Finite Group Databasebeta

← #211942

Subgroups — #211942

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128C211yesmaximal, characteristic
2#17956C211yesmaximal, characteristic, complemented
3#17940C212yesmaximal, complemented
4#17328C211yesmaximal, characteristic, complemented
5#9724C211yesmaximal, characteristic, complemented
6#9634C211yesmaximal, characteristic, complemented
7#9588C212yesmaximal, complemented
8#9213C212yesmaximal, complemented
9#8671C211yesmaximal, characteristic, complemented
10#8293C211yesmaximal, characteristic, complemented
11#8243C212yesmaximal, complemented
12#3169C2211yescharacteristic
13#3106C2212yes—
14#2965C2211yescharacteristic
15C8×#41C2211yescharacteristic
16#2931C2212yes—
17#2922C2211yescharacteristic
18C4×#108C2211yescharacteristic
19#1678C412yes—
20#1677C412yes—
21#1676C411yescharacteristic
22#1675C411yescharacteristic
23#1668C411yescharacteristic
24#1667C411yescharacteristic
25#1594—22no—
26#1592—22no—
27#1589—21no—
28#1589—21no—
29#1582—22no—
30#1581—22no—
31C8×D8—21no—
32C8×D8—21no—
33#1574C2212yes—
34#1567C2211yescharacteristic
35#1564C2211yescharacteristic
36#1543C2212yes—
37#1525C2212yes—
38#1492C2212yes—
39#1489C2212yes—
40#1484C2212yes—
41#1454—21no—
42#1453—21no—
43#1452C2211yescharacteristic
44#1451C2212yes—
45#1451C2211yescharacteristic
46#1450—21no—
47C8×M16—21no—
48C8×M16C2211yescharacteristic
49C2×C82C2211yescharacteristic
50#1299C2212yes—
51#1299C2212yes—
52#1278C2212yes—
53C2×#131—22no—
54#532C2312yes—
55C4×#41C2311yescharacteristic
56#500C2×C411yescharacteristic
57#493C2×C411yescharacteristic
58#486C2×C412yes—
59#482—22no—
60#477C2×C412yes—
61#476C2×C412yes—
62#466—21no—
63#466—21no—
64#460C2×C412yes—
65#459C2×C412yes—
66#449—22no—
67#446D411yescharacteristic
68#446D411yescharacteristic
69C8×Q8C2×C411yescharacteristic
70C8×Q8C2×C411yescharacteristic
71C4×SD16—22no—
72C4×D8—22no—
73C4×D8—21no—
74C4×D8—21no—
75#436—22no—
76#436—22no—
77#436D411yescharacteristic
78#436D411yescharacteristic
79#435—21no—
80#435—21no—
81C8×D4—21no—
82C8×D4—21no—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4C2×C411yescharacteristic
86C8×D4C2×C411yescharacteristic
87#433—22no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431C2311yescharacteristic
93#431C2312yes—
94#431C2311yescharacteristic
95C2×#108—21no—
96#424—21no—
97#423C2312yes—
98#408—21no—
99#408—22no—
100C4×M16—21no—
101C4×M16C2312yes—
102C4×M16C2311yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#341—21no—
108#335—22no—
109#335—21no—
110#335—21no—
111#335C2×C411yescharacteristic
112#335C2×C411yescharacteristic
113#334—22no—
114#334C2×C412yes—
115#333C2×C412yes—
116#333C2×C412yes—
117#331C2×C412yes—
118#331C2×C412yes—
119#329—24no—
120Q8⋊C8—22no—
121D4⋊C8—22no—
122D4⋊C8—22no—
123D4⋊C8—22no—
124D4⋊C8—22no—
125D4⋊C8—22no—
126#322—22no—
127#322—22no—
128#322—21no—
129#322—21no—
130#322—21no—
131#322—21no—
132#322—21no—
133C82—21no—
134C82D411yescharacteristic
135C82D411yescharacteristic
136C82D411yescharacteristic
137C82D411yescharacteristic
138C2×#41—22no—
139C2×#41#4111yescharacteristic
140C2×SD16—22no—
141C2×D8—22no—
142C2×D8—21no—
143C2×D8—21no—
144#131—84no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—42no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—42no—
155C22×C8#4111yescharacteristic
156C22×C8#4111yescharacteristic
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×Q8C2×D411yescharacteristic
