Finite Group Databasebeta

← #232158

Subgroups — #232158

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19747C211yesmaximal, characteristic
2#9428C211yesmaximal, characteristic, complemented
3#9428C211yesmaximal, characteristic, complemented
4#9425C211yesmaximal, characteristic, complemented
5#9424C211yesmaximal, characteristic, complemented
6#9423C211yesmaximal, characteristic, complemented
7#9423C211yesmaximal, characteristic, complemented
8#9418C211yesmaximal, characteristic, complemented
9#9418C211yesmaximal, characteristic, complemented
10#9417C211yesmaximal, characteristic, complemented
11#9417C211yesmaximal, characteristic, complemented
12#9415C211yesmaximal, characteristic, complemented
13#9415C211yesmaximal, characteristic, complemented
14#8804C211yesmaximal, characteristic, complemented
15#8803C211yesmaximal, characteristic, complemented
16#3184C2211yescharacteristic
17#3182C2211yescharacteristic
18#3182C2211yescharacteristic
19#3151C2211yescharacteristic
20#3151C2211yescharacteristic
21#3138C2211yescharacteristic
22#2987C2211yescharacteristic
23#2980C2211yescharacteristic
24#2980C2211yescharacteristic
25#2380C2211yescharacteristic
26#1721C411yescharacteristic, complemented
27#1721C411yescharacteristic, complemented
28#1717C411yescharacteristic, complemented
29#1717C411yescharacteristic, complemented
30#1714C411yescharacteristic, complemented
31#1714C411yescharacteristic, complemented
32#1706C411yescharacteristic, complemented
33#1706C411yescharacteristic, complemented
34#1555C2211yescharacteristic
35#1555C2211yescharacteristic
36#1555C2211yescharacteristic
37#1537C2211yescharacteristic
38#1536C2211yescharacteristic
39#1536C2211yescharacteristic
40#1515C2211yescharacteristic
41#1515C2211yescharacteristic
42#1514C2211yescharacteristic
43#1514C2211yescharacteristic
44#1514C2211yescharacteristic
45#1514C2211yescharacteristic
46#1455C2211yescharacteristic
47#1291C2211yescharacteristic
48#1291C2211yescharacteristic
49#1291C2211yescharacteristic
50#1291C2211yescharacteristic
51#1291C2211yescharacteristic
52#1291C2211yescharacteristic
53#1288C2211yescharacteristic
54#1288C2211yescharacteristic
55#1288C2211yescharacteristic
56#1288C2211yescharacteristic
57#1288C2211yescharacteristic
58#1288C2211yescharacteristic
59C2×AGL(1,ℤ/8)—41no—
60C2×AGL(1,ℤ/8)—41no—
61#533C2311yescharacteristic
62#532C2311yescharacteristic
63#532C2311yescharacteristic
64C2×#127C2311yescharacteristic
65#500C2×C411yescharacteristic, complemented
66#500C2×C411yescharacteristic, complemented
67#498C2×C411yescharacteristic, complemented
68#496—41no—
69#496—41no—
70#493C2×C411yescharacteristic
71#493C2×C411yescharacteristic
72#493C2×C411yescharacteristic
73#493C2×C411yescharacteristic
74#493C2×C411yescharacteristic, complemented
75#492C2×C411yescharacteristic
76#492C2×C411yescharacteristic
77#492C2×C411yescharacteristic
78#492C2×C411yescharacteristic
79#492C2×C411yescharacteristic, complemented
80#492C2×C411yescharacteristic, complemented
81#490—41no—
82#487C2×C411yescharacteristic, complemented
83#487C2×C411yescharacteristic, complemented
84#486C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#486C2×C411yescharacteristic, complemented
89#486C2×C411yescharacteristic, complemented
90#486C2×C411yescharacteristic, complemented
91#486C2×C411yescharacteristic, complemented
92#469—41no—
93#468—41no—
94#468—41no—
95#446—41no—
96#446—41no—
97#442—41no—
98#440—41no—
99#440—41no—
100#435—41no—
101#435—41no—
102#435—41no—
103#435—41no—
104#433—41no—
105#432C2311yescharacteristic
106#432C2311yescharacteristic
107#432C2311yescharacteristic
108#432C2311yescharacteristic
109#432C2311yescharacteristic
110#432C2311yescharacteristic
111#418—41no—
112#408—41no—
113#408—41no—
114C4×M16C2311yescharacteristic
115C4×M16C2311yescharacteristic
116C4×M16C2311yescharacteristic
117#386—41no—
118#386—41no—
119#386—41no—
120#383C2311yescharacteristic
121#383C2311yescharacteristic
122#331—41no—
123#331—41no—
124#331—41no—
125#330—41no—
126#329—41no—
127#329—41no—
128#328—41no—
129#328—41no—
130#327—41no—
131#327—41no—
132#327—41no—
133#327—41no—
134#322C2×C411yescharacteristic, complemented
135#322C2×C411yescharacteristic, complemented
136#322C2×C411yescharacteristic, complemented
137C82C2×C411yescharacteristic, complemented
138C2×#41—21no—
