Finite Group Databasebeta

← #211894

Subgroups — #211894

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21627C212yesmaximal
2#19158C211yesmaximal, characteristic
3#17958C211yesmaximal, characteristic, complemented
4#17335C211yesmaximal, characteristic
5#9720C211yesmaximal, characteristic, complemented
6#9626C211yesmaximal, characteristic, complemented
7#9591C212yesmaximal, complemented
8#9209C212yesmaximal
9#8669C211yesmaximal, characteristic
10C8×#106C211yesmaximal, characteristic, complemented
11#8252C212yesmaximal, complemented
12#3170C2211yescharacteristic
13#3076C2212yes—
14#2965C2211yescharacteristic
15#2965C2211yescharacteristic
16#2960C2212yes—
17C4×#131C2211yescharacteristic
18#1986C2212yes—
19#1950C2212yes—
20#1922—22no—
21#1866—22no—
22#1806C2212yes—
23C4×#106C2211yescharacteristic
24#1772—22no—
25C8×C4⋊C4C2212yes—
26#1764—22no—
27#1704C412yescomplemented
28#1703C412yescomplemented
29#1696C411yescharacteristic, complemented
30#1695C411yescharacteristic, complemented
31#1690C411yescharacteristic, complemented
32#1689C411yescharacteristic, complemented
33#1591—21no—
34#1591—21no—
35C8×Q16—21no—
36C8×Q16—21no—
37#1567C2211yescharacteristic
38C2×#334C2211yescharacteristic
39#1542C2212yes—
40#1503C2212yes—
41#1493C2212yes—
42#1455—21no—
43#1453—21no—
44#1451C2211yescharacteristic
45#1451C2211yescharacteristic
46#1450C2211yescharacteristic
47C8×M16—21no—
48C8×M16—21no—
49C2×C82C2211yescharacteristic
50#1303C2212yes—
51#1301C2212yes—
52#1279C2212yes—
53C2×#131D411yescharacteristic
54C2×#131D411yescharacteristic
55#533C2312yes—
56C4×#41C2311yescharacteristic
57#500C2×C411yescharacteristic
58#494C2×C411yescharacteristic
59#487C2×C412yes—
60#479—22no—
61#478—22no—
62#477C2×C412yes—
63#476C2×C412yes—
64#475C2×C412yes—
65#474C2×C412yes—
66#467—21no—
67#467—21no—
68#446—21no—
69#446—22no—
70#446—22no—
71#446C2×C411yescharacteristic
72#446C2×C411yescharacteristic
73C8×Q8—22no—
74C8×Q8—22no—
75C4×Q16—21no—
76C4×Q16—21no—
77#436—21no—
78#436C2×C411yescharacteristic
79#436C2×C411yescharacteristic
80#435—21no—
81#435—21no—
82#435—22no—
83#431—21no—
84#431—21no—
85#431C2311yescharacteristic
86#431C2311yescharacteristic
87C2×#108—22no—
88#427—22no—
89C2×#106—21no—
90#424—22no—
91#423C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#104—22no—
94#416—22no—
95#405—21no—
96#405—21no—
97#405—22no—
98#405D411yescharacteristic
99#405D411yescharacteristic
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16C2311yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C4×C4⋊C4C2312yes—
110#341—22no—
111#340—21no—
112#339—22no—
113#339—22no—
114#339—22no—
115#336—22no—
116#335—21no—
117#335—21no—
118#334C2×C411yescharacteristic
119#334C2×C411yescharacteristic
120#334C2×C412yes—
121#333C2×C412yes—
122#333C2×C412yes—
123#332C2×C412yes—
124#332C2×C412yes—
125#328—24no—
126Q8⋊C8—22no—
127Q8⋊C8—22no—
128#322—21no—
129#322—21no—
130#322—21no—
131#322—21no—
132#322—21no—
133C82—21no—
134C82D411yescharacteristic
135C82D411yescharacteristic
136C82D411yescharacteristic
137C82D411yescharacteristic
138C2×#41C2×D411yescharacteristic
139C2×Q16—21no—
140C2×Q16—21no—
141#131—24no—
142#131—42no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—22no—
149C2×M16C2×D411yescharacteristic
150C2×M16C2×D411yescharacteristic
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8#4111yescharacteristic
158C22×C8#4111yescharacteristic
159C4⋊Q8C22×C412yes—
160C4⋊Q8C22×C412yes—
161#125—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8C22×C411yescharacteristic
167C4×Q8#4111yescharacteristic
168C4×D4—21no—
169C4×D4—21no—
170C4×D4C22×C411yescharacteristic
171C4×D4#4111yescharacteristic
172#117—21no—
