Finite Group Databasebeta

← #219672

Subgroups — #219672

Order 512.

479 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19670C211yesmaximal, characteristic
2#17990C211yesmaximal, characteristic, complemented
3#17989C211yesmaximal, characteristic, complemented
4#9766C211yesmaximal, characteristic, complemented
5#9764C211yesmaximal, characteristic, complemented
6#9285C211yesmaximal, characteristic, complemented
7#9284C211yesmaximal, characteristic, complemented
8#9283C211yesmaximal, characteristic, complemented
9#9283C211yesmaximal, characteristic, complemented
10#9282C211yesmaximal, characteristic, complemented
11#9281C211yesmaximal, characteristic, complemented
12#9281C211yesmaximal, characteristic, complemented
13#9279C211yesmaximal, characteristic, complemented
14#8796C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#3228C2211yescharacteristic
17#3210C2211yescharacteristic
18#3207C2211yescharacteristic
19#3207C2211yescharacteristic
20#3169C2211yescharacteristic
21#3167C2211yescharacteristic
22#2985C2211yescharacteristic
23#2981C2211yescharacteristic
24#2981C2211yescharacteristic
25#2981C2211yescharacteristic
26#1746C2211yescharacteristic
27#1723C411yescharacteristic, complemented
28#1721C411yescharacteristic, complemented
29#1715C411yescharacteristic, complemented
30#1714C411yescharacteristic, complemented
31#1711C411yescharacteristic, complemented
32#1709C411yescharacteristic, complemented
33#1708C411yescharacteristic, complemented
34#1706C411yescharacteristic, complemented
35#1588—21no—
36#1586—21no—
37#1586—21no—
38#1585—21no—
39#1583—21no—
40#1582—21no—
41#1581—21no—
42#1581—21no—
43#1568C2211yescharacteristic
44#1568C2211yescharacteristic
45#1531C2211yescharacteristic
46#1530C2211yescharacteristic
47#1530C2211yescharacteristic
48#1530C2211yescharacteristic
49#1506C2211yescharacteristic
50#1505C2211yescharacteristic
51#1505C2211yescharacteristic
52#1505C2211yescharacteristic
53#1497C2211yescharacteristic
54#1496C2211yescharacteristic
55#1496C2211yescharacteristic
56#1496C2211yescharacteristic
57#1452C2211yescharacteristic
58#1450C2211yescharacteristic
59#1287C2211yescharacteristic
60#1287C2211yescharacteristic
61#1287C2211yescharacteristic
62#1287C2211yescharacteristic
63#1275C2211yescharacteristic
64#1275C2211yescharacteristic
65#1274C2211yescharacteristic
66#1274C2211yescharacteristic
67#533C2311yescharacteristic
68#532C2311yescharacteristic
69#532C2311yescharacteristic
70#532C2311yescharacteristic
71#500C2×C411yescharacteristic
72#500C2×C411yescharacteristic
73#498C2×C411yescharacteristic
74#498C2×C411yescharacteristic
75#494C2×C411yescharacteristic
76#493C2×C411yescharacteristic
77#493C2×C411yescharacteristic
78#493C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#492C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#492C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#487C2×C411yescharacteristic
86#487C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#486C2×C411yescharacteristic
89#486C2×C411yescharacteristic
90#486C2×C411yescharacteristic
91#486C2×C411yescharacteristic
92#486C2×C411yescharacteristic
93#486C2×C411yescharacteristic
94#486C2×C411yescharacteristic
95#482—41no—
96#478—41no—
97#466—21no—
98#466—21no—
99#465—21no—
100#465—21no—
101#464—41no—
102#463—41no—
103#463—41no—
104#452—41no—
105#449—41no—
106#449—41no—
107#446—21no—
108#446—41no—
109C8×Q8—21no—
110C4×SD16—21no—
111C4×SD16—21no—
112C4×D8—21no—
113C4×D8—21no—
114#436—21no—
115#436—21no—
116#436—21no—
117#435—41no—
118#435—41no—
119#435—41no—
120C8×D4—21no—
121C8×D4—21no—
122C8×D4—21no—
123#433—41no—
124#432C2311yescharacteristic
125#432C2311yescharacteristic
126#432C2311yescharacteristic
127#432C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#408—41no—
133#408—41no—
134#408—41no—
135C2×#97C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#376C2311yescharacteristic
138#336—21no—
139#336—21no—
140#335—21no—
141#335—21no—
142#334—21no—
143#334—21no—
144#330—41no—
145#329—41no—
146#329—41no—
147#329—41no—
148#328—41no—
149#327—41no—
150#327—41no—
151#327—41no—
152Q8⋊C8—21no—
153Q8⋊C8—21no—
154D4⋊C8—21no—
155D4⋊C8—21no—
156D4⋊C8—21no—
157D4⋊C8—21no—
158D4⋊C8—21no—
159D4⋊C8—21no—
160#322C2×C411yescharacteristic
161#322C2×C411yescharacteristic
162#322D411yescharacteristic
163#322D411yescharacteristic
