Finite Group Databasebeta

← #501507

Subgroups — #501507

Order 512.

688 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2208C211yesmaximal, characteristic, complemented
2C2×#2163C211yesmaximal, characteristic, complemented
3#33172C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic
5#13383C212yesmaximal, complemented
6#13305C212yesmaximal
7#13253C211yesmaximal, characteristic, complemented
8#13238C212yesmaximal, complemented
9#13176C211yesmaximal, characteristic, complemented
10#13150C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C2×#494C2211yescharacteristic
14C2×#467—21no—
15C2×#467C2212yes—
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2857—21no—
20C4×#121—21no—
21#2208—24no—
22#2163—24no—
23C2×#366C2212yes—
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2212yes—
27C2×#358C2212yes—
28C2×#358C2212yes—
29C2×#357C2212yes—
30C2×C22⋊C16C2212yes—
31C2×#346C2212yes—
32#1970C2211yescharacteristic
33#1969—21no—
34#1969C2212yes—
35#1956—21no—
36#1942C2211yescharacteristic
37#1934—21no—
38#1893—21no—
39#1845C2211yescharacteristic
40#1808C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#102—21no—
43#1380C2212yes—
44#1380C2212yes—
45C22×Q16—21no—
46C22×Q16C2312yes—
47C22×D8C2311yescharacteristic
48C23×C8D411yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×#121—21no—
53C2×C4×D4—21no—
54C2×C4×D4—41no—
55C22×C42D411yescharacteristic
56C2×SD32—24no—
57C2×SD32—24no—
58C2×SD32—42no—
59C22×C16C2312yes—
60C22×C16C2312yes—
61#494—22no—
62#494—22no—
63#467—42no—
64#467—42no—
65#467—44no—
66#466—22no—
67#466—22no—
68C4×D8—22no—
69C4×D8—22no—
70C2×#107—21no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2312yes—
74C2×#107C2311yescharacteristic
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—21no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#391—22no—
87#390—21no—
88#389—21no—
89#389—41no—
90#389—41no—
91#388—41no—
92#386—41no—
93#384—21no—
94#380—21no—
95#380—41no—
96#380—41no—
97C4×C4⋊C4—21no—
98C4×C22⋊C4—41no—
99#366—42no—
100C4⋊C16—22no—
101C4⋊C16D414yescomplemented
102#358—24no—
103#358—24no—
104#358—42no—
105#357—42no—
106C22⋊C16—24no—
107#346—24no—
108#340—21no—
109#340—21no—
110#340—21no—
111C22×Q8—22no—
112C22×D4—21no—
113C22×D4—41no—
114C23×C4—41no—
115C23×C4D811yescharacteristic
116C23×C4D811yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×Q16—22no—
119C2×Q16—22no—
120C2×Q16—22no—
121C2×Q16—22no—
122C2×Q16—22no—
123C2×Q16—22no—
124C2×Q16—42no—
125C2×Q16—42no—
126C2×Q16—42no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8#4112yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—41no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C22⋊Q8—44no—
149C22⋊Q8—44no—
150#121—24no—
151#121—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—84no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42D811yescharacteristic
175C2×C42D811yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178SD32—48no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16C2×D414yes—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—22no—
189#107—21no—
190#107—21no—
191#107—22no—
192#107—42no—
193#107#4112yes—
194#103—44no—
195#103—44no—
196#102—24no—
197#102—42no—
198#102—42no—
199C4×C8—22no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—22no—
203C4×C8C2×D412yes—
204C4×C8#4112yes—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—42no—
209#95—42no—
210C24—41no—
211C24C2×D811yescharacteristic
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×D4—22no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—82no—
226C2×D4—82no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C22×D411yescharacteristic
253Q16—42no—
254Q16—42no—
255Q16—42no—
256Q16—42no—
257Q16—84no—
258D8—22no—
259D8—22no—
260D8—41no—
261D8—41no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—44no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8#12112yes—
275C2×C8AGL(1,ℤ/8)12yes—
276C2×C8C22×D411yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—84no—
301C22⋊C4—82no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—82no—
307C42#12112yes—
308C42#13512yes—
309C16—24no—
310C16—42no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23#51012yes—
325C23C22×D811yescharacteristic
326Q8—82no—
327Q8—82no—
328Q8—82no—
329Q8—82no—
330D4—42no—
331D4—42no—
332D4—88no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—164no—
368C2×C4#46612yes—
369C2×C4#46612yes—
370C2×C4#50912yes—
371C2×C4#51012yes—
372C2×C4C2×#12111yescharacteristic
373C2×C4#52511yescharacteristic
374C2×C4#53511yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
377C2×C4#57511yescharacteristic
378C8—22no—
379C8—22no—
380C8—41no—
381C8#54611yescharacteristic
382C8#55011yescharacteristic
383C8#55311yescharacteristic
384C8#55511yescharacteristic
385C22—22no—
386C22—22no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22C2×#46611yescharacteristic
400C22#310011yescharacteristic
401C22#320511yescharacteristic
402C22C2×#50911yescharacteristic
403C22C2×#51011yescharacteristic
404C22C2×#51011yescharacteristic
405C22#341411yescharacteristic
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—82no—
411C4—161no—
412C4—162no—
413C4—161no—
414C4—162no—
415C4#222014yes—
416C4#241411yescharacteristic
417C4#305511yescharacteristic
418C4#310611yescharacteristic
419C4#321211yescharacteristic
420C2—42no—
421C2—162no—
422C2#2465011yescharacteristic
423C2C2×#222011yescharacteristic
424C2#3360611yescharacteristic
425C2#3367011yescharacteristic
426C2#3367611yescharacteristic
427C2#3378011yescharacteristic
428C2#3378411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2208C2semidirect
2. C2×#2163C2semidirect
3. #33172C2semidirect
5. #13383C2semidirect
7. #13253C2semidirect
8. #13238C2semidirect
9. #13176C2semidirect
10. #13150C2semidirect
11. #12811C2semidirect
101. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22082
C2×#21632
#331722
C4×#4662
#133832
#133052
#132532
#132382
#131762
#131502
#128112