173#117—22no—
174#117C2×D411yescharacteristic
175C2×C4⋊C4—22no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42C2411yescharacteristic
179#108—44no—
180#107—21no—
181#107—21no—
182#107—22no—
183#107—22no—
184#106—24no—
185#106—22no—
186#106—22no—
187C4⋊C8—22no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8C22×C411yescharacteristic
197C4⋊C8C22×C411yescharacteristic
198C4⋊C8C22×C412yes—
199C4⋊C8C22×C412yes—
200#104—44no—
201#103—22no—
202#103—22no—
203#103—22no—
204#103—22no—
205#102—22no—
206#102—22no—
207C22⋊C8—22no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8#4112yes—
212#97—22no—
213#97—21no—
214#97—21no—
215#97—21no—
216#97—21no—
217#97—21no—
218#97—42no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—42no—
226C4×C8—44no—
227C4×C8C22×C411yescharacteristic
228C4×C8C22×C411yescharacteristic
229C4×C8C2×D411yescharacteristic
230C4×C8C2×D411yescharacteristic
231C4×C8C2×D411yescharacteristic
232C4×C8C2×D411yescharacteristic
233C4×C8C2×D411yescharacteristic
234C4×C8C2×D411yescharacteristic
235C4×C8#4111yescharacteristic
236C4×C8#4111yescharacteristic
237#95—22no—
238#41—22no—
239#41—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C4×D411yescharacteristic
243C2×D4—21no—
244C2×D4C4×D411yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252Q16—42no—
253Q16—42no—
254M16—22no—
255M16—22no—
256M16—44no—
257M16—42no—
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—42no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8C4×D411yescharacteristic
282C2×C8C4×D411yescharacteristic
283C2×C8C4×D411yescharacteristic
284C2×C8C4×D411yescharacteristic
285C2×C8#12111yescharacteristic
286C2×C8#12111yescharacteristic
287C2×C8#12111yescharacteristic
288C2×C8#12111yescharacteristic
289C2×C8#13714yes—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4C4×D411yescharacteristic
301C4⋊C4C4×D411yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—42no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42C23×C411yescharacteristic
313C42C22×D411yescharacteristic
314C42C2×#4111yescharacteristic
315C42C2×#4111yescharacteristic
316C23—21no—
317C23#53211yescharacteristic
318Q8—22no—
319Q8—42no—
320Q8—42no—
321Q8—82no—
322D4—22no—
323D4—42no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4C2×C4×D411yescharacteristic
341C2×C4C2×C4×D411yescharacteristic
342C2×C4C4×#4111yescharacteristic
343C2×C4C2×#12111yescharacteristic
344C2×C4#52511yescharacteristic
345C2×C4#52911yescharacteristic
346C2×C4C2×#13712yes—
347C2×C4#57512yes—
348C8—22no—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—41no—
355C8—42no—
356C8—42no—
357C8—42no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—82no—
363C8#44312yes—
364C8#47012yes—
365C8#47012yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22C4×#12111yescharacteristic
372C22#317212yes—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—84no—
380C4—82no—
381C4—82no—
382C4C4×#13712yes—
383C4C2×#44311yescharacteristic
384C4#295511yescharacteristic
385C4C2×#47011yescharacteristic
386C4#305411yescharacteristic
387C2—22no—
388C2—81no—
389C2#2151711yescharacteristic
390C2C4×#47011yescharacteristic
391C2#2155811yescharacteristic
subgroupG/H
centerC2×C8#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17958C2semidirect
5. #9720C2semidirect
6. #9626C2semidirect
7. #9591C2semidirect
10. C8×#106C2semidirect
11. #8252C2semidirect
27. #1704C4semidirect
28. #1703C4semidirect
29. #1696C4semidirect
30. #1695C4semidirect
31. #1690C4semidirect
32. #1689C4semidirect

Maximal subgroups

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

subgroupindex
#216272
#191582
#179582
#173352
#97202
#96262
#95912
#92092
#86692
C8×#1062
#82522