8340C212yesmaximal
8#8236C212yesmaximal, complemented
9#8233C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#433—21no—
12#2369C414yes—
13#2365—22no—
14#2345C414yes—
15Q8×C22⋊C4—22no—
16#2331—21no—
17#2331—21no—
18#2316—21no—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2212yes—
21#2285C2211yescharacteristic
22#2274—21no—
23C4×#117C2211yescharacteristic
24#1924—22no—
25#1917—22no—
26#1842C2212yes—
27#1842C2211yescharacteristic
28#1832—21no—
29#1831C2211yescharacteristic
30#1798—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1768—21no—
34#1767C2211yescharacteristic
35#1767C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C23×Q8—21no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×#117—21no—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C412yes—
59C22×C42C2311yescharacteristic
60#433—44no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—21no—
71C2×#97—22no—
72C2×C4×C8—21no—
73#391—22no—
74#391—22no—
75#391—24no—
76#391—24no—
77#389—24no—
78#389—22no—
79#389—22no—
80#389C2×C414yes—
81#389C2×C414yes—
82#387—24no—
83#387—24no—
84#384C2312yes—
85#382—21no—
86#380—21no—
87#380—24no—
88#380C2×C412yes—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×C414yes—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101#376D411yescharacteristic
102#376D411yescharacteristic
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—21no—
108#336—21no—
109#336—22no—
110#336—22no—
111#332—24no—
112#332—24no—
113#332—24no—
114#332—24no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—82no—
120C22×Q8—82no—
121C23×C4—22no—
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—24no—
130C4⋊Q8—24no—
131C4⋊Q8—28no—
132C4⋊Q8—28no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8#4114yes—
136C4⋊Q8#4114yes—
137C4×Q8—82no—
138C4×Q8—82no—
139C4×Q8—82no—
140C4×Q8—82no—
141#117—22no—
142#117—22no—
143#117—24no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C22×C414yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C412yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2411yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—44no—
188C4⋊C8—44no—
189C4⋊C8—82no—
190C22⋊C8—44no—
191C22⋊C8—82no—
192#97—42no—
193#97—82no—
194C4×C8—42no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205C24C2×C22⋊C411yescharacteristic
206C2×Q8—28no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—84no—
214C2×Q8—88no—
215C2×Q8—84no—
216C2×Q8—84no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—84no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C412yes—
236C22×C4AGL(1,ℤ/8)12yes—
237C22×C4#13712yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4#11718yes—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—44no—
271C42—22no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—21no—
276C42—21no—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42—42no—
286C42—44no—
287C42—84no—
288C42—84no—
289C42#13714yes—
290C42C2×#4112yes—
291C42C2×#4112yes—
292C23—22no—
293C23—21no—
294C23—22no—
295C23—21no—
296C23#41712yes—
297C23C22×C22⋊C411yescharacteristic
298Q8—44no—
299Q8—44no—
300Q8—44no—
301Q8—44no—
302Q8—88no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—28no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—88no—
334C2×C4#41712yes—
335C2×C4#41914yes—
336C2×C4C2×#11712yes—
337C2×C4#52512yes—
338C2×C4#52511yescharacteristic
339C2×C4#52511yescharacteristic
340C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
341C2×C4C2×#13712yes—
342C2×C4C2×#13711yescharacteristic
343C2×C4#57612yes—
344C2×C4#57712yes—
345C8—82no—
346C8—82no—
347C8—82no—
348C8—82no—
349C22—22no—
350C22—24no—
351C22—41no—
352C22—42no—
353C22—41no—
354C22#228011yescharacteristic
355C22#288912yes—
356C22C2×#41911yescharacteristic
357C22C2×#41711yescharacteristic
358C22#289712yes—
359C4—22no—
360C4—24no—
361C4—22no—
362C4—41no—
363C4—48no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—41no—
368C4—84no—
369C4#311212yes—
370C4#311312yes—
371C4#311812yes—
372C4#312211yescharacteristic
373C4#312311yescharacteristic
374C2—42no—
375C2#1776112yes—
376C2#1776512yes—
377C2#2073011yescharacteristic
378C2#2074011yescharacteristic
379C2#2074411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8591C2semidirect
6. #8586C2semidirect
8. #8236C2semidirect
9. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#342162
#219262
#208122
#87262
#85912
#85862
#83402
#82362
#82332