Finite Group Databasebeta

← #213065

Subgroups — #213065

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22268C211yesmaximal, characteristic
2#22246C211yesmaximal, characteristic, complemented
3#21727C211yesmaximal, characteristic, complemented
4C4×#498C211yesmaximal, characteristic, complemented
5#21692C211yesmaximal, characteristic
6C4×#492C211yesmaximal, characteristic
7C2×#1705C211yesmaximal, characteristic
8#9842C211yesmaximal, characteristic
9#9841C211yesmaximal, characteristic, complemented
10#9829C211yesmaximal, characteristic
11#9828C211yesmaximal, characteristic, complemented
12#9820C211yesmaximal, characteristic
13#9819C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#492C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#440—22no—
23C2×C4×SD16—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1973—22no—
35#1968—22no—
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1944—22no—
43#1941—22no—
44#1920—22no—
45#1919—22no—
46#1880—22no—
47#1877—22no—
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1776—22no—
55C4×#106—22no—
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1705C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×SD16—22no—
65C22×SD16—22no—
66C22×SD16—22no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#498—22no—
78#498—22no—
79#498—22no—
80#498C2×C414yescomplemented
81#498C2×C414yescomplemented
82#498C2×C414yescomplemented
83#492—22no—
84#492—22no—
85#492—22no—
86#492C2×C414yes—
87#492C2×C414yes—
88#492C2×C414yes—
89#440—48no—
90C4×SD16—48no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#97D412yes—
105C2×#97D412yes—
106C2×C4×C8D412yes—
107C2×C4×C8D412yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—21no—
116#391—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C43C2311yescharacteristic
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#336D412yes—
146#336D412yes—
147#322—22no—
148#322—22no—
149C82—22no—
150C82C2×C414yescomplemented
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C2×SD16—28no—
159C2×SD16—28no—
160C2×SD16—28no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—44no—
164C2×SD16—44no—
165C2×SD16—82no—
166C2×SD16—82no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184C4⋊Q8C22×C412yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8—82no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—82no—
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229#107—44no—
230#107—44no—
231#106—28no—
232#106—28no—
233#106—28no—
234#106—44no—
235#106—44no—
236#103—44no—
237#103—44no—
238#103—82no—
239#102—44no—
240#102—44no—
241#102—82no—
242#97—24no—
243#97SD1618yescomplemented
244C4×C8—24no—
245C4×C8—24no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254C4×C8C22×C412yes—
255C4×C8#4112yes—
256C4×C8#4112yes—
257C4×C8#4112yes—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—41no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4#12711yescharacteristic
300C22×C4#12711yescharacteristic
301C22×C4#12711yescharacteristic
302C22×C4#12711yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306SD16—416no—
307SD16—416no—
308SD16—88no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8C4×D414yes—
320C2×C8C4×D414yes—
321C2×C8C4×D414yes—
322C2×C8C2×SD1614yes—
323C2×C8C2×SD1614yes—
324C2×C8#13514yes—
325C2×C8AGL(1,ℤ/8)14yes—
326C2×C8#13714yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—82no—
363C42—82no—
364C42—82no—
365C42C2×SD1614yes—
366C42C23×C411yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C2×#12711yescharacteristic
378Q8—44no—
379Q8—44no—
380Q8—44no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384D4—44no—
385D4—44no—
386D4—44no—
387D4—82no—
388D4—82no—
389D4—82no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
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—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—164no—
422C2×C4#49212yes—
423C2×C4#49212yes—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4#53211yescharacteristic
428C2×C4#53211yescharacteristic
429C2×C4#53211yescharacteristic
430C2×C4C22×SD1612yes—
431C2×C4C2×#13512yes—
432C2×C4C2×AGL(1,ℤ/8)12yes—
433C2×C4C2×#13712yes—
434C2×C4#57712yes—
435C2×C4#57712yes—
436C8—24no—
437C8—24no—
438C8—42no—
439C8—42no—
440C8C4×SD1618yescomplemented
441C8#44018yescomplemented
442C22—82no—
443C22—82no—
444C22—82no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22C4×#12711yescharacteristic
450C22C2×#49211yescharacteristic
451C22#314711yescharacteristic
452C22#315111yescharacteristic
453C22#315211yescharacteristic
454C22#315411yescharacteristic
455C22#315411yescharacteristic
456C4—22no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4C2×C4×SD1612yes—
468C4C2×#44012yes—
469C4#294912yes—
470C4#316712yes—
471C4#317412yes—
472C4#318212yes—
473C2—162no—
474C2#1962111yescharacteristic
475C2#1962211yescharacteristic
476C2#1964111yescharacteristic
477C2#1964211yescharacteristic
478C2C4×#49211yescharacteristic
479C2#2169211yescharacteristic
480C2#2169511yescharacteristic
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
2. #22246C2semidirect
3. #21727C2semidirect
4. C4×#498C2semidirect
9. #9841C2semidirect
11. #9828C2semidirect
13. #9819C2semidirect
14. #8814C2semidirect
15. C8×#97C2semidirect
59. #1705C4semidirect
80. #498C2×C4semidirect
81. #498C2×C4semidirect
82. #498C2×C4semidirect
150. C82C2×C4semidirect
243. #97SD16semidirect
440. C8C4×SD16semidirect
441. C8#440semidirect

Maximal subgroups

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

subgroupindex
#222682
#222462
#217272
C4×#4982
#216922
C4×#4922
C2×#17052
#98422
#98412
#98292
#98282
#98202
#98192
#88142
C8×#972