Finite Group Databasebeta

← #499340

Subgroups — #499340

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33409C211yesmaximal, characteristic
2C2×#2153C211yesmaximal, characteristic
3C22×#362C211yesmaximal, characteristic
4C2×#2141C211yesmaximal, characteristic
5#21232C211yesmaximal, characteristic
6#21163C211yesmaximal, characteristic
7#13479C218yesmaximal, complemented
8#12569C211yesmaximal, characteristic
9C23×Q16C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C22×M32C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#427C2211yescharacteristic
15#2905—21no—
16C2×#419—21no—
17C2×#405C2211yescharacteristic
18#2310—21no—
19#2159—22no—
20#2159C2214yes—
21#2158—22no—
22#2153—24no—
23C2×#364C2214yes—
24C2×#362—24no—
25C2×#362C2214yes—
26#2141—22no—
27#2141C2214yes—
28C2×#358—22no—
29#1971—22no—
30#1969C2214yes—
31#1946—22no—
32#1945C2214yes—
33#1733—21no—
34#1387C2214yes—
35C23×Q8—21no—
36C22×Q16—21no—
37C22×Q16—21no—
38C22×Q16—22no—
39C22×Q16—21no—
40C22×Q16—22no—
41C22×Q16—21no—
42C22×Q16—21no—
43C22×Q16—22no—
44C22×Q16Q814yes—
45C22×Q16C2312yes—
46C22×M16—21no—
47C22×M16D411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117—21no—
50C2×#117—21no—
51C2×#117D411yescharacteristic
52C2×M32—21no—
53C2×M32—22no—
54C2×M32—21no—
55C2×M32—22no—
56C2×M32—22no—
57C2×M32D414yes—
58C2×M32C2312yes—
59C2×M32C2312yes—
60C22×C16—21no—
61C22×C16—21no—
62C2×#108—22no—
63C2×#108—22no—
64C2×#108—21no—
65C2×#108C2312yes—
66#427—21no—
67#427—21no—
68#427—21no—
69#427—21no—
70#427—21no—
71#427—21no—
72#427D412yes—
73#427C2312yes—
74C2×#107D412yes—
75C2×#107C2312yes—
76C2×#106—21no—
77C2×#106—21no—
78#424—24no—
79#424—42no—
80#423—42no—
81#419—42no—
82#419—42no—
83C2×#103—24no—
84#405—22no—
85#405—22no—
86C2×#97D412yes—
87C2×C4×C8C2312yes—
88#391—24no—
89#391—42no—
90#381—42no—
91#367—24no—
92#366—24no—
93#364—28no—
94#362—28no—
95#358—28no—
96#339—24no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—42no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8—41no—
105C22×Q8—41no—
106C22×Q8—42no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C2×Q16—24no—
110C2×Q16—24no—
111C2×Q16—24no—
112C2×Q16—24no—
113C2×Q16—24no—
114C2×Q16—22no—
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—22no—
119C2×Q16—42no—
120C2×Q16—42no—
121C2×Q16—42no—
122C2×Q16—41no—
123C2×Q16—41no—
124C2×Q16—42no—
125C2×Q16—42no—
126C2×Q16C2×Q814yes—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—41no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16D812yes—
134C2×M16D812yes—
135C22×C8—41no—
136C22×C8C2×D412yes—
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D412yes—
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×Q812yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144C22×C8#4111yescharacteristic
145C22×C8#4111yescharacteristic
146C22×C8C2411yescharacteristic
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42C2×D412yes—
166M32—24no—
167M32—22no—
168M32—22no—
169M32—22no—
170M32—42no—
171M32C2×D414yes—
172C2×C16—24no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—42no—
176#108—22no—
177#108—22no—
178#108—42no—
179#107—22no—
180#107—22no—
181#107—22no—
182#107C2×D414yes—
183#106—22no—
184#106—22no—
185#106—41no—
186#106—41no—
187C4⋊C8—48no—
188#103—48no—
189#97—24no—
190C4×C8—22no—
191C4×C8—22no—
192#95—48no—
193C24AGL(1,ℤ/8)11yescharacteristic
194C2×Q8—24no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12112yes—
223C22×C4C22⋊Q812yes—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4#13512yes—
227C22×C4AGL(1,ℤ/8)11yescharacteristic
228C22×C4#13712yes—
229C22×C4C22×D411yescharacteristic
230Q16—24no—
231Q16—24no—
232Q16—42no—
233Q16—42no—
234Q16—44no—
235Q16—44no—
236M16—24no—
237M16—24no—
238M16—42no—
239M16—42no—
240M16—82no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—42no—
244C2×C8—82no—
245C2×C8#12112yes—
246C2×C8#12112yes—
247C2×C8#12112yes—
248C2×C8#12111yescharacteristic
249C2×C8#12111yescharacteristic
250C2×C8C22⋊Q812yes—
251C2×C8C22⋊Q812yes—
252C2×C8C22⋊Q812yes—
253C2×C8C22⋊Q812yes—
254C2×C8C22⋊Q812yes—
255C2×C8#12612yes—
256C2×C8#12612yes—
257C2×C8C2×D814yes—
258C2×C8C22×D411yescharacteristic
259C2×C8C22×D411yescharacteristic
260C2×C8C22×Q811yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8C2×#4111yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—42no—
278C42—81no—
279C42—81no—
280C16—24no—
281C16—42no—
282C23—42no—
283C23#45911yescharacteristic
284C23#45911yescharacteristic
285C23#46812yes—
286C23#47812yes—
287C23C2×AGL(1,ℤ/8)11yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—84no—
291Q8—84no—
292Q8—82no—
293Q8—82no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—162no—
305C2×C4#39612yes—
306C2×C4#39612yes—
307C2×C4#45911yescharacteristic
308C2×C4#45911yescharacteristic
309C2×C4#46412yes—
310C2×C4#46412yes—
311C2×C4#46612yes—
312C2×C4#46612yes—
313C2×C4#46812yes—
314C2×C4#47412yes—
315C2×C4#47412yes—
316C2×C4#47812yes—
317C2×C4C2×#12111yescharacteristic
318C2×C4C2×#12111yescharacteristic
319C2×C4C2×C22⋊Q811yescharacteristic
320C2×C4C22×D811yescharacteristic
321C2×C4C2×#13511yescharacteristic
322C2×C4C2×#13711yescharacteristic
323C8—24no—
324C8—82no—
325C8#39612yes—
326C8#39612yes—
327C8C2×#12111yescharacteristic
328C8C2×C22⋊Q811yescharacteristic
329C8C2×C22⋊Q811yescharacteristic
330C8C2×#12611yescharacteristic
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22#206712yes—
337C22#206712yes—
338C22C2×#45911yescharacteristic
339C22C2×#46811yescharacteristic
340C22C2×#47811yescharacteristic
341C4—82no—
342C4—84no—
343C4—84no—
344C4—162no—
345C4#206712yes—
346C4#206712yes—
347C4C2×#39611yescharacteristic
348C4C2×#46411yescharacteristic
349C4C2×#46611yescharacteristic
350C4C2×#47411yescharacteristic
351C2—21no—
352C2—21no—
353C2—22no—
354C2—22no—
355C2C2×#206711yescharacteristic
356C2#3359612yes—
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #13479C2semidirect

Maximal subgroups

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

subgroupindex
#334092
C2×#21532
C22×#3622
C2×#21412
#212322
#211632
#134792
#125692