Finite Group Databasebeta

← #211913

Subgroups — #211913

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22166C212yesmaximal, complemented
2#19157C211yesmaximal, characteristic
3#17962C211yesmaximal, characteristic, complemented
4#17331C211yesmaximal, characteristic, complemented
5#9719C211yesmaximal, characteristic, complemented
6#9623C211yesmaximal, characteristic, complemented
7#9592C212yesmaximal, complemented
8#9208C212yesmaximal, complemented
9#8668C211yesmaximal, characteristic, complemented
10#8287C211yesmaximal, characteristic, complemented
11#8255C212yesmaximal, complemented
12#3169C2211yescharacteristic
13#3035C2212yes—
14#2965C2211yescharacteristic
15C8×#41C2211yescharacteristic
16#2931C2212yes—
17#2922C2211yescharacteristic
18#1980C2212yes—
19#1975—22no—
20#1950C2212yes—
21#1938—22no—
22#1876—22no—
23#1866—22no—
24#1806C2212yes—
25C4×#107C2211yescharacteristic
26#1770C2212yes—
27#1702C412yescomplemented
28#1701C412yescomplemented
29#1694C411yescharacteristic, complemented
30#1693C411yescharacteristic, complemented
31#1688C411yescharacteristic, complemented
32#1687C411yescharacteristic, complemented
33#1594—21no—
34#1594—21no—
35#1580—21no—
36#1580—21no—
37#1566C2211yescharacteristic
38C2×#334C2211yescharacteristic
39#1540C2212yes—
40#1502C2212yes—
41#1492C2212yes—
42#1454—21no—
43#1453—21no—
44#1452—21no—
45#1451C2211yescharacteristic
46#1450C2211yescharacteristic
47C8×M16—21no—
48C8×M16C2211yescharacteristic
49C2×#322C2211yescharacteristic
50#1303C2212yes—
51#1301C2212yes—
52#1279C2212yes—
53C2×#135—22no—
54C2×#131—22no—
55#532C2312yes—
56C4×#41C2311yescharacteristic
57#500C2×C411yescharacteristic
58#493C2×C411yescharacteristic
59#486C2×C412yes—
60#466—21no—
61#466—21no—
62#464—22no—
63#463—22no—
64#462C2×C412yes—
65#461C2×C412yes—
66#460C2×C412yes—
67#459C2×C412yes—
68#446C2×C411yescharacteristic
69#446C2×C411yescharacteristic
70C8×Q8—21no—
71C4×D8—21no—
72C4×D8—21no—
73#436—22no—
74#436—22no—
75#436C2×C411yescharacteristic
76#436C2×C411yescharacteristic
77#435—21no—
78#435—21no—
79C8×D4—21no—
80C8×D4—22no—
81C8×D4—22no—
82C8×D4—22no—
83#433—22no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90C2×#108—22no—
91C2×#107—21no—
92#424D412yes—
93#423C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—22no—
98#416—22no—
99#408—22no—
100#408D412yes—
101C4×M16—21no—
102C4×M16C2311yescharacteristic
103C2×#97—22no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C4×C4⋊C4C2312yes—
109#341—22no—
110#340—21no—
111#339—22no—
112#336—22no—
113#334—21no—
114#334—21no—
115#334C2×C411yescharacteristic
116#334C2×C411yescharacteristic
117#334C2×C412yes—
118#332C2×C412yes—
119#332C2×C412yes—
120#331C2×C412yes—
121#331C2×C412yes—
122#327—24no—
123D4⋊C8—22no—
124D4⋊C8—22no—
125#322—21no—
126#322—21no—
127#322—21no—
128#322—21no—
129#322D411yescharacteristic
130#322D411yescharacteristic
131#322D411yescharacteristic
132#322D411yescharacteristic
133C82—21no—
134C82—21no—
135C2×#41—22no—
136C2×#41C2×D411yescharacteristic
137#135—82no—
138#135—82no—
139C2×Q16—22no—
140C2×SD16—22no—
141C2×SD16—22no—
142C2×D8—22no—
143C2×D8—21no—
144C2×D8—21no—
145#131—82no—
146#131—82no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—42no—
151C2×M16C2×D412yes—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—42no—
156C22×C8C2×D412yes—
157C22×C8#4111yescharacteristic
158C22×C8#4111yescharacteristic
159C4⋊Q8C22×C412yes—
160#127C22×C412yes—
161#124—22no—
162#121—22no—
163#121—22no—
164C4×Q8C22×C411yescharacteristic
165C4×Q8#4111yescharacteristic
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4C22×C411yescharacteristic
