Finite Group Databasebeta

← #229232

Subgroups — #229232

Order 512.

604 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23179C211yesmaximal, characteristic
2Q8×#108C211yesmaximal, characteristic, complemented
3#19114C211yesmaximal, characteristic
4#18004C212yesmaximal, complemented
5#9856C211yesmaximal, characteristic
6#9627C211yesmaximal, characteristic, complemented
7#9592C212yesmaximal, complemented
8#8992C212yesmaximal, complemented
9#8930C212yesmaximal, complemented
10#8293C211yesmaximal, characteristic, complemented
11C8×#103C211yesmaximal, characteristic, complemented
12#3147C2211yescharacteristic
13#3103C2212yes—
14Q8×M16C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×#443—21no—
17#2935C2212yes—
18#1980C2212yes—
19#1974—21no—
20#1950—22no—
21#1950C2212yes—
22#1876—22no—
23#1848—21no—
24#1848C2211yescharacteristic
25C4×#108—21no—
26C4×#108C2211yescharacteristic
27C8×C4⋊C4C2211yescharacteristic
28C4×#103C2211yescharacteristic
29#1670C411yescharacteristic, complemented
30#1670C411yescharacteristic, complemented
31#1668C411yescharacteristic, complemented
32#1668C411yescharacteristic, complemented
33#1666C412yescomplemented
34#1666C412yescomplemented
35#1592—22no—
36#1588—22no—
37#1585—22no—
38SD16⋊C8—22no—
39#1571C2212yes—
40#1564C2211yescharacteristic
41#1540C2212yes—
42#1492C2212yes—
43#1492C2212yes—
44C2×Q8⋊C8C2211yescharacteristic
45#1451C2212yes—
46C2×C82C2211yescharacteristic
47#1303C2212yes—
48#1279C2211yescharacteristic
49#1278C2212yes—
50#1277C2211yescharacteristic
51C2×#135—21no—
52C2×#135—21no—
53C2×#131—22no—
54#532C2312yes—
55C2×C4×Q8C2311yescharacteristic
56#495—21no—
57#495—21no—
58#494C2×C411yescharacteristic
59#493C2×C411yescharacteristic
60#492C2×C412yes—
61#481—22no—
62#475C2×C412yes—
63#475C2×C412yes—
64#460C2×C412yes—
65#460C2×C412yes—
66#448—22no—
67#446—22no—
68#446—42no—
69C8×Q8—21no—
70C8×Q8—21no—
71C8×Q8—22no—
72C8×Q8C2×C411yescharacteristic
73C8×Q8C2×C411yescharacteristic
74C8×Q8C2×C411yescharacteristic
75C8×Q8C2×C411yescharacteristic
76#443—44no—
77C4×SD16—22no—
78C4×SD16—22no—
79#436—22no—
80#435—42no—
81C8×D4—22no—
82#431C2312yes—
83C2×#108—21no—
84C2×#108—22no—
85C2×#108—21no—
86C2×#108D411yescharacteristic
87C2×#108D411yescharacteristic
88#423—22no—
89#423C2312yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#104—22no—
94C2×#104—22no—
95C2×#103—21no—
96#408—22no—
97C4×M16—22no—
98C4×M16C2312yes—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105#341—22no—
106#341D412yes—
107#340—21no—
108#336—21no—
109#335—22no—
110#335—22no—
111#335C2×C411yescharacteristic
112#335C2×C411yescharacteristic
113#334C2×C412yes—
114#332C2×C412yes—
115#332C2×C412yes—
116#331C2×C412yes—
117#331C2×C412yes—
118Q8⋊C8D411yescharacteristic
119Q8⋊C8D411yescharacteristic
120Q8⋊C8D411yescharacteristic
121Q8⋊C8D411yescharacteristic
122D4⋊C8—22no—
123D4⋊C8—22no—
124D4⋊C8—42no—
125#322—22no—
126#322—22no—
127C82—21no—
128C82—21no—
129C2×#41—22no—
130C22×Q8#4111yescharacteristic
131#135—42no—
132#135—42no—
133#135—81no—
134#135—81no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×SD16—22no—
138C2×SD16—22no—
139C2×SD16—22no—
140C2×SD16—22no—
141C2×D8—21no—
142C2×D8—21no—
143#131—44no—
144#131—82no—
145C2×M16—22no—
146C2×M16—42no—
147C2×M16C2×D412yes—
148C2×M16C2×D412yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—42no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C4⋊Q8C22×C412yes—
157#127C22×C412yes—
158#124—22no—
159#121—22no—
160#121—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8C22×C411yescharacteristic
164C4×Q8C22×C411yescharacteristic
165C4×Q8C2×D411yescharacteristic
166C4×Q8C2×D411yescharacteristic
167C4×D4—22no—
168C4×D4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4#4111yescharacteristic
