Finite Group Databasebeta

← #512284

Subgroups — #512284

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C212yesmaximal, complemented
2#33171C211yesmaximal, characteristic
3C4×#427C211yesmaximal, characteristic, complemented
4#13252C211yesmaximal, characteristic, complemented
5#13252C211yesmaximal, characteristic, complemented
6#13177C211yesmaximal, characteristic, complemented
7#13176C211yesmaximal, characteristic, complemented
8#13131C212yesmaximal, complemented
9#13130C212yesmaximal, complemented
10#12992C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2856—21no—
19C4×#117—21no—
20#2234—28no—
21C2×#367C2212yes—
22C2×#366C2212yes—
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2212yes—
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×C22⋊C16C2212yes—
30#1970—21no—
31#1970C2212yes—
32#1957—21no—
33#1845C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#106—21no—
39C4×#106—21no—
40#1735—21no—
41#1380C2212yes—
42#1380C2212yes—
43#1374C2212yes—
44C22×D8—21no—
45C22×D8C2312yes—
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#127—21no—
49C2×#121—22no—
50C2×#121—41no—
51C2×#121—41no—
52C2×#117—21no—
53C2×#117—21no—
54C22×C42D411yescharacteristic
55C22×C16C2312yes—
56C22×C16C2312yes—
57#493—22no—
58#493—22no—
59#466—24no—
60#466—24no—
61#466—44no—
62#427—22no—
63#427—22no—
64#427—22no—
65#427D414yescomplemented
66C2×#107—21no—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#106—21no—
73C2×#106—21no—
74C2×#102—22no—
75C2×#102—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8D411yescharacteristic
79C2×C4×C8D411yescharacteristic
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#390—22no—
83#390—41no—
84#390—41no—
85#384—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92#376—41no—
93C43—41no—
94#367—24no—
95#366—24no—
96C4⋊C16—22no—
97C4⋊C16—22no—
98#357—24no—
99#357—24no—
100#357—24no—
101#357—24no—
102C22⋊C16—24no—
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—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—41no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D412yes—
133C22×C8C2×D411yescharacteristic
134C22×C8#4112yes—
135C22×C8C2411yescharacteristic
136#127—42no—
137#127—42no—
138#121—44no—
139#121—44no—
140#121—84no—
141#121—84no—
142#117—24no—
143#117—24no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
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×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42D811yescharacteristic
167C2×C42D811yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—22no—
176#107—21no—
177#107—21no—
178#107—21no—
179#107—21no—
180#107—22no—
181#107—22no—
182#107—42no—
183#107C2×D412yes—
184#107C2×D412yes—
185#107#4112yes—
186#107#4112yes—
187#106—24no—
188#106—24no—
189#102—44no—
190#102—44no—
191C4×C8—22no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—22no—
195C4×C8#4112yes—
196C4×C8#4112yes—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201C24—42no—
202C24C2×D811yescharacteristic
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—82no—
210C2×D4—82no—
211C2×D4—84no—
212C2×D4—84no—
213C2×D4—84no—
214C2×D4—82no—
215C2×D4—82no—
216C2×D4—82no—
217C2×D4—82no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4#12112yes—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4AGL(1,ℤ/8)12yes—
240C22×C4C22×D411yescharacteristic
241D8—42no—
242D8—42no—
243D8—42no—
244D8—42no—
245D8—84no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—44no—
258C2×C8#12112yes—
259C2×C8#12312yes—
260C2×C8C22×D411yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8C2×#4111yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C2×C8C2×#4111yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—84no—
281C22⋊C4—84no—
282C22⋊C4—84no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—44no—
288C42—44no—
289C42—82no—
290C42—82no—
291C42—82no—
292C42—82no—
293C42#12312yes—
294C42#13712yes—
295C16—42no—
296C16—42no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—82no—
304C23—82no—
305C23—82no—
306C23—82no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23#45912yes—
311C23C22×D811yescharacteristic
312D4—82no—
313D4—82no—
314D4—82no—
315D4—82no—
316D4—164no—
317D4—168no—
318D4—164no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#45912yes—
350C2×C4#48012yes—
351C2×C4#48012yes—
352C2×C4#50812yes—
353C2×C4#50912yes—
354C2×C4#50912yes—
355C2×C4C2×#12111yescharacteristic
356C2×C4C2×#12311yescharacteristic
357C2×C4#52511yescharacteristic
358C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C2×#13711yescharacteristic
360C2×C4#57511yescharacteristic
361C8—22no—
362C8—22no—
363C8—41no—
364C8#52911yescharacteristic
365C8#52911yescharacteristic
366C8#53211yescharacteristic
367C8#53211yescharacteristic
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—162no—
376C22—162no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22—162no—
381C22—162no—
382C22C2×#45911yescharacteristic
383C22C2×#48011yescharacteristic
384C22#310011yescharacteristic
385C22C2×#50811yescharacteristic
386C22C2×#50911yescharacteristic
387C22C2×#50911yescharacteristic
388C22#341411yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—82no—
394C4—82no—
395C4—82no—
396C4—82no—
397C4—162no—
398C4#223414yescomplemented
399C4#285411yescharacteristic
400C4#303711yescharacteristic
401C4#309411yescharacteristic
402C4#311211yescharacteristic
403C2—42no—
404C2—322no—
405C2#3310611yescharacteristic
406C2#3357911yescharacteristic
407C2#3358211yescharacteristic
408C2C2×#223411yescharacteristic
409C2#3364811yescharacteristic
410C2#3366611yescharacteristic
411C2#3367511yescharacteristic
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×#2234C2semidirect
3. C4×#427C2semidirect
4. #13252C2semidirect
5. #13252C2semidirect
6. #13177C2semidirect
7. #13176C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #12992C2semidirect
11. #12783C2semidirect
65. #427D4semidirect
398. C4#2234semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
#331712
C4×#4272
#132522
#131772
#131762
#131312
#131302
#129922
#127832