Finite Group Databasebeta

← #190737

Subgroups — #190737

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#8726C212yesmaximal
5#8593C212yesmaximal, complemented
6#8589C212yesmaximal, complemented
7#8340C212yesmaximal
8#8237C212yesmaximal, complemented
9C4×#332C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11#2369C414yes—
12#2365—22no—
13#2345C414yes—
14Q8×C22⋊C4—22no—
15#2331—21no—
16#2331—21no—
17#2316—21no—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2212yes—
20#2285C2211yescharacteristic
21#2274—21no—
22C4×#117C2211yescharacteristic
23#1923—22no—
24#1916—22no—
25#1842C2211yescharacteristic
26#1840—21no—
27#1831C2212yes—
28#1831C2211yescharacteristic
29#1818—21no—
30#1797C2212yes—
31#1797C2212yes—
32#1767C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×C4⋊C8C2212yes—
35#1750—21no—
36#1746—21no—
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
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—22no—
70#391—22no—
71#391—22no—
72#391—24no—
73#391—24no—
74#389—24no—
75#389—22no—
76#389—22no—
77#389C2×C414yes—
78#389C2×C414yes—
79#387—24no—
80#387—24no—
81#384C2312yes—
82#382—21no—
83#380—21no—
84#380—24no—
85#380C2×C412yes—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C414yes—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D412yes—
98#376D411yescharacteristic
99#376D411yescharacteristic
100C43C2311yescharacteristic
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—22no—
105#336—21no—
106#336—21no—
107#336—22no—
108#336—22no—
109#336—22no—
110#332—24no—
111#332—24no—
112#332—24no—
113#332—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—22no—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8#4114yes—
135C4⋊Q8#4114yes—
136C4×Q8—82no—
137C4×Q8—82no—
138C4×Q8—82no—
139C4×Q8—82no—
140#117—24no—
141#117—22no—
142#117—22no—
143#117—44no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C414yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2411yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—82no—
186C22⋊C8—44no—
187C22⋊C8—82no—
188#97—42no—
189C4×C8—42no—
190C4×C8—82no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201C24C2×C22⋊C411yescharacteristic
202C2×Q8—28no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—84no—
210C2×Q8—88no—
211C2×Q8—84no—
212C2×Q8—84no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—84no—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4#13514yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—84no—
261C4⋊C4—84no—
262C4⋊C4#11718yes—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—44no—
266C42—22no—
267C42—21no—
268C42—21no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—42no—
279C42—42no—
280C42—44no—
281C42—84no—
282C42—84no—
283C42#13712yes—
284C42#13712yes—
285C42C2×#4112yes—
286C42C2×#4112yes—
287C23—22no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23#41612yes—
292C23C22×C22⋊C411yescharacteristic
293Q8—44no—
294Q8—44no—
295Q8—44no—
296Q8—44no—
297Q8—88no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—28no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—88no—
329C2×C4#41612yes—
330C2×C4#41914yes—
331C2×C4C2×#11712yes—
332C2×C4#52512yes—
333C2×C4#52511yescharacteristic
334C2×C4#52511yescharacteristic
335C2×C4C2×#13512yes—
336C2×C4C2×#13711yescharacteristic
337C2×C4C2×#13711yescharacteristic
338C2×C4#57612yes—
339C2×C4#57712yes—
340C8—82no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—24no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22#228011yescharacteristic
350C22#288912yes—
351C22C2×#41611yescharacteristic
352C22C2×#41911yescharacteristic
353C22#289712yes—
354C4—24no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—48no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—41no—
363C4—84no—
364C4#310812yes—
365C4#310812yes—
366C4#311812yes—
367C4#312211yescharacteristic
368C4#312311yescharacteristic
369C2—42no—
370C2#1772712yes—
371C2#1776112yes—
372C2#2073011yescharacteristic
373C2#2073511yescharacteristic
374C2#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. #8593C2semidirect
6. #8589C2semidirect
8. #8237C2semidirect
9. C4×#332C2semidirect

Maximal subgroups

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

subgroupindex
#342162
#219222
#207862
#87262
#85932
#85892
#83402
#82372
C4×#3322