Finite Group Databasebeta

← #302366

Subgroups — #302366

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#33310C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic, complemented
5#10580C214yesmaximal, complemented
6#9896C211yesmaximal, characteristic, complemented
7#9870C211yesmaximal, characteristic, complemented
8#8886C214yesmaximal
9C4×D4⋊C8C211yesmaximal, characteristic
10C2×#554—21no—
11C2×#498—21no—
12C2×#498—22no—
13C2×#474—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2857—21no—
17#2567—21no—
18#2552—21no—
19#2552C2212yes—
20#2502—22no—
21#2440—22no—
22#2440—22no—
23D4×C4⋊C4—21no—
24D4×C4⋊C4C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26C2×#384—21no—
27C42×D4C2211yescharacteristic
28#2084—22no—
29#1990—21no—
30#1988—21no—
31#1948C2214yes—
32#1947—21no—
33#1947C2212yes—
34#1844—22no—
35#1844C2214yes—
36#1842C2211yescharacteristic
37#1802—21no—
38#1802C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45#554—44no—
46#554—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D412yes—
54C2×#125—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—24no—
57C2×C4×Q8—21no—
58C2×C4×D4—22no—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C42—21no—
66#498—48no—
67#498—44no—
68#474—44no—
69C2×#107—22no—
70C2×#106—24no—
71C2×#106—28no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#102—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#398—24no—
82#398—24no—
83#395—24no—
84#391—22no—
85#391—21no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—21no—
91#389D412yes—
92#385—24no—
93#385—24no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D412yes—
104#384C2312yes—
105#384C2312yes—
106#380—22no—
107#380—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2311yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C43C2311yescharacteristic
117#340—22no—
118#340—22no—
119#340—24no—
120#336—21no—
121D4⋊C8Q818yescomplemented
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×D4C2×D411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—24no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—82no—
144#125—44no—
145#125—44no—
146C22⋊Q8—416no—
147C22⋊Q8—48no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4SD1614yes—
156C4×D4SD1614yes—
157C4×D4C2×Q814yes—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203#107—44no—
204#106—48no—
205#106—416no—
206C4⋊C8—22no—
207C4⋊C8—24no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210C4⋊C8C2×Q814yes—
211#102—44no—
212C4×C8—24no—
213C4×C8—24no—
214C4×C8—22no—
215C4×C8—42no—
216C4×C8C2×Q814yes—
217#95—24no—
218#95—24no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—21no—
223#95—22no—
224C24—21no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4C4⋊Q812yes—
238C2×D4C2×SD1612yes—
239C2×D4C2×SD1612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4C4⋊Q812yes—
306C4⋊C4C2×SD1612yes—
307C4⋊C4C2×SD1612yes—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—48no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—42no—
316C42—41no—
317C42—41no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42C2×SD1612yes—
322C42C2×SD1612yes—
323C42#13512yes—
324C42AGL(1,ℤ/8)12yes—
325C42AGL(1,ℤ/8)12yes—
326C42#13712yes—
327C42C22×Q812yes—
328C422−1+412yes—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23C2×#12011yescharacteristic
337Q8—84no—
338Q8—164no—
339Q8—164no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4#49814yescomplemented
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—168no—
376C2×C4#44812yes—
377C2×C4#44812yes—
378C2×C4#44912yes—
379C2×C4#44912yes—
380C2×C4#45012yes—
381C2×C4#45012yes—
382C2×C4#45312yes—
383C2×C4#45612yes—
384C2×C4C2×C4⋊Q811yescharacteristic
385C2×C4#53611yescharacteristic
386C2×C4D4×Q812yes—
387C2×C4D4×Q812yes—
388C2×C4C22×SD1611yescharacteristic
389C2×C4C22×SD1611yescharacteristic
390C2×C4C2×#13511yescharacteristic
391C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4#57712yes—
395C8—28no—
396C8—44no—
397C8—82no—
398C22—24no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22#243411yescharacteristic
406C22C2×#45011yescharacteristic
407C22C2×#44811yescharacteristic
408C22C2×#44911yescharacteristic
409C22#300911yescharacteristic
410C22C2×#45311yescharacteristic
411C22C2×#45611yescharacteristic
412C4—22no—
413C4—22no—
414C4—22no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—82no—
420C4—84no—
421C4—162no—
422C4#162214yes—
423C4#165414yes—
424C4C2×#49811yescharacteristic
425C4#316511yescharacteristic
426C4#322711yescharacteristic
427C4#322911yescharacteristic
428C4#338112yes—
429C4#338412yes—
430C2—44no—
431C2C2×#162211yescharacteristic
432C2C2×#165411yescharacteristic
433C2#1862111yescharacteristic
434C2#1865211yescharacteristic
435C2#2581111yescharacteristic
436C2#2583011yescharacteristic
437C2#2583511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33310C2semidirect
4. #28687C2semidirect
5. #10580C2semidirect
6. #9896C2semidirect
7. #9870C2semidirect
121. D4⋊C8Q8semidirect
343. D4#498semidirect

Maximal subgroups

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

subgroupindex
#361802
#333102
#331952
#286872
#105802
#98962
#98702
#88862
C4×D4⋊C82