Finite Group Databasebeta

← #508435

Subgroups — #508435

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2244C211yesmaximal, characteristic, complemented
2C2×#2163C211yesmaximal, characteristic, complemented
3#33208C211yesmaximal, characteristic, complemented
4#33171C211yesmaximal, characteristic
5#13476C212yesmaximal, complemented
6#13474C212yesmaximal, complemented
7#13381C212yesmaximal, complemented
8#13380C212yesmaximal, complemented
9#13284C211yesmaximal, characteristic, complemented
10#13206C211yesmaximal, characteristic, complemented
11#12908C211yesmaximal, characteristic, complemented
12C2×#499C2211yescharacteristic
13C2×#493C2211yescharacteristic
14C2×#466—21no—
15C2×#466C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2856—21no—
18#2851—21no—
19#2244—24no—
20#2163—24no—
21C2×#367C2212yes—
22C2×#366C2212yes—
23C2×C4⋊C16C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×C22⋊C16C2212yes—
29C2×#346C2211yescharacteristic
30C2×#346C2211yescharacteristic
31#1991—21no—
32#1970—21no—
33#1970C2212yes—
34#1957—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1941—21no—
38#1846C2211yescharacteristic
39#1845C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1807—21no—
43#1380C2212yes—
44#1380C2212yes—
45C22×D8—21no—
46C22×D8C2312yes—
47C23×C8D411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#127—21no—
50C2×#125—21no—
51C2×#121—22no—
52C2×#121—41no—
53C2×#121—41no—
54C22×C42D411yescharacteristic
55C22×C16C2312yes—
56C22×C16C2312yes—
57#499—22no—
58#499—22no—
59#493—22no—
60#493—22no—
61#466—42no—
62#466—42no—
63#466—44no—
64C2×#107—21no—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2312yes—
70C2×#106—21no—
71C2×#106—21no—
72C2×#102—22no—
73C2×#102—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#390—22no—
81#390—41no—
82#390—41no—
83#389—21no—
84#389—21no—
85#387—41no—
86#387—41no—
87#385—41no—
88#385—41no—
89#384—21no—
90#382—21no—
91#380—21no—
92#380—21no—
93#367—42no—
94#366—42no—
95C4⋊C16—24no—
96#357—24no—
97#357—24no—
98#357—42no—
99#357—42no—
100C22⋊C16—24no—
101#346—22no—
102#346D414yescomplemented
103#340—21no—
104#340—21no—
105#340—21no—
106#340—21no—
107C22×D4—22no—
108C22×D4—42no—
109C22×D4—41no—
110C22×D4—41no—
111C23×C4D811yescharacteristic
112C23×C4D811yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×D8—22no—
115C2×D8—22no—
116C2×D8—22no—
117C2×D8—22no—
118C2×D8—22no—
119C2×D8—22no—
120C2×D8—42no—
121C2×D8—42no—
122C2×D8—42no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134#127—42no—
135#127—42no—
136#125—42no—
137#125—42no—
138#121—44no—
139#121—44no—
140#121—84no—
141#121—84no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42D811yescharacteristic
161C2×C42D811yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16C2×D414yes—
170#107—21no—
171#107—21no—
172#107—21no—
173#107—21no—
174#107—22no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—42no—
179#106—42no—
180#106—42no—
181#102—44no—
182#102—44no—
183C4×C8—22no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—22no—
187C4×C8C2×D412yes—
188C4×C8#4112yes—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194#95—41no—
195C24—42no—
196C24C2×D811yescharacteristic
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—84no—
208C2×D4—84no—
209C2×D4—84no—
210C2×D4—82no—
211C2×D4—82no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C2×D811yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4C22×D411yescharacteristic
235D8—42no—
236D8—42no—
237D8—42no—
238D8—42no—
239D8—84no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8#12112yes—
250C2×C8AGL(1,ℤ/8)12yes—
251C2×C8#13712yes—
252C2×C8C22×D411yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8H(5,𝔽2)12yes—
255C2×C82−1+412yes—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C22⋊C4—84no—
277C42—21no—
278C42—21no—
279C42—22no—
280C42—22no—
281C42#12112yes—
282C42#13512yes—
283C16—24no—
284C16—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23#50912yes—
299C23C22×D811yescharacteristic
300D4—82no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4—164no—
305D4—164no—
306D4—168no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4#46612yes—
341C2×C4#46612yes—
342C2×C4#50912yes—
343C2×C4C2×#12111yescharacteristic
344C2×C4#53511yescharacteristic
345C2×C4#53611yescharacteristic
346C2×C4C2×#13511yescharacteristic
347C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C4C2×#13711yescharacteristic
349C8—41no—
350C8—41no—
351C8#46814yes—
352C8#53712yes—
353C8#54212yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—162no—
362C22—162no—
363C22—162no—
364C22—162no—
365C22—162no—
366C22—162no—
367C22—162no—
368C22C2×#46611yescharacteristic
369C22#320511yescharacteristic
370C22#322611yescharacteristic
371C22C2×#50911yescharacteristic
372C22C2×#50911yescharacteristic
373C22#341411yescharacteristic
374C22#341411yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—161no—
381C4—161no—
382C4—161no—
383C4—162no—
384C4—161no—
385C4#262011yescharacteristic
386C4C2×#46811yescharacteristic
387C4#321011yescharacteristic
388C4#322811yescharacteristic
389C2—42no—
390C2—322no—
391C2#2984211yescharacteristic
392C2#3360511yescharacteristic
393C2#3360511yescharacteristic
394C2#3378311yescharacteristic
395C2#3378511yescharacteristic
396C2#3380011yescharacteristic
397C2#3380111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33171C2
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×#2244C2semidirect
2. C2×#2163C2semidirect
3. #33208C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13284C2semidirect
10. #13206C2semidirect
11. #12908C2semidirect
102. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22442
C2×#21632
#332082
#331712
#134762
#134742
#133812
#133802
#132842
#132062
#129082