Finite Group Databasebeta

← #302395

Subgroups — #302395

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic
5#10582C214yesmaximal
6#9898C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8896C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×#557—21no—
11C2×#498—21no—
12C2×#475—21no—
13#2868—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2552—22no—
17#2552—22no—
18#2552—21no—
19#2552C2212yes—
20#2546—21no—
21#2546—22no—
22Q8×C4⋊C4—21no—
23Q8×C4⋊C4C2212yes—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#2084—22no—
28#1990—22no—
29#1990—21no—
30#1990—22no—
31#1988—21no—
32#1948—21no—
33#1948C2214yes—
34#1948C2212yes—
35#1842C2211yescharacteristic
36#1834C2214yes—
37#1803—21no—
38#1803C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45#557—44no—
46#557—44no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8D411yescharacteristic
55C2×C4⋊Q8D412yes—
56C2×#125—24no—
57C2×#125—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64#498—44no—
65#475—44no—
66C2×#106—22no—
67C2×#106—24no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2312yes—
77#398—24no—
78#398—24no—
79#398—24no—
80#391—22no—
81#391—21no—
82#389—22no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—24no—
88#389—21no—
89#389D412yes—
90#384—22no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384D412yes—
101#384C2312yes—
102#384C2312yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4C2312yes—
112C4×C4⋊C4C2311yescharacteristic
113C43—21no—
114C43C2311yescharacteristic
115#340—22no—
116#340—22no—
117#340—24no—
118#340—28no—
119#336—21no—
120Q8⋊C8Q818yescomplemented
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8C2×D411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—42no—
141C4⋊Q8—84no—
142#125—416no—
143#125—44no—
144#125—44no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—48no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8SD1614yes—
154C4×Q8SD1614yes—
155C4×Q8C2×Q814yes—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D414yes—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200#106—48no—
201#106—44no—
202C4⋊C8—22no—
203C4⋊C8—24no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8C2×Q814yes—
207#103—44no—
208C4×C8—22no—
209C4×C8—42no—
210C4×C8—42no—
211C4×C8—42no—
212C4×C8C2×Q814yes—
213#95—24no—
214#95—24no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—22no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8C4⋊Q812yes—
233C2×Q8C2×SD1612yes—
234C2×Q8C2×SD1612yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4C4⋊Q812yes—
294C4⋊C4C2×SD1612yes—
295C4⋊C4C2×SD1612yes—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—48no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42C2×SD1612yes—
309C42C2×SD1612yes—
310C42#13512yes—
311C42AGL(1,ℤ/8)12yes—
312C42AGL(1,ℤ/8)12yes—
313C42#13712yes—
314C42C22×Q812yes—
315C422−1+412yes—
316C23C2×#12011yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—22no—
320Q8—84no—
321Q8—88no—
322Q8—164no—
323Q8#49814yescomplemented
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#44812yes—
355C2×C4#44812yes—
356C2×C4#44912yes—
357C2×C4#44912yes—
358C2×C4#45012yes—
359C2×C4#45012yes—
360C2×C4#45412yes—
361C2×C4#45512yes—
362C2×C4C2×C4⋊Q811yescharacteristic
363C2×C4#53611yescharacteristic
364C2×C4D4×Q812yes—
365C2×C4D4×Q812yes—
366C2×C4C22×SD1611yescharacteristic
367C2×C4C22×SD1611yescharacteristic
368C2×C4C2×#13511yescharacteristic
369C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
370C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4#57812yes—
373C8—44no—
374C8—44no—
375C8—82no—
376C22#243411yescharacteristic
377C22C2×#45011yescharacteristic
378C22C2×#44811yescharacteristic
379C22C2×#44911yescharacteristic
380C22#301111yescharacteristic
381C22C2×#45511yescharacteristic
382C22C2×#45411yescharacteristic
383C4—22no—
384C4—22no—
385C4—24no—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—84no—
392C4—82no—
393C4—162no—
394C4#162614yes—
395C4#164214yes—
396C4C2×#49811yescharacteristic
397C4#316511yescharacteristic
398C4#322711yescharacteristic
399C4#322911yescharacteristic
400C4#338312yes—
401C4#338612yes—
402C2C2×#162611yescharacteristic
403C2C2×#164211yescharacteristic
404C2#1863211yescharacteristic
405C2#1864111yescharacteristic
406C2#2581111yescharacteristic
407C2#2582911yescharacteristic
408C2#2584011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
120. Q8⋊C8Q8semidirect
323. Q8#498semidirect

Maximal subgroups

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

subgroupindex
#365932
#332892
#331962
#286872
#105822
#98982
#98702
#88962
C4×Q8⋊C82