Finite Group Databasebeta

← #500896

Subgroups — #500896

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D8C211yesmaximal, characteristic, complemented
2C2×#2224C211yesmaximal, characteristic, complemented
3C2×C4×D16C211yesmaximal, characteristic
4#28263C211yesmaximal, characteristic, complemented
5#13451C212yesmaximal, complemented
6#13363C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13263C211yesmaximal, characteristic, complemented
9#13012C212yesmaximal
10#12935C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C2×D4×Q8—21no—
13C22×D16C2211yescharacteristic
14Q8×D8—24no—
15Q8×D8—24no—
16C2×#500C2212yes—
17C2×#500C2211yescharacteristic
18C2×#474—21no—
19C2×#474—22no—
20C2×C8×Q8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2212yes—
23C2×C4×D8C2211yescharacteristic
24#2538—21no—
25#2224—24no—
26C4×D16—24no—
27C2×#366C2212yes—
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#357C2212yes—
31C2×#357C2211yescharacteristic
32C2×#357C2212yes—
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2067—22no—
36#1970C2212yes—
37#1942C2212yes—
38#1845C2211yescharacteristic
39#1802—21no—
40#1380C2212yes—
41#1380C2212yes—
42#1380C2212yes—
43#1374C2212yes—
44#1374C2212yes—
45C23×Q8—41no—
46C22×D8C2311yescharacteristic
47C22×D8C2311yescharacteristic
48D4×Q8—28no—
49D4×Q8—44no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C22⋊Q8—41no—
53C2×C22⋊Q8—41no—
54C2×C22⋊Q8—42no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×D16—22no—
60C2×D16—22no—
61C2×D16Q814yescomplemented
62C22×C16C2311yescharacteristic
63C22×C16C2312yes—
64C22×C16C2311yescharacteristic
65#500—24no—
66#500—24no—
67#500—22no—
68#500D414yes—
69#474—28no—
70#474—216no—
71C8×Q8—22no—
72C8×Q8—22no—
73C4×D8—24no—
74C4×D8—24no—
75C4×D8—22no—
76C4×D8—22no—
77C4×D8—22no—
78C4×D8D414yes—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107C2312yes—
83C2×#107C2311yescharacteristic
84C2×#107C2312yes—
85C2×#107C2311yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D412yes—
88C2×#102—22no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—22no—
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94#396—42no—
95#390—22no—
96#389—22no—
97#384—21no—
98#380—41no—
99#366—22no—
100#366—24no—
101C4⋊C16—22no—
102#357—22no—
103#357—24no—
104#357—22no—
105#357—24no—
106C4×C16—22no—
107#340—22no—
108C22×Q8—48no—
109C22×Q8—42no—
110C22×Q8—41no—
111C22×Q8—84no—
112C22×Q8—84no—
113C22×Q8D811yescharacteristic
114C22×D4—21no—
115C22×D4—21no—
116C23×C4—42no—
117C23×C4—41no—
118C23×C4—41no—
119C2×D8—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8C2×D412yes—
128C2×D8C2×Q812yes—
129C2×D8C2×Q812yescomplemented
130C2×D8#4112yes—
131C22×C8C2×D412yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134C4⋊Q8—28no—
135C4⋊Q8—24no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C22⋊Q8—48no—
139C22⋊Q8—416no—
140C22⋊Q8—84no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×D4—28no—
144C4×D4—24no—
145C4×D4—44no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4D811yescharacteristic
159C2×C4⋊C4D812yes—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C42C2×D412yes—
166C2×C42C2×D411yescharacteristic
167D16—28no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—22no—
176C2×C16C2×Q812yescomplemented
177#107—28no—
178#107—24no—
179#107—22no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—22no—
184#107—22no—
185#107—81no—
186#107C2×D414yes—
187#107C2×D412yes—
188C4⋊C8—24no—
189C4⋊C8—22no—
190#102—28no—
191#102—24no—
192#102—42no—
193#102—82no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8C2×D412yes—
197C4×C8C2×D412yes—
198#95—42no—
199C24—41no—
200C24—41no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—48no—
204C2×Q8—44no—
205C2×Q8—84no—
206C2×Q8—832no—
207C2×Q8—84no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8D1614yescomplemented
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—48no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—44no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C2×D812yes—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D812yes—
239C22×C4C2×D811yescharacteristic
240C22×C4C22×D411yescharacteristic
241D8—22no—
242D8—22no—
243D8—22no—
244D8—22no—
245D8—22no—
246D8—41no—
247D8—41no—
248D8C22⋊Q814yescomplemented
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8#12012yes—
255C2×C8C22×D411yescharacteristic
256C2×C8C22×D411yescharacteristic
257C2×C8C22×Q811yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—84no—
275C4⋊C4—161no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—82no—
279C22⋊C4—84no—
280C22⋊C4—162no—
281C42—22no—
282C42—22no—
283C42#12012yes—
284C42AGL(1,ℤ/8)12yes—
285C16—22no—
286C16—41no—
287C16—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23C22×D811yescharacteristic
303Q8—28no—
304Q8—816no—
305Q8—88no—
306D4—24no—
307D4—42no—
308D4—42no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—816no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—88no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—164no—
336C2×C4#44712yes—
337C2×C4#44712yes—
338C2×C4C2×D1614yes—
339C2×C4C2×D1612yes—
340C2×C4#50912yes—
341C2×C4C2×#12011yescharacteristic
342C2×C4#54811yescharacteristic
343C2×C4D4×Q811yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4#57711yescharacteristic
346C2×C4#57811yescharacteristic
347C8—22no—
348C8—42no—
349C8C2×C22⋊Q811yescharacteristic
350C8#53611yescharacteristic
351C8#54811yescharacteristic
352C8D4×Q811yescharacteristic
353C22—44no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22—161no—
367C22C2×#44711yescharacteristic
368C22#335811yescharacteristic
369C22Q8×D811yescharacteristic
370C22C22×D1611yescharacteristic
371C22C2×#50911yescharacteristic
372C22#341611yescharacteristic
373C22#341711yescharacteristic
374C4—24no—
375C4—22no—
376C4—42no—
377C4—84no—
378C4—82no—
379C4—162no—
380C4—161no—
381C4#218414yes—
382C4#240911yescharacteristic
383C4#301311yescharacteristic
384C4#336711yescharacteristic
385C4#338411yescharacteristic
386C2—84no—
387C2—162no—
388C2#2432611yescharacteristic
389C2C2×#218411yescharacteristic
390C2#3352711yescharacteristic
391C2#3392511yescharacteristic
392C2#3393411yescharacteristic
393C2Q8×D1611yescharacteristic
394C2#3395111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D8C2semidirect
2. C2×#2224C2semidirect
4. #28263C2semidirect
5. #13451C2semidirect
6. #13363C2semidirect
7. #13282C2semidirect
8. #13263C2semidirect
10. #12935C2semidirect
11. #12811C2semidirect
61. C2×D16Q8semidirect
129. C2×D8C2×Q8semidirect
176. C2×C16C2×Q8semidirect
210. C2×Q8D16semidirect
248. D8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D82
C2×#22242
C2×C4×D162
#282632
#134512
#133632
#132822
#132632
#130122
#129352
#128112