139C2×#41—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C23×C4—41no—
144AGL(1,ℤ/8)—82no—
145AGL(1,ℤ/8)—82no—
146AGL(1,ℤ/8)—82no—
147AGL(1,ℤ/8)—82no—
148C2×SD16—41no—
149C2×SD16—41no—
150C2×SD16—41no—
151C2×SD16—41no—
152C2×SD16—41no—
153C2×SD16—41no—
154C2×D8—41no—
155C2×D8—41no—
156C2×D8—41no—
157C2×D8—41no—
158C2×D8—41no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—41no—
165C22×C8—41no—
166C4⋊Q8C22×C411yescharacteristic
167C4⋊Q8C22×C411yescharacteristic
168C4⋊Q8C22×C411yescharacteristic
169C4⋊Q8C22×C411yescharacteristic
170#127—21no—
171#127—21no—
172#127—21no—
173#127C22×C411yescharacteristic
174#127C22×C411yescharacteristic
175#127C22×C411yescharacteristic
176#127C22×C411yescharacteristic
177#125—21no—
178#124—21no—
179#124—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192#117—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42C2411yescharacteristic
197#107—41no—
198#107—41no—
199#106—41no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—42no—
207#103—41no—
208#103—41no—
209#102—41no—
210#102—41no—
211#102—41no—
212#102—41no—
213#102—41no—
214#102—41no—
215#102—41no—
216#102—41no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#97—21no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97—21no—
230#97—21no—
231#97—21no—
232#97—21no—
233#97—21no—
234#97—41no—
235C4×C8—41no—
236C4×C8C22×C411yescharacteristic
237C4×C8C22×C411yescharacteristic
238C4×C8C22×C411yescharacteristic
239C4×C8C22×C411yescharacteristic
240C4×C8C22×C411yescharacteristic
241C4×C8C22×C411yescharacteristic
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248C24—41no—
249#41—81no—
250#41—81no—
251#41—81no—
252#41—81no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—161no—
285C22×C4—161no—
286C22×C4—161no—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C22×C4H(5,𝔽2)11yescharacteristic
290SD16—82no—
291SD16—82no—
292SD16—82no—
293SD16—82no—
294SD16—82no—
295SD16—82no—
296D8—82no—
297D8—82no—
298D8—82no—
299D8—82no—
300D8—82no—
301M16—42no—
302M16—42no—
303M16—81no—
304M16—161no—
305M16—161no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—161no—
319C2×C8—161no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—161no—
336C22⋊C4—161no—
337C22⋊C4—161no—
338C42—41no—
339C42C23×C411yescharacteristic
340C42H(5,𝔽2)11yescharacteristic
341C42H(5,𝔽2)11yescharacteristic
342C42H(5,𝔽2)11yescharacteristic
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23#56011yescharacteristic
354Q8—81no—
355Q8—81no—
356Q8—81no—
357Q8—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—161no—
371D4—161no—
372D4—161no—
373D4—161no—
374D4—161no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—164no—
393C2×C4#51811yescharacteristic
394C2×C4#51811yescharacteristic
395C2×C4#51811yescharacteristic
396C2×C4#56011yescharacteristic
397C2×C4#56011yescharacteristic
398C2×C4#56011yescharacteristic
399C8—22no—
400C8—22no—
401C8—41no—
402C8—41no—
403C8—41no—
404C8—41no—
405C8—81no—
406C8—81no—
407C8—81no—
408C8—81no—
409C8—81no—
410C8—81no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22#238011yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—161no—
442C4—161no—
443C2—41no—
444C2—161no—
445C2—161no—
446C2—161no—
447C2—161no—
448C2#2336111yescharacteristic
449C2#2336111yescharacteristic
450C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19747C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9428C2semidirect
3. #9428C2semidirect
4. #9425C2semidirect
5. #9424C2semidirect
6. #9423C2semidirect
7. #9423C2semidirect
8. #9418C2semidirect
9. #9418C2semidirect
10. #9417C2semidirect
11. #9417C2semidirect
12. #9415C2semidirect
13. #9415C2semidirect
14. #8804C2semidirect
15. #8803C2semidirect
26. #1721C4semidirect
27. #1721C4semidirect
28. #1717C4semidirect
29. #1717C4semidirect
30. #1714C4semidirect
31. #1714C4semidirect
32. #1706C4semidirect
33. #1706C4semidirect
65. #500C2×C4semidirect
66. #500C2×C4semidirect
67. #498C2×C4semidirect
74. #493C2×C4semidirect
79. #492C2×C4semidirect
80. #492C2×C4semidirect
82. #487C2×C4semidirect
83. #487C2×C4semidirect
88. #486C2×C4semidirect
89. #486C2×C4semidirect
90. #486C2×C4semidirect
91. #486C2×C4semidirect
134. #322C2×C4semidirect
135. #322C2×C4semidirect
136. #322C2×C4semidirect
137. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#197472
#94282
#94252
#94242
#94232
#94182
#94172
#94152
#88042
#88032