Finite Group Databasebeta

← #213077

Subgroups — #213077

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22221C211yesmaximal, characteristic, complemented
2#22220C211yesmaximal, characteristic
3C4×#498C211yesmaximal, characteristic, complemented
4C4×#492C211yesmaximal, characteristic
5#21647C211yesmaximal, characteristic, complemented
6#21646C211yesmaximal, characteristic
7C2×#1708C211yesmaximal, characteristic
8#9840C211yesmaximal, characteristic, complemented
9#9838C211yesmaximal, characteristic
10#9833C211yesmaximal, characteristic, complemented
11#9830C211yesmaximal, characteristic
12#9820C211yesmaximal, characteristic
13#9819C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×C4×SD16—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1968—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1956C2211yescharacteristic
38#1956C2211yescharacteristic
39#1956C2211yescharacteristic
40#1941—22no—
41#1936—22no—
42#1935—22no—
43#1928—22no—
44#1925—22no—
45#1863—22no—
46#1862—22no—
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53C4×#106—22no—
54#1763—22no—
55#1762—22no—
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1708C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×SD16—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#498—22no—
76#498C2×C414yescomplemented
77#492—22no—
78#492C2×C414yes—
79#487—22no—
80#487—22no—
81#487C2×C414yescomplemented
82#487C2×C414yescomplemented
83#486—22no—
84#486—22no—
85#486C2×C414yes—
86#486C2×C414yes—
87C4×SD16—48no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#97D412yes—
104C2×#97D412yes—
105C2×C4×C8D412yes—
106C2×C4×C8D412yes—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C43C2311yescharacteristic
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#336D412yes—
145#336D412yes—
146#322—22no—
147#322—22no—
148C82—22no—
149C82C2×C414yescomplemented
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C23×C4—41no—
157C2×SD16—28no—
158C2×SD16—44no—
159C2×SD16—44no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C22×C412yes—
178#127—21no—
179#127—21no—
180#127—21no—
181#127C22×C412yes—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×Q8—82no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—82no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222#107—82no—
223#107—82no—
224#106—28no—
225#106—44no—
226#106—44no—
227#103—28no—
228#103—28no—
229#103—44no—
230#103—44no—
231#103—82no—
232#102—28no—
233#102—28no—
234#102—44no—
235#102—44no—
236#102—82no—
237#97—24no—
238#97—24no—
239C4×C8—24no—
240C4×C8—24no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C22×C412yes—
248C4×C8C22×C412yes—
249C4×C8C22×C412yes—
250C4×C8#4112yes—
251C4×C8#4112yes—
252C4×C8#4112yes—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—41no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#12711yescharacteristic
295C22×C4#12711yescharacteristic
296C22×C4#12711yescharacteristic
297C22×C4#12711yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301SD16—416no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8C4×D414yes—
315C2×C8C4×D414yes—
316C2×C8C4×D414yes—
317C2×C8#13514yes—
318C2×C8AGL(1,ℤ/8)14yes—
319C2×C8#13714yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—82no—
356C42—82no—
357C42—82no—
358C42#13514yes—
359C42C23×C411yescharacteristic
360C42C2×#4111yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C23—42no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23C2×#12711yescharacteristic
371Q8—44no—
372Q8—44no—
373Q8—44no—
374Q8—82no—
375Q8—82no—
376Q8—82no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—82no—
381D4—82no—
382D4—82no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—164no—
415C2×C4#49512yes—
416C2×C4#49512yes—
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×C4×D411yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4#53211yescharacteristic
421C2×C4#53211yescharacteristic
422C2×C4#53211yescharacteristic
423C2×C4C2×#13512yes—
424C2×C4C2×#13512yes—
425C2×C4C2×AGL(1,ℤ/8)12yes—
426C2×C4C2×#13712yes—
427C2×C4#57712yes—
428C2×C4#57712yes—
429C8—24no—
430C8—24no—
431C8—24no—
432C8—42no—
433C8—42no—
434C8#44018yes—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22C4×#12711yescharacteristic
443C22C2×#49511yescharacteristic
444C22#314711yescharacteristic
445C22#315111yescharacteristic
446C22#315211yescharacteristic
447C22#315411yescharacteristic
448C22#315411yescharacteristic
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4C4×#13512yes—
461C4C2×#44012yes—
462C4#295212yes—
463C4#316812yes—
464C4#317312yes—
465C4#318212yes—
466C2—162no—
467C2#1962511yescharacteristic
468C2#1962611yescharacteristic
469C2#1964111yescharacteristic
470C2#1964211yescharacteristic
471C2C4×#49511yescharacteristic
472C2#2169211yescharacteristic
473C2#2169411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22221C2semidirect
3. C4×#498C2semidirect
5. #21647C2semidirect
8. #9840C2semidirect
10. #9833C2semidirect
13. #9819C2semidirect
14. #8814C2semidirect
15. C8×#97C2semidirect
59. #1708C4semidirect
76. #498C2×C4semidirect
81. #487C2×C4semidirect
82. #487C2×C4semidirect
149. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222212
#222202
C4×#4982
C4×#4922
#216472
#216462
C2×#17082
#98402
#98382
#98332
#98302
#98202
#98192
#88142
C8×#972