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C2411yescharacteristic
175#108—22no—
176#108—22no—
177#108—42no—
178#108—81no—
179#108—82no—
180#106—22no—
181#106—22no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C4⋊C8C22×C411yescharacteristic
193C4⋊C8C22×C411yescharacteristic
194C4⋊C8C22×C412yes—
195C4⋊C8C22×C412yes—
196C4⋊C8C2×D411yescharacteristic
197C4⋊C8C2×D411yescharacteristic
198#104—44no—
199#104—82no—
200#103—24no—
201#103—22no—
202#103—22no—
203#102—22no—
204#102—22no—
205C22⋊C8—22no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208#97—22no—
209#97—42no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—42no—
220C4×C8—42no—
221C4×C8C22×C411yescharacteristic
222C4×C8C22×C411yescharacteristic
223C4×C8C2×D411yescharacteristic
224C4×C8C2×D411yescharacteristic
225C4×C8#4111yescharacteristic
226C4×C8#4111yescharacteristic
227#95—21no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8C4×D411yescharacteristic
237C2×Q8C4×D411yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C42−1+411yescharacteristic
249Q16—42no—
250Q16—81no—
251SD16—44no—
252SD16—44no—
253SD16—82no—
254SD16—82no—
255D8—42no—
256D8—81no—
257M16—22no—
258M16—22no—
259M16—22no—
260M16—22no—
261M16—42no—
262M16—42no—
263M16—84no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—84no—
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C2×C8C4×D411yescharacteristic
289C2×C8C4×D411yescharacteristic
290C2×C8AGL(1,ℤ/8)12yes—
291C2×C8#13712yes—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4C4×D411yescharacteristic
302C4⋊C4C4×D411yescharacteristic
303C22⋊C4—42no—
304C22⋊C4—42no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—41no—
310C42—41no—
311C42C23×C411yescharacteristic
312C42C22×D411yescharacteristic
313C42C2×#4111yescharacteristic
314C42H(5,𝔽2)11yescharacteristic
315C23—42no—
316C23#54811yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—42no—
321Q8—42no—
322Q8—42no—
323Q8#44312yescomplemented
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4C2×C4×D411yescharacteristic
347C2×C4C2×C4×D411yescharacteristic
348C2×C4#52011yescharacteristic
349C2×C4D4211yescharacteristic
350C2×C4#54711yescharacteristic
351C2×C4#55311yescharacteristic
352C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
353C2×C4C2×#13711yescharacteristic
354C2×C4#57611yescharacteristic
355C2×C4#57711yescharacteristic
356C8—22no—
357C8—21no—
358C8—21no—
359C8—41no—
360C8—42no—
361C8—41no—
362C8—41no—
363C8—42no—
364C8—42no—
365C8—41no—
366C8—41no—
367C8—42no—
368C8—42no—
369C8—82no—
370C8#44012yes—
371C8#44012yes—
372C22—21no—
373C22—21no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22#236111yescharacteristic
378C22#336811yescharacteristic
379C22#337511yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—82no—
388C4—82no—
389C4C2×#44011yescharacteristic
390C4#294611yescharacteristic
391C4C2×#44311yescharacteristic
392C4#295711yescharacteristic
393C4#327411yescharacteristic
394C4#327811yescharacteristic
395C2—21no—
396C2—21no—
397C2—162no—
398C2#2300311yescharacteristic
399C2#2301811yescharacteristic
400C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3147C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#108C2semidirect
4. #18004C2semidirect
6. #9627C2semidirect
7. #9592C2semidirect
8. #8992C2semidirect
9. #8930C2semidirect
10. #8293C2semidirect
11. C8×#103C2semidirect
29. #1670C4semidirect
30. #1670C4semidirect
31. #1668C4semidirect
32. #1668C4semidirect
33. #1666C4semidirect
34. #1666C4semidirect
323. Q8#443semidirect

Maximal subgroups

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

subgroupindex
#231792
Q8×#1082
#191142
#180042
#98562
#96272
#95922
#89922
#89302
#82932
C8×#1032