Finite Group Databasebeta

← #6477759

Subgroups — #6477759

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic
2C2×#3094C212yesmaximal, complemented
3#60124C211yesmaximal, characteristic, complemented
4#33125C214yesmaximal, complemented
5#29791C214yesmaximal, complemented
6#24623C218yesmaximal, complemented
7#24520C218yesmaximal, complemented
8#21949C211yesmaximal, characteristic, complemented
9#21752C212yesmaximal, complemented
10C22×#121C2212yes—
11C22×#117C2211yescharacteristic
12#3192—22no—
13#3192—22no—
14#3192—24no—
15#3094—24no—
16#3094—22no—
17#3094—22no—
18#3094C2218yes—
19#3094C2214yescomplemented
20#3094C2214yescomplemented
21C2×#481—22no—
22C2×#481C2214yes—
23C2×#482—24no—
24C2×#480—22no—
25C2×#469C2214yes—
26C2×#447C2214yes—
27#2924—24no—
28#2924C2218yescomplemented
29#2921—22no—
30#2920—22no—
31C2×#428C2214yescomplemented
32#2905C2212yescomplemented
33C2×#418C2214yescomplemented
34C2×#407C2212yes—
35C2×#407C2211yescharacteristic
36C2×#407C2211yescharacteristic
37#2873C2212yescomplemented
38#2090C2218yescomplemented
39#2063C22116yescomplemented
40#2045C2218yescomplemented
41#1978C2218yescomplemented
42#1951C2218yescomplemented
43#1931C22116yescomplemented
44#1900C2218yes—
45#1900C2218yescomplemented
46#1855C2218yescomplemented
47#1821C2214yescomplemented
48#1811C2214yescomplemented
49#1783C2214yescomplemented
50C23×D4—22no—
51C23×D4C2312yes—
52C24×C4C2311yescharacteristic
53C22×D8C2314yes—
54C22×M16C2312yes—
55C22×M16C2314yes—
56C22×M16C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2314yes—
68C2×#121C2314yes—
69C2×#117—21no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—21no—
73C2×#117D414yes—
74C2×#117D412yes—
75C2×#117C2314yes—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83#482—28no—
84#482—28no—
85#482—28no—
86#482—28no—
87#481—28no—
88#481—24no—
89#481—24no—
90#481—28no—
91#481—28no—
92#480—28no—
93#480—24no—
94#480—24no—
95#469—28no—
96#469—28no—
97#447—28no—
98#447—28no—
99#433—216no—
100#432—28no—
101#431—28no—
102C2×#108C2318yes—
103#428—24no—
104#428—24no—
105#428C2318yes—
106#428C2318yes—
107C2×#107—24no—
108C2×#106—24no—
109C2×#106C2314yes—
110#424—24no—
111#424C2318yes—
112#423—24no—
113#418—24no—
114#418—24no—
115#418C2318yes—
116#418C2318yes—
117C2×#102—24no—
118C2×#102—24no—
119C2×#102C2318yes—
120C2×#100C23116yes—
121#407—22no—
122#407—22no—
123#407—21no—
124#407—21no—
125#407D412yes—
126#407D412yes—
127#407D414yes—
128#407C2314yes—
129#407C2314yes—
130#407C2314yes—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8C2314yes—
135#405—24no—
136#405C2318yes—
137C4×M16—24no—
138#343C23116yes—
139#341C2318yes—
140C25—22no—
141C25#4111yescharacteristic
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—48no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—88no—
155C22×D4C2414yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—44no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4112yes—
166C23×C4#4111yescharacteristic
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C2×D8—28no—
170C2×D8—48no—
171C2×M16—22no—
172C2×M16—24no—
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—48no—
176C2×M16C2×D414yes—
177C2×M16C2×D414yes—
178C2×M16C2418yes—
179C2×M16C2414yes—
180C22×C8—22no—
181C22×C8—24no—
182C22×C8C2414yes—
183#121—24no—
184#121—24no—
185#121—24no—
186#121—24no—
187#121—24no—
188#121—24no—
189#121—24no—
190#121—24no—
191#121—42no—
192#121—42no—
193#121—42no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—42no—
199#117—24no—
200#117—22no—
201#117—22no—
202#117—24no—
203#117—21no—