171C4×D4#4111yescharacteristic
172#117—21no—
173#117C2×D411yescharacteristic
174C2×C4⋊C4—22no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42C2411yescharacteristic
178#108—82no—
179#107—21no—
180#107—21no—
181#107—42no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—22no—
191C4⋊C8C22×C411yescharacteristic
192C4⋊C8C22×C411yescharacteristic
193C4⋊C8C22×C412yes—
194C4⋊C8C22×C412yes—
195#104—82no—
196#104—82no—
197#104—82no—
198#103—22no—
199#103—22no—
200#102—22no—
201#102—22no—
202#102—22no—
203#102—22no—
204C22⋊C8—22no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8#4112yes—
213#97—21no—
214#97—21no—
215#97—21no—
216#97—82no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8C22×C411yescharacteristic
223C4×C8C22×C411yescharacteristic
224C4×C8C2×D411yescharacteristic
225C4×C8C2×D411yescharacteristic
226C4×C8C2×D411yescharacteristic
227C4×C8C2×D411yescharacteristic
228C4×C8C2×D411yescharacteristic
229C4×C8C2×D411yescharacteristic
230C4×C8#4112yes—
231#95—22no—
232#41—22no—
233#41—22no—
234#41—42no—
235#41—42no—
236#41—42no—
237#41—42no—
238C2×Q8—22no—
239C2×Q8C4×D411yescharacteristic
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—42no—
245C2×D4C4×D411yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254Q16—82no—
255SD16—82no—
256SD16—82no—
257D8—42no—
258D8—42no—
259D8—82no—
260M16—22no—
261M16—22no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—84no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—84no—
283C2×C8—81no—
284C2×C8—82no—
285C2×C8C4×D411yescharacteristic
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C2×C8C4×D411yescharacteristic
289C2×C8#12112yes—
290C2×C8#12112yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4C4×D411yescharacteristic
298C4⋊C4C4×D411yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—42no—
308C42—42no—
309C42C23×C411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C23—21no—
314C23—42no—
315C23#53211yescharacteristic
316Q8—22no—
317Q8—42no—
318Q8—42no—
319D4—22no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—42no—
326D4—82no—
327D4—82no—
328D4—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4C2×C4×D411yescharacteristic
346C2×C4C2×C4×D411yescharacteristic
347C2×C4C4×#4111yescharacteristic
348C2×C4C2×#12111yescharacteristic
349C2×C4#52511yescharacteristic
350C2×C4#52911yescharacteristic
351C2×C4#57612yes—
352C2×C4#57712yes—
353C8—21no—
354C8—21no—
355C8—21no—
356C8—22no—
357C8—22no—
358C8—42no—
359C8—41no—
360C8—41no—
361C8—42no—
362C8—42no—
363C8—41no—
364C8—41no—
365C8—42no—
366C8—82no—
367C8#44412yes—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22C4×#12111yescharacteristic
377C22#317311yescharacteristic
378C22#317611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—162no—
387C4#294612yes—
388C4C2×#44411yescharacteristic
389C4#295511yescharacteristic
390C4#306212yes—
391C2—21no—
392C2—21no—
393C2—81no—
394C2—162no—
395C2#2151811yescharacteristic
396C2#2154111yescharacteristic
397C2#2155911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19157C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22166C2semidirect
3. #17962C2semidirect
4. #17331C2semidirect
5. #9719C2semidirect
6. #9623C2semidirect
7. #9592C2semidirect
8. #9208C2semidirect
9. #8668C2semidirect
10. #8287C2semidirect
11. #8255C2semidirect
27. #1702C4semidirect
28. #1701C4semidirect
29. #1694C4semidirect
30. #1693C4semidirect
31. #1688C4semidirect
32. #1687C4semidirect

Maximal subgroups

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

subgroupindex
#221662
#191572
#179622
#173312
#97192
#96232
#95922
#92082
#86682
#82872
#82552