Finite Group Databasebeta

← #229237

Subgroups — #229237

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23171C211yesmaximal, characteristic, complemented
2D4×#108C211yesmaximal, characteristic
3#19105C211yesmaximal, characteristic
4#18000C212yesmaximal
5#9861C211yesmaximal, characteristic, complemented
6#9630C211yesmaximal, characteristic, complemented
7#9591C212yesmaximal, complemented
8#8989C212yesmaximal
9#8933C212yesmaximal, complemented
10#8295C211yesmaximal, characteristic, complemented
11#8202C211yesmaximal, characteristic
12#3160C2211yescharacteristic
13#3101C2212yes—
14#2962C2212yes—
15D4×M16C2212yes—
16C2×C8×D4C2211yescharacteristic
17C22×#108—21no—
18#1986C2212yes—
19#1950C2212yes—
20#1946—21no—
21#1922—22no—
22#1855—22no—
23#1848C2211yescharacteristic
24#1810—21no—
25C4×#108C2211yescharacteristic
26#1773—21no—
27#1770C2211yescharacteristic
28C4×#102C2211yescharacteristic
29#1675C411yescharacteristic, complemented
30#1675C411yescharacteristic, complemented
31#1673C411yescharacteristic, complemented
32#1673C411yescharacteristic, complemented
33#1671C412yescomplemented
34#1671C412yescomplemented
35#1592—22no—
36#1588—22no—
37#1585—22no—
38SD16⋊C8—22no—
39#1571C2212yes—
40#1564C2211yescharacteristic
41#1542C2212yes—
42#1493C2212yes—
43#1493C2212yes—
44C2×D4⋊C8C2211yescharacteristic
45#1451C2212yes—
46C2×#322C2211yescharacteristic
47#1303C2212yes—
48#1279C2211yescharacteristic
49#1278C2212yes—
50#1277C2211yescharacteristic
51C22×M16—22no—
52C23×C8—21no—
53#533C2312yes—
54C2×C4×D4C2311yescharacteristic
55#500C2×C411yescharacteristic
56#500C2×C411yescharacteristic
57#499—21no—
58#499—21no—
59#498C2×C412yes—
60#483—22no—
61#476C2×C412yes—
62#476C2×C412yes—
63#461C2×C412yes—
64#461C2×C412yes—
65#450—22no—
66#446—22no—
67#446—42no—
68C8×Q8—22no—
69C4×SD16—22no—
70C4×SD16—22no—
71#436—22no—
72#435—42no—
73C8×D4—22no—
74C8×D4—21no—
75C8×D4—21no—
76C8×D4C2×C411yescharacteristic
77C8×D4C2×C411yescharacteristic
78C8×D4C2×C411yescharacteristic
79C8×D4C2×C411yescharacteristic
80#431C2312yes—
81C2×#108—22no—
82C2×#108—21no—
83C2×#108—21no—
84C2×#108—44no—
85C2×#108D411yescharacteristic
86C2×#108D411yescharacteristic
87#427—21no—
88#427—21no—
89#424—22no—
90#423C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#102—21no—
94#407—22no—
95C2×C22⋊C8—21no—
96#405—22no—
97C4×M16C2312yes—
98C2×#97—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103#341—22no—
104#341D412yes—
105#340—21no—
106#339—22no—
107#339—22no—
108#336—21no—
109#335—22no—
110#335—22no—
111#335C2×C411yescharacteristic
112#335C2×C411yescharacteristic
113#334C2×C412yes—
114#333C2×C412yes—
115#333C2×C412yes—
116#332C2×C412yes—
117#332C2×C412yes—
118Q8⋊C8—22no—
119Q8⋊C8—22no—
120Q8⋊C8—42no—
121D4⋊C8D412yes—
122D4⋊C8D412yes—
123#322—22no—
124#322—22no—
125#322—21no—
126#322—21no—
127C22×D4#4111yescharacteristic
128C23×C4—21no—
129C2×SD16—22no—
130C2×SD16—22no—
131C2×M16—22no—
132C2×M16—42no—
133C2×M16—44no—
134C2×M16—82no—
135C2×M16C2×D412yes—
136C2×M16C2×D412yes—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C4⋊Q8C22×C412yes—
151C4⋊Q8C22×C412yes—
152#125—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4C22×C411yescharacteristic
160C4×D4C22×C411yescharacteristic
161C4×D4C2×D411yescharacteristic
162C4×D4C2×D411yescharacteristic
163#117—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4#4111yescharacteristic
166C2×C22⋊C4—21no—
167C2×C42—21no—
168C2×C42C2411yescharacteristic
169#108—22no—