204#117—21no—
205#117—24no—
206#117—21no—
207#117—21no—
208#117—24no—
209#117C2×D414yes—
210#117C2×D414yes—
211#117C2×D412yes—
212#117C2×D412yes—
213#117C2412yes—
214#117C2412yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—24no—
244#108—48no—
245#107—28no—
246#107—28no—
247#106—28no—
248#106—28no—
249#106—28no—
250C4⋊C8—216no—
251#102—216no—
252#102—28no—
253#102—28no—
254#102—28no—
255#102—28no—
256#100—416no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8C2×D418yes—
264C22⋊C8C2×D418yes—
265#97—28no—
266C4×C8—28no—
267C24—24no—
268C24—42no—
269C24—44no—
270C24—42no—
271C24—42no—
272C24—84no—
273C24—84no—
274C24—84no—
275C24#12111yescharacteristic
276C24#12112yes—
277C24#12111yescharacteristic
278C24C2×#4112yes—
279C24C2×#4111yescharacteristic
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—216no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—84no—
294C2×D4—88no—
295C2×D4—88no—
296C2×D4—88no—
297C2×D4—88no—
298C2×D4—88no—
299C2×D4—88no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4#12112yes—
320C22×C4#12111yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12111yescharacteristic
325C22×C4#12112yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2511yescharacteristic
337D8—88no—
338M16—28no—
339M16—216no—
340M16—44no—
341M16—44no—
342M16—48no—
343M16—44no—
344C2×C8—28no—
345C2×C8—24no—
346C2×C8—28no—
347C2×C8—48no—
348C2×C8—44no—
349C2×C8C22×D418yes—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4C22×D414yes—
359C4⋊C4C22×D414yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C42—22no—
371C42—22no—
372C42—24no—
373C42—24no—
374C42—22no—
375C42—22no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—48no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23—84no—
397C23—88no—
398C23—84no—
399C23—1616no—
400C23C2×#12112yes—
401C23C2×#12111yescharacteristic
402C23C2×#12111yescharacteristic
403C23C2×#12112yes—
404C23C22×#4111yescharacteristic
405D4—48no—
406D4—48no—
407D4—84no—
408D4—84no—
409D4—88no—
410D4—88no—
411D4—88no—
412D4—88no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12112yes—
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12112yes—
444C2×C4C23×D411yescharacteristic
445C2×C4C23×D411yescharacteristic
446C2×C4#58514yes—
447C8—28no—
448C8—48no—
449C8—44no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—44no—
461C22—48no—
462C22—82no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—168no—
469C22—168no—
470C22—168no—
471C22#306814yes—
472C22C22×#12111yescharacteristic
473C22#349112yes—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4#344912yes—
482C4#348512yes—
483C2—22no—
484C2—24no—
485C2—41no—
486C2—42no—
487C2—41no—
488C2—164no—
489C2#3508911yescharacteristic
490C2C2×#306812yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3192C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3094C2semidirect
3. #60124C2semidirect
4. #33125C2semidirect
5. #29791C2semidirect
6. #24623C2semidirect
7. #24520C2semidirect
8. #21949C2semidirect
9. #21752C2semidirect
19. #3094C22semidirect
20. #3094C22semidirect
28. #2924C22semidirect
31. C2×#428C22semidirect
32. #2905C22semidirect
33. C2×#418C22semidirect
37. #2873C22semidirect
38. #2090C22semidirect
39. #2063C22semidirect
40. #2045C22semidirect
41. #1978C22semidirect
42. #1951C22semidirect
43. #1931C22semidirect
45. #1900C22semidirect
46. #1855C22semidirect
47. #1821C22semidirect
48. #1811C22semidirect
49. #1783C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C2×#30942
#601242
#331252
#297912
#246232
#245202
#219492
#217522