Finite Group Databasebeta

← #500897

Subgroups — #500897

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3380C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic, complemented
3C2×C4×D16C211yesmaximal, characteristic
4#28263C211yesmaximal, characteristic, complemented
5#13453C212yesmaximal, complemented
6#13365C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13263C211yesmaximal, characteristic, complemented
9#13012C212yesmaximal
10#12937C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C2×#554—21no—
13C22×D16C2211yescharacteristic
14#3380—24no—
15#3380—24no—
16C2×#500C2211yescharacteristic
17C2×#499C2212yes—
18C2×#478—22no—
19C2×#476—21no—
20C2×C8×Q8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2212yes—
23C2×C4×D8C2211yescharacteristic
24#2538—21no—
25#2226—24no—
26C4×D16—24no—
27C2×#367C2212yes—
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—
45C22×D8C2311yescharacteristic
46C22×D8C2311yescharacteristic
47#554—28no—
48#554—44no—
49C2×C4⋊Q8—21no—
50C2×#125—22no—
51C2×C22⋊Q8—41no—
52C2×C22⋊Q8—41no—
53C2×C22⋊Q8—42no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4—22no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C22×C4⋊C4—41no—
59C2×D16—22no—
60C2×D16—22no—
61C2×D16Q814yescomplemented
62C22×C16C2311yescharacteristic
63C22×C16C2312yes—
64C22×C16C2311yescharacteristic
65#500—22no—
66#500D414yes—
67#499—24no—
68#499—24no—
69#478—216no—
70#476—28no—
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—21no—
80C2×#107—21no—
81C2×#107C2312yes—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2312yes—
85C2×#106—22no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D411yescharacteristic
88C2×#102—21no—
89C2×#102—22no—
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#367—24no—
100#366—22no—
101C4⋊C16—22no—
102#357—22no—
103#357—24no—
104#357—22no—
105#357—24no—
106C4×C16—22no—
107#340—22no—
108C22×Q8—41no—
109C22×Q8D811yescharacteristic
110C22×D4—21no—
111C22×D4—21no—
112C23×C4—41no—
113C23×C4—42no—
114C23×C4—41no—
115C2×D8—21no—
116C2×D8—21no—
117C2×D8—21no—
118C2×D8—21no—
119C2×D8—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8C2×D412yes—
124C2×D8C2×Q812yes—
125C2×D8C2×Q812yes—
126C2×D8#4112yes—
127C22×C8C2×D412yes—
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130C4⋊Q8—24no—
131C4⋊Q8—42no—
132#125—28no—
133#125—44no—
134C22⋊Q8—416no—
135C22⋊Q8—48no—
136C22⋊Q8—84no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×D4—28no—
140C4×D4—24no—
141C4×D4—44no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4D811yescharacteristic
157C2×C4⋊C4D812yes—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C42C2×D412yes—
164C2×C42C2×D411yescharacteristic
165D16—28no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—22no—
174C2×C16C2×Q812yes—
175#107—24no—
176#107—21no—
177#107—22no—
178#107—21no—
179#107—21no—
180#107—22no—
181#107—22no—
182#107—81no—
183#107C2×D412yes—
184#107C2×D414yes—
185#106—28no—
186C4⋊C8—24no—
187C4⋊C8—22no—
188#102—28no—
189#102—24no—
190#102—42no—
191#102—82no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8C2×D412yes—
195C4×C8C2×D412yes—
196#95—42no—
197C24—41no—
198C24—41no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×Q8—161no—
204C2×Q8—161no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C22×C4—48no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—162no—
229C22×C4—162no—
230C22×C4C2×D812yes—
231C22×C4C2×D812yes—
232C22×C4C2×D811yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4C22×D411yescharacteristic
235D8—22no—
236D8—22no—
237D8—22no—
238D8—22no—
239D8—22no—
240D8—41no—
241D8—41no—
242D8C22⋊Q814yescomplemented
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8#12012yes—
249C2×C8C22×D411yescharacteristic
250C2×C8C22×D411yescharacteristic
251C2×C8C22×Q811yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C82−1+411yescharacteristic
254C2×C82−1+411yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—84no—
269C4⋊C4—82no—
270C4⋊C4—161no—
271C22⋊C4—48no—
272C22⋊C4—44no—
273C22⋊C4—84no—
274C22⋊C4—82no—
275C22⋊C4—162no—
276C42—22no—
277C42—22no—
278C42#12012yes—
279C42AGL(1,ℤ/8)12yes—
280C16—22no—
281C16—41no—
282C16—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23C22×D811yescharacteristic
298Q8—44no—
299Q8—164no—
300D4—24no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—81no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—88no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—168no—
330C2×C4—164no—
331C2×C4#44712yes—
332C2×C4#44712yes—
333C2×C4#50812yes—
334C2×C4#50912yes—
335C2×C4C2×#12011yescharacteristic
336C2×C4#54811yescharacteristic
337C2×C4D4×Q811yescharacteristic
338C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
339C2×C4#57711yescharacteristic
340C2×C4#57811yescharacteristic
341C8—22no—
342C8—42no—
343C8C2×C22⋊Q811yescharacteristic
344C8#53611yescharacteristic
345C8#54811yescharacteristic
346C8D4×Q811yescharacteristic
347C22—44no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22C2×#44711yescharacteristic
362C22#335811yescharacteristic
363C22Q8×D811yescharacteristic
364C22C2×#50811yescharacteristic
365C22C2×#50911yescharacteristic
366C22#341511yescharacteristic
367C22#341611yescharacteristic
368C4—22no—
369C4—42no—
370C4—42no—
371C4—82no—
372C4—84no—
373C4—162no—
374C4—161no—
375C4#218614yes—
376C4#240911yescharacteristic
377C4#301311yescharacteristic
378C4#336711yescharacteristic
379C4#338411yescharacteristic
380C2—84no—
381C2—162no—
382C2#2432611yescharacteristic
383C2C2×#218611yescharacteristic
384C2#3352611yescharacteristic
385C2#3392711yescharacteristic
386C2#3393311yescharacteristic
387C2#3394711yescharacteristic
388C2#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×#3380C2semidirect
2. C2×#2226C2semidirect
4. #28263C2semidirect
5. #13453C2semidirect
6. #13365C2semidirect
7. #13282C2semidirect
8. #13263C2semidirect
10. #12937C2semidirect
11. #12811C2semidirect
61. C2×D16Q8semidirect
242. D8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#22262
C2×C4×D162
#282632
#134532
#133652
#132822
#132632
#130122
#129372
#128112