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4C22×C411yescharacteristic
171C4×D4C2×D411yescharacteristic
172#117—21no—
173#117#4111yescharacteristic
174C2×C42—21no—
175C2×C42C2411yescharacteristic
176#108—42no—
177#107—21no—
178#107—21no—
179#107—22no—
180#106—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8C22×C411yescharacteristic
190C4⋊C8C22×C411yescharacteristic
191C4⋊C8C22×C412yes—
192C4⋊C8C22×C412yes—
193C4⋊C8C2×D411yescharacteristic
194C4⋊C8C2×D411yescharacteristic
195#103—22no—
196#102—22no—
197#102—22no—
198#102—22no—
199#102—22no—
200#102—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—22no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212#97—21no—
213#97—21no—
214#97—22no—
215#97#4112yes—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8C22×C411yescharacteristic
225C4×C8C22×C411yescharacteristic
226C4×C8C2×D411yescharacteristic
227C4×C8C2×D411yescharacteristic
228C4×C8C2×D411yescharacteristic
229C4×C8C2×D411yescharacteristic
230C4×C8C2×D411yescharacteristic
231C4×C8C2×D411yescharacteristic
232C4×C8C2×D411yescharacteristic
233C4×C8C2×D411yescharacteristic
234#41—41no—
235#41—41no—
236#41—82no—
237#41—82no—
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×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253SD16—82no—
254D8—42no—
255D8—42no—
256D8—82no—
257M16—42no—
258M16—41no—
259M16—41no—
260M16—42no—
261M16—42no—
262M16—42no—
263M16—84no—
264M16—82no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—84no—
285C2×C8C4×D411yescharacteristic
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C2×C8C4×D411yescharacteristic
289C2×C8#12111yescharacteristic
290C2×C8#12111yescharacteristic
291C2×C8#12111yescharacteristic
292C2×C8#12111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
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—
307C42C23×C411yescharacteristic
308C42C22×D411yescharacteristic
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C23—21no—
312C23—42no—
313C23#52911yescharacteristic
314Q8—21no—
315Q8—21no—
316Q8—82no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—42no—
323D4—42no—
324D4—82no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4C2×C4×D411yescharacteristic
344C2×C4C2×C4×D411yescharacteristic
345C2×C4C4×#4111yescharacteristic
346C2×C4C2×#12111yescharacteristic
347C2×C4#52511yescharacteristic
348C2×C4#53211yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—41no—
356C8—41no—
357C8—42no—
358C8—41no—
359C8—41no—
360C8—42no—
361C8—42no—
362C8—41no—
363C8—41no—
364C8—82no—
365C8—82no—
366C8#44412yes—
367C8#47212yes—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22C4×#12111yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—82no—
385C4C2×#44411yescharacteristic
386C4#295511yescharacteristic
387C4#295711yescharacteristic
388C4#295711yescharacteristic
389C4C2×#47211yescharacteristic
390C4#306711yescharacteristic
391C2—41no—
392C2—81no—
393C2—162no—
394C2#2151811yescharacteristic
395C2#2154511yescharacteristic
396C2C4×#47211yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19128C2
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
2. #17956C2semidirect
3. #17940C2semidirect
4. #17328C2semidirect
5. #9724C2semidirect
6. #9634C2semidirect
7. #9588C2semidirect
8. #9213C2semidirect
9. #8671C2semidirect
10. #8293C2semidirect
11. #8243C2semidirect

Maximal subgroups

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

subgroupindex
#191282
#179562
#179402
#173282
#97242
#96342
#95882
#92132
#86712
#82932
#82432