Finite Group Databasebeta

← #326597

Subgroups — #326597

Order 512.

451 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22302C211yesmaximal, characteristic, complemented
2#22282C211yesmaximal, characteristic
3#10667C211yesmaximal, characteristic, complemented
4#10666C211yesmaximal, characteristic, complemented
5#10372C211yesmaximal, characteristic, complemented
6#10372C211yesmaximal, characteristic, complemented
7#9753C211yesmaximal, characteristic
8#9750C211yesmaximal, characteristic
9#9632C211yesmaximal, characteristic, complemented
10#9629C211yesmaximal, characteristic, complemented
11#8880C211yesmaximal, characteristic, complemented
12#8880C211yesmaximal, characteristic, complemented
13#8878C211yesmaximal, characteristic, complemented
14#8878C211yesmaximal, characteristic, complemented
15#8782C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#2932C2211yescharacteristic
18#2932C2211yescharacteristic
19#2402C2211yescharacteristic
20#1963C2211yescharacteristic
21#1963C2211yescharacteristic
22#1940—21no—
23#1940C2211yescharacteristic
24#1940C2211yescharacteristic
25#1939—21no—
26#1917—21no—
27#1916—21no—
28#1876—21no—
29#1876—21no—
30#1876—21no—
31#1876—21no—
32#1843C2211yescharacteristic
33#1837C2211yescharacteristic
34#1770C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1746C2211yescharacteristic
37#1571C2211yescharacteristic
38#1571C2211yescharacteristic
39#1570C2211yescharacteristic
40#1570C2211yescharacteristic
41#1564C2211yescharacteristic
42#1564C2211yescharacteristic
43#1502C2211yescharacteristic
44#1502C2211yescharacteristic
45#1502C2211yescharacteristic
46#1502C2211yescharacteristic
47#1480C2211yescharacteristic
48#1480C2211yescharacteristic
49#1480C2211yescharacteristic
50#1480C2211yescharacteristic
51#1303C2211yescharacteristic
52#1303C2211yescharacteristic
53#1303C2211yescharacteristic
54#1303C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1274C2211yescharacteristic
58#1274C2211yescharacteristic
59C2×#135—21no—
60C2×#135—21no—
61C2×#131—21no—
62C2×#131—21no—
63#532C2311yescharacteristic
64#532C2311yescharacteristic
65#495—21no—
66#495—21no—
67#494Q811yescharacteristic
68#493Q811yescharacteristic
69#492Q811yescharacteristic
70#492Q811yescharacteristic
71#435—41no—
72#435—41no—
73#435—41no—
74#435—41no—
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79#423C2311yescharacteristic
80#423C2311yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#104—21no—
88C2×#104—21no—
89C2×#104—21no—
90C2×#104—21no—
91#408—21no—
92#408—21no—
93C2×#97—21no—
94C2×#97C2311yescharacteristic
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97#400—21no—
98#400—21no—
99#398—21no—
100#398—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2311yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105#376C2311yescharacteristic
106#341—21no—
107#341—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336D411yescharacteristic
113#336D411yescharacteristic
114#336D411yescharacteristic
115#336D411yescharacteristic
116#335—21no—
117#335—21no—
118#335—41no—
119#335—41no—
120#334—21no—
121#334—21no—
122#334—41no—
123#334—41no—
124#327—21no—
125#327—21no—
126#327—21no—
127#327—21no—
128#327—21no—
129#327—21no—
130#327—21no—
131#327—21no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134D4⋊C8—41no—
135D4⋊C8—41no—
136C2×#41—21no—
137C2×#41—21no—
138#135—81no—
139#135—81no—
140#135—81no—
141#135—81no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×SD16—21no—
145C2×SD16—21no—
146C2×SD16—21no—
147C2×SD16—21no—
148C2×D8—21no—
149C2×D8—21no—
150#131—81no—
151#131—81no—
152#131—81no—
153#131—81no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—41no—
157C2×M16—41no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8C2×Q811yescharacteristic
167C4⋊Q8C2×Q811yescharacteristic
168#127C2×Q811yescharacteristic
169#127C2×Q811yescharacteristic
170#124—21no—
171#124—21no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—81no—
204C4⋊C8—81no—
205C4⋊C8—81no—
206C4⋊C8—81no—
207#104—81no—
208#104—81no—
209#104—81no—
210#104—81no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—42no—
216C22⋊C8—42no—
217#97—21no—
218#97—21no—
219#97—81no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—81no—
225C4×C8C2×Q811yescharacteristic
226C4×C8C2×Q811yescharacteristic
227C4×C8#4111yescharacteristic
228C4×C8#4111yescharacteristic
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95#4111yescharacteristic
236#95#4111yescharacteristic
237#95#4111yescharacteristic
238#95#4111yescharacteristic
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268Q16—81no—
269Q16—81no—
270SD16—81no—
271SD16—81no—
272SD16—81no—
273SD16—81no—
274D8—81no—
275D8—81no—
276M16—41no—
277M16—41no—
278M16—41no—
279M16—41no—
280M16—82no—
281M16—82no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8C22⋊Q811yescharacteristic
307C2×C8C22⋊Q811yescharacteristic
308C2×C8C22⋊Q811yescharacteristic
309C2×C8C22⋊Q811yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42C22×Q811yescharacteristic
333C42C2×#4111yescharacteristic
334C42H(5,𝔽2)11yescharacteristic
335C42H(5,𝔽2)11yescharacteristic
336C23—41no—
337C23—41no—
338C23#52511yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4#53511yescharacteristic
383C2×C4#55411yescharacteristic
384C2×C4#55411yescharacteristic
385C2×C4#55911yescharacteristic
386C2×C4#55911yescharacteristic
387C2×C4#57611yescharacteristic
388C2×C4#57611yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C8—21no—
392C8—21no—
393C8—21no—
394C8—21no—
395C8—41no—
396C8—41no—
397C8—81no—
398C8—81no—
399C8—81no—
400C8—81no—
401C8—81no—
402C8—81no—
403C8—81no—
404C8—81no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#258011yescharacteristic
414C22#311911yescharacteristic
415C22#311911yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—42no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4#308111yescharacteristic
429C4#308111yescharacteristic
430C4#322011yescharacteristic
431C4#322211yescharacteristic
432C2—21no—
433C2—21no—
434C2—161no—
435C2—161no—
436C2#2926511yescharacteristic
437C2#2927011yescharacteristic
438C2#2927111yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#3147C22
LCSG ▷ C2×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. #22302C2semidirect
3. #10667C2semidirect
4. #10666C2semidirect
5. #10372C2semidirect
6. #10372C2semidirect
9. #9632C2semidirect
10. #9629C2semidirect
11. #8880C2semidirect
12. #8880C2semidirect
13. #8878C2semidirect
14. #8878C2semidirect
15. #8782C2semidirect

Maximal subgroups

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

subgroupindex
#223022
#222822
#106672
#106662
#103722
#97532
#97502
#96322
#96292
#88802
#88782
#87822