Finite Group Databasebeta

← #192206

Subgroups — #192206

Order 512.

1059 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2#21949C212yesmaximal, complemented
3#21740C211yesmaximal, characteristic
4#17362C211yesmaximal, characteristic
5#8860C218yesmaximal, complemented
6#8057C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8#2975—24no—
9#2975—22no—
10#2975C2214yes—
11C2×#436—24no—
12C2×#431—21no—
13C2×#407—21no—
14C2×#407C2212yes—
15C2×#407C2212yes—
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#1870C2214yes—
19#1869—22no—
20#1855—24no—
21#1855C2218yescomplemented
22#1817—21no—
23#1811—24no—
24#1779—21no—
25#1465C414yes—
26#1463C412yes—
27#1462C412yes—
28#1459C2214yes—
29#1298C2218yescomplemented
30C23×D4Q811yescharacteristic
31C24×C4D411yescharacteristic
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C22×M16—22no—
35C22×M16D414yes—
36C22×M16C2312yes—
37C23×C8—21no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4Q812yes—
45C2×C4×D4C2312yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4Q811yescharacteristic
49C22×C42D411yescharacteristic
50#436—216no—
51#436—216no—
52#431—24no—
53#431—24no—
54C2×#108—28no—
55C2×C4⋊C8—22no—
56#407—24no—
57#407—24no—
58#407—22no—
59#407—22no—
60#407—22no—
61#407—22no—
62#407—22no—
63#407—22no—
64#407C2×C414yes—
65#407C2×C414yes—
66#407D414yes—
67#407C2314yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2314yes—
80C2×C4×C8—22no—
81#341—28no—
82#336—24no—
83#324C2×C418yes—
84#323C2×C418yes—
85C25C4⋊C411yescharacteristic
86C25C2×Q811yescharacteristic
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—44no—
90C22×D4C4⋊C412yes—
91C22×D4C2×Q812yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×Q811yescharacteristic
110C23×C4#4112yes—
111C23×C4#4112yes—
112C23×C4C2411yescharacteristic
113C2×M16—28no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—82no—
119C2×M16—84no—
120C2×M16—84no—
121C2×M16—84no—
122C2×M16C2×D418yes—
123C22×C8—24no—
124C22×C8—24no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—42no—
128C22×C8—82no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4C4⋊C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4C4⋊C412yes—
146C2×C22⋊C4C4⋊C412yes—
147C2×C22⋊C4C22×C412yes—
148C2×C22⋊C4C2×Q812yes—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C4⋊C412yes—
154C2×C42C2×D412yes—
155#108—88no—
156C4⋊C8—28no—
157C22⋊C8—28no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C22⋊C8—41no—
167C22⋊C8—42no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—84no—
172C22⋊C8C22×C418yes—
173C4×C8—28no—
174C24—21no—
175C24—21no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—81no—
181C24—81no—
182C24C2×C4⋊C411yescharacteristic
183C24C2×C4⋊C412yes—
184C24C2×C4⋊C411yescharacteristic
185C24C22⋊Q812yes—
186C24C22⋊Q812yes—
187C24C22×Q811yescharacteristic
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—48no—
195C2×D4—48no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—81no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×C4⋊C412yes—
226C22×C4C2×C4⋊C412yes—
227C22×C4C2×C4⋊C412yes—
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4C4×D412yes—
231C22×C4C4×D412yes—
232C22×C4C4×D412yes—
233C22×C4C4×D412yes—
234C22×C4#12011yescharacteristic
235C22×C4#12012yes—
236C22×C4#12011yescharacteristic
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12312yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4112yes—
246M16—48no—
247M16—48no—
248M16—84no—
249M16—84no—
250M16—84no—
251M16—88no—
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—48no—
255C2×C8—48no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—84no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—81no—
296C23—81no—
297C23—88no—
298C23—88no—
299C23#38512yes—
300C23#38511yescharacteristic
301C23#38511yescharacteristic
302C23C22×C4⋊C411yescharacteristic
303C23C2×C22⋊Q812yes—
304D4—48no—
305D4—48no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38512yes—
343C2×C4#43014yes—
344C2×C4C2×C4×D412yes—
345C2×C4C2×#12011yescharacteristic
346C2×C4C2×#12311yescharacteristic
347C8—48no—
348C8—48no—
349C8—82no—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—84no—
372C22—82no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22#185812yes—
377C22C2×#38511yescharacteristic
378C22C2×#43012yes—
379C4—24no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—22no—
385C4—42no—
386C4—42no—
387C4#302612yes—
388C2—22no—
389C2—22no—
390C2—42no—
391C2—41no—
392C2—42no—
393C2—82no—
394C2#2085111yescharacteristic
395C2C2×#185811yescharacteristic
396C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21949C2semidirect
5. #8860C2semidirect
6. #8057C2semidirect
21. #1855C22semidirect
29. #1298C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#219492
#217402
#173622
#88602
#80572