170#108—22no—
171#108—44no—
172#108—42no—
173#108—81no—
174#107—21no—
175#107—21no—
176#107—21no—
177#107—21no—
178#106—22no—
179#106—22no—
180#106—22no—
181#106—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8C22×C411yescharacteristic
189C4⋊C8C22×C411yescharacteristic
190C4⋊C8C22×C412yes—
191C4⋊C8C22×C412yes—
192C4⋊C8C2×D412yes—
193#103—22no—
194#103—22no—
195#102—22no—
196#102—22no—
197#102—42no—
198C22⋊C8—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205#97—22no—
206#97—42no—
207#97—42no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—42no—
214C4×C8C22×C411yescharacteristic
215C4×C8C22×C411yescharacteristic
216C4×C8C2×D412yes—
217C4×C8#4111yescharacteristic
218C4×C8#4111yescharacteristic
219#95—21no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—41no—
226C2×D4C4×D411yescharacteristic
227C2×D4C4×D411yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4111yescharacteristic
240C22×C42−1+411yescharacteristic
241SD16—44no—
242SD16—82no—
243M16—22no—
244M16—22no—
245M16—22no—
246M16—22no—
247M16—42no—
248M16—42no—
249M16—84no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—81no—
273C2×C8—84no—
274C2×C8C4×D411yescharacteristic
275C2×C8C4×D411yescharacteristic
276C2×C8C4×D411yescharacteristic
277C2×C8C4×D411yescharacteristic
278C2×C8AGL(1,ℤ/8)12yes—
279C2×C8#13712yes—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4C4×D411yescharacteristic
290C4⋊C4C4×D411yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—42no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42C23×C411yescharacteristic
300C42C22×D411yescharacteristic
301C42C2×#4111yescharacteristic
302C42H(5,𝔽2)11yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23—81no—
309C23#54811yescharacteristic
310Q8—42no—
311Q8—42no—
312Q8—82no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4#44412yes—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C2×C4×D411yescharacteristic
339C2×C4C2×C4×D411yescharacteristic
340C2×C4#52011yescharacteristic
341C2×C4D4211yescharacteristic
342C2×C4#54711yescharacteristic
343C2×C4#55311yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4C2×#13711yescharacteristic
346C2×C4#57711yescharacteristic
347C2×C4#57811yescharacteristic
348C8—21no—
349C8—21no—
350C8—42no—
351C8—41no—
352C8—41no—
353C8—42no—
354C8—42no—
355C8—42no—
356C8—42no—
357C8—42no—
358C8—81no—
359C8—82no—
360C8#44012yes—
361C8#44012yes—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—82no—
371C22—82no—
372C22#236111yescharacteristic
373C22#336711yescharacteristic
374C22#337211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—82no—
381C4—82no—
382C4—161no—
383C4C2×#44011yescharacteristic
384C4#294711yescharacteristic
385C4C2×#44411yescharacteristic
386C4#295711yescharacteristic
387C4#327211yescharacteristic
388C4#327611yescharacteristic
389C2—21no—
390C2—21no—
391C2—42no—
392C2—81no—
393C2#2300211yescharacteristic
394C2#2301911yescharacteristic
395C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC22×D4#41
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
1. #23171C2semidirect
5. #9861C2semidirect
6. #9630C2semidirect
7. #9591C2semidirect
9. #8933C2semidirect
10. #8295C2semidirect
29. #1675C4semidirect
30. #1675C4semidirect
31. #1673C4semidirect
32. #1673C4semidirect
33. #1671C4semidirect
34. #1671C4semidirect

Maximal subgroups

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

subgroupindex
#231712
D4×#1082
#191052
#180002
#98612
#96302
#95912
#89892
#89332
#82952
#82022