Finite Group Databasebeta

← #238540

Subgroups — #238540

Order 512.

1450 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
1#35166C211yesmaximal, characteristic
2#21859C212yesmaximal, complemented
3C2×#1502C212yesmaximal, complemented
4#13911C211yesmaximal, characteristic, complemented
5#10403C218yesmaximal, complemented
6#7908C211yesmaximal, characteristic, complemented
7C2×#537—21no—
8C2×#537—21no—
9C2×#532C2212yes—
10C2×#431C2212yes—
11#2856—22no—
12#2656C2214yes—
13#2542—22no—
14#2541—22no—
15#2540—24no—
16C2×#384C2211yescharacteristic
17C2×#382—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1990—22no—
21#1957—22no—
22#1874C2218yes—
23#1802—24no—
24#1502—28no—
25C2×#327C2218yes—
26#1442C2214yes—
27C4⋊#95—22no—
28C4⋊#95—21no—
29#1280C2214yes—
30C22×#41—22no—
31#537—44no—
32#537—44no—
33#537—44no—
34#537—44no—
35#532—24no—
36#532—24no—
37#532—24no—
38#532—24no—
39C2×C4⋊Q8D412yes—
40C2×#127—22no—
41C2×#127D412yes—
42C2×#124—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×#121—24no—
46C2×#121—24no—
47C2×#121—22no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121C2314yes—
51C2×C4×D4—22no—
52C2×C4×D4D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62#431—24no—
63#431—24no—
64C2×C4⋊C8—22no—
65C2×#102—28no—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#399—28no—
69#392—28no—
70#390—24no—
71#390—24no—
72#390—24no—
73#384—22no—
74#384—22no—
75#384—22no—
76#382—44no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4D414yescomplemented
81C2×#95—24no—
82C2×#95—24no—
83C2×#95C2312yes—
84C2×#95C2312yes—
85#340—28no—
86#327—216no—
87C2×#41—44no—
88C2×#41—44no—
89C2×#41—44no—
90C2×#41—44no—
91C2×#41—44no—
92C2×#41—44no—
93C22×Q8—22no—
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4C2×D414yes—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111#127—24no—
112#127—24no—
113#127—84no—
114#124—44no—
115#124—44no—
116C22⋊Q8—416no—
117C22⋊Q8—416no—
118#121—24no—
119#121—24no—
120#121—24no—
121#121—24no—
122#121—416no—
123#121—416no—
124#121—48no—
125#121—48no—
126#121—44no—
127#121—44no—
128#121—48no—
129#121—48no—
130C4×D4—28no—
131C4×D4—28no—
132C4×D4—44no—
133C4×D4—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4D814yes—
153C2×C4⋊C4SD1614yes—
154C2×C4⋊C4C2×D414yes—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—44no—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42#4114yes—
169C2×C42#4112yes—
170C4⋊C8—44no—
171#102—88no—
172C22⋊C8—24no—
173C22⋊C8—24no—
174C4×C8—44no—
175#95—24no—
176#95—22no—
177#95—48no—
178#95—84no—
179#95C2×D414yes—
180C24—24no—
181C24C22×D411yescharacteristic
182#41—88no—
183#41—88no—
184#41—88no—
185#41—88no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C2×D4—48no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—84no—
208C2×D4—84no—
209C2×D4—84no—
210C2×D4—84no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4#12011yescharacteristic
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12112yes—
243C22×C4#12112yes—
244C22×C4#12412yes—
245C22×C4#12412yes—
246C22×C4C2×D814yes—
247C22×C4C2×D812yes—
248C22×C4C2×SD1614yes—
249C22×C4C2×SD1612yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4C22×D412yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4111yescharacteristic
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—48no—
267C4⋊C4—44no—
268C4⋊C4—82no—
269C4⋊C4—84no—
270C4⋊C4—82no—
271C22⋊C4—416no—
272C22⋊C4—416no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—48no—
276C42—44no—
277C42—48no—
278C42—82no—
279C42#12114yes—
280C42#12114yes—
281C42#12314yes—
282C42#12314yes—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23C2×#12011yescharacteristic
295C23C2×#12111yescharacteristic
296C23C2×#12411yescharacteristic
297C23C22×D811yescharacteristic
298C23C22×SD1611yescharacteristic
299C23C2×AGL(1,ℤ/8)11yescharacteristic
300C23C2×AGL(1,ℤ/8)11yescharacteristic
301Q8—88no—
302D4—48no—
303D4—84no—
304D4—84no—
305D4—84no—
306D4—84no—
307D4—84no—
308D4—88no—
309D4—88no—
310D4—168no—
311D4—168no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—816no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#39412yes—
348C2×C4#39412yes—
349C2×C4#39414yes—
350C2×C4#44714yes—
351C2×C4#45014yes—
352C2×C4#45412yes—
353C2×C4#45512yes—
354C2×C4#45914yes—
355C2×C4#46014yes—
356C2×C4#48614yes—
357C2×C4#49014yes—
358C2×C4C2×#12112yes—
359C2×C4C2×#12312yes—
360C8—84no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22—84no—
378C22—84no—
379C22—84no—
380C22C2×#39411yescharacteristic
381C22C2×#44711yescharacteristic
382C22C2×#45011yescharacteristic
383C22C2×#45511yescharacteristic
384C22C2×#45411yescharacteristic
385C22#302711yescharacteristic
386C22#302811yescharacteristic
387C22C2×#45911yescharacteristic
388C22C2×#46011yescharacteristic
389C22C2×#48611yescharacteristic
390C22C2×#49011yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—44no—
396C4—84no—
397C4—84no—
398C4—162no—
399C4#204014yes—
400C4#204118yes—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—164no—
406C2#1838811yescharacteristic
407C2#1839411yescharacteristic
408C2#1840111yescharacteristic
409C2#1840911yescharacteristic
410C2C2×#204011yescharacteristic
411C2C2×#204111yescharacteristic
412C2#2351611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35166C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21859C2semidirect
3. C2×#1502C2semidirect
4. #13911C2semidirect
5. #10403C2semidirect
6. #7908C2semidirect
80. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351662
#218592
C2×#15022
#139112
#104032
#79082