164#322D411yescharacteristic
165#322D411yescharacteristic
166C82C2×C411yescharacteristic
167C82C2×C411yescharacteristic
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×Q16—41no—
172C2×SD16—21no—
173C2×SD16—21no—
174C2×SD16—41no—
175C2×SD16—41no—
176C2×SD16—41no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—41no—
180C2×D8—41no—
181C2×M16—41no—
182C2×M16—41no—
183C2×M16—41no—
184C22×C8—21no—
185C22×C8—41no—
186C22×C8—41no—
187C22×C8—41no—
188C22×C8#4111yescharacteristic
189C22×C8#4111yescharacteristic
190C4⋊Q8C22×C411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194C4⋊Q8C22×C411yescharacteristic
195#127C22×C411yescharacteristic
196#127C22×C411yescharacteristic
197#127C22×C411yescharacteristic
198#125—21no—
199#124—21no—
200#124—21no—
201#124—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218#117—21no—
219C2×C42—21no—
220C2×C42C2411yescharacteristic
221#107—21no—
222#107—21no—
223#107—41no—
224#106—21no—
225#106—21no—
226#106—41no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237#103—21no—
238#103—21no—
239#103—41no—
240#103—41no—
241#103—41no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—21no—
246#102—21no—
247#102—21no—
248#102—41no—
249#102—41no—
250#102—41no—
251#102—41no—
252#102—41no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267#97—21no—
268#97—21no—
269#97—21no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97—41no—
274C4×C8—41no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8C22×C411yescharacteristic
278C4×C8C22×C411yescharacteristic
279C4×C8C22×C411yescharacteristic
280C4×C8C22×C411yescharacteristic
281C4×C8C2×D411yescharacteristic
282C4×C8C2×D411yescharacteristic
283C4×C8C2×D411yescharacteristic
284C4×C8C2×D411yescharacteristic
285C4×C8C2×D411yescharacteristic
286C4×C8C2×D411yescharacteristic
287#95—21no—
288#95—21no—
289#41—161no—
290#41—161no—
291#41—161no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×#4111yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316Q16—161no—
317SD16—81no—
318SD16—81no—
319SD16—161no—
320SD16—161no—
321SD16—161no—
322D8—81no—
323D8—81no—
324D8—161no—
325D8—161no—
326M16—161no—
327M16—161no—
328M16—161no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—161no—
347C2×C8—161no—
348C2×C8—161no—
349C2×C8C4×D411yescharacteristic
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8C4×D411yescharacteristic
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42C23×C411yescharacteristic
376C42C22×D411yescharacteristic
377C42H(5,𝔽2)11yescharacteristic
378C42H(5,𝔽2)11yescharacteristic
379C23—41no—
380C23—41no—
381C23—41no—
382C23#54011yescharacteristic
383Q8—41no—
384Q8—41no—
385Q8—81no—
386Q8—81no—
387Q8—81no—
388Q8—81no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—161no—
402D4—161no—
403D4—161no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4C2×C4×D411yescharacteristic
423C2×C4#51811yescharacteristic
424C2×C4#51811yescharacteristic
425C2×C4#53511yescharacteristic
426C2×C4#53511yescharacteristic
427C2×C4#54011yescharacteristic
428C8—21no—
429C8—21no—
430C8—21no—
431C8—21no—
432C8—41no—
433C8—41no—
434C8—41no—
435C8—41no—
436C8—41no—
437C8—41no—
438C8—41no—
439C8—41no—
440C8—81no—
441C8—81no—
442C8—81no—
443C8—81no—
444C8—81no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22#232811yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—161no—
469C4#295711yescharacteristic
470C4#295711yescharacteristic
471C2—41no—
472C2—161no—
473C2—161no—
474C2—161no—
475C2#2228211yescharacteristic
476C2#2228411yescharacteristic
477C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#19670C2
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. #17990C2semidirect
3. #17989C2semidirect
4. #9766C2semidirect
5. #9764C2semidirect
6. #9285C2semidirect
7. #9284C2semidirect
8. #9283C2semidirect
9. #9283C2semidirect
10. #9282C2semidirect
11. #9281C2semidirect
12. #9281C2semidirect
13. #9279C2semidirect
14. #8796C2semidirect
15. #8141C2semidirect
27. #1723C4semidirect
28. #1721C4semidirect
29. #1715C4semidirect
30. #1714C4semidirect
31. #1711C4semidirect
32. #1709C4semidirect
33. #1708C4semidirect
34. #1706C4semidirect

Maximal subgroups

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

subgroupindex
#196702
#179902
#179892
#97662
#97642
#92852
#92842
#92832
#92822
#92812
#92792
#87962
#81412