Finite Group Databasebeta

← #220084

Subgroups — #220084

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36149C211yesmaximal, characteristic
2#22648C212yesmaximal, complemented
3C2×#1547C212yesmaximal, complemented
4#9708C212yesmaximal, complemented
5#9137C212yesmaximal, complemented
6#9131C212yesmaximal, complemented
7#8585C211yesmaximal, characteristic, complemented
8#8341C212yesmaximal, complemented
9#8231C211yesmaximal, characteristic, complemented
10C2×#532C2212yes—
11C2×#435—22no—
12C2×#423C2212yes—
13C22×C4⋊C8C2212yes—
14#2862C2211yescharacteristic
15#2857C2212yes—
16#2535C412yes—
17#2447—22no—
18#2427—22no—
19#2420C414yes—
20#2407—22no—
21#2349C412yes—
22#2339—22no—
23C4×#127C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1940—22no—
26#1851—22no—
27#1850—22no—
28#1842C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1767C2212yes—
32#1547—28no—
33C2×#332C2212yes—
34C2×#332C2212yes—
35C2×#332C2212yes—
36C2×#332C2212yes—
37C2×#331C2211yescharacteristic
38C2×#331C2212yes—
39C2×#331C2211yescharacteristic
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C22×#41—22no—
44C22×M16—22no—
45C23×C8—22no—
46#532—24no—
47#532—24no—
48#532—42no—
49#532—42no—
50C2×C4⋊Q8C2312yes—
51C2×#127C2311yescharacteristic
52C2×#124—22no—
53C2×C22⋊Q8—22no—
54C2×#121—22no—
55C2×#121C2×C414yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C414yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C22×C42C2312yes—
63#435—48no—
64#423—24no—
65#423—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79#397—22no—
80#397—24no—
81#394—24no—
82#392—22no—
83#391—22no—
84#391—22no—
85#391—22no—
86#390—22no—
87#390C2×C412yes—
88#389—22no—
89#389C2×C414yes—
90#386—24no—
91#386—24no—
92#385—24no—
93#384C2311yescharacteristic
94#382C2×C412yes—
95#382C2×C412yes—
96#381—24no—
97#380—22no—
98#380—22no—
99#380C2×C414yes—
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C4⋊C4D412yes—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4D412yes—
104C4×C22⋊C4D412yes—
105C2×#95—22no—
106C43C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#332—24no—
111#332—24no—
112#332—42no—
113#332—42no—
114#331—24no—
115#331—22no—
116#331D414yescomplemented
117C2×#41—44no—
118C2×#41—44no—
119C2×#41—44no—
120C2×#41—44no—
121C2×#41—82no—
122C2×#41—82no—
123C22×Q8—22no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22×C412yes—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C2×M16—44no—
134C2×M16—44no—
135C2×M16—82no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—82no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147#127—22no—
148#127C2×D412yes—
149#127#4112yes—
150#124—44no—
151#124—44no—
152C22⋊Q8—82no—
153C22⋊Q8—82no—
154#121—28no—
155#121—44no—
156#121—44no—
157#121—44no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—28no—
161C4×D4—42no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—82no—
167C4×D4—82no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—44no—
203C4⋊C8—44no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8C2×D414yes—
207C22⋊C8—44no—
208C22⋊C8—44no—
209#97—44no—
210#95—24no—
211#95—22no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95C22⋊C411yescharacteristic
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C411yescharacteristic
219C24—22no—
220C24C2×C22⋊C412yes—
221#41—88no—
222#41—88no—
223#41—88no—
224#41—88no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—84no—
245C2×D4C4×D414yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4AGL(1,ℤ/8)14yes—
274C22×C4#13714yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D412yes—
277M16—88no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—88no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—84no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—48no—
305C22⋊C4—48no—
306C22⋊C4—44no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42C22×D411yescharacteristic
322C42C2×#4111yescharacteristic
323C42H(5,𝔽2)12yes—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—44no—
335C23#41714yes—
336C23C22×C22⋊C411yescharacteristic
337Q8—82no—
338Q8—82no—
339Q8—82no—
340Q8—82no—
341D4—44no—
342D4—44no—
343D4—44no—
344D4—44no—
345D4—44no—
346D4—44no—
347D4—44no—
348D4—816no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—816no—
381C2×C4—164no—
382C2×C4#41312yes—
383C2×C4#41714yes—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4#51811yescharacteristic
386C2×C4D4212yes—
387C2×C4#54612yes—
388C2×C4C2×AGL(1,ℤ/8)12yes—
389C2×C4C2×#13712yes—
390C2×C4#57612yes—
391C2×C4#57712yes—
392C8—44no—
393C8—82no—
394C8—82no—
395C8—82no—
396C22—24no—
397C22—24no—
398C22—24no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—44no—
404C22—48no—
405C22—82no—
406C22—84no—
407C22—82no—
408C22D4×C22⋊C411yescharacteristic
409C22C2×#41311yescharacteristic
410C22#288911yescharacteristic
411C22C2×#41712yes—
412C22#289712yes—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—24no—
417C4—41no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—82no—
422C4—84no—
423C4#326412yes—
424C4#326612yes—
425C4#330612yes—
426C4#331412yes—
427C2—44no—
428C2—84no—
429C2#1775211yescharacteristic
430C2#1776512yes—
431C2#1777111yescharacteristic
432C2#2241111yescharacteristic
433C2#2241912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36149C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22648C2semidirect
3. C2×#1547C2semidirect
4. #9708C2semidirect
5. #9137C2semidirect
6. #9131C2semidirect
7. #8585C2semidirect
8. #8341C2semidirect
9. #8231C2semidirect
116. #331D4semidirect

Maximal subgroups

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

subgroupindex
#361492
#226482
C2×#15472
#97082
#91372
#91312
#85852
#83412
#82312