Finite Group Databasebeta

← #318860

Subgroups — #318860

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29289C211yesmaximal, characteristic, complemented
2#28286C211yesmaximal, characteristic
3#22303C211yesmaximal, characteristic, complemented
4#22219C211yesmaximal, characteristic
5#10601C211yesmaximal, characteristic, complemented
6#10599C211yesmaximal, characteristic
7#10593C211yesmaximal, characteristic, complemented
8#10591C211yesmaximal, characteristic
9#9747C211yesmaximal, characteristic
10#9747C211yesmaximal, characteristic
11#9618C211yesmaximal, characteristic, complemented
12#9617C211yesmaximal, characteristic, complemented
13#8810C211yesmaximal, characteristic, complemented
14#8587C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic
16C2×#486C2211yescharacteristic
17C2×#478—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2088—21no—
22#2086—21no—
23#2067—21no—
24#2065—21no—
25#1990C2211yescharacteristic
26#1989—21no—
27#1987C2211yescharacteristic
28#1987C2211yescharacteristic
29#1982C2211yescharacteristic
30#1982C2211yescharacteristic
31#1957C2211yescharacteristic
32#1949—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1940—21no—
38#1940—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1926—21no—
42#1926—21no—
43#1917—21no—
44#1917—21no—
45#1864—21no—
46#1864—21no—
47#1844C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1842C2211yescharacteristic
51#1804—21no—
52#1770C2211yescharacteristic
53#1770C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125D411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#486—22no—
75#486Q814yes—
76#478—44no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389D411yescharacteristic
116#389D411yescharacteristic
117#387—21no—
118#387—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382D411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#335—22no—
149#335—41no—
150#335—41no—
151#333—41no—
152#333—41no—
153#331—41no—
154#331—41no—
155C22×Q8—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C23×C4—41no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Q812yes—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C2×Q812yes—
178#125—22no—
179#125—22no—
180C4×Q8—82no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—81no—
184C4×D4—81no—
185C4×D4—82no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
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—41no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—81no—
218#106—42no—
219#106—81no—
220#106—81no—
221#106—81no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242#102—24no—
243#102—24no—
244#102—42no—
245#102—42no—
246#102—81no—
247#102—81no—
248#102—81no—
249#102—81no—
250#97—81no—
251#97—81no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8C2×Q812yes—
255C4×C8#4112yes—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264C24—41no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—81no—
268C2×Q8—81no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
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—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8C22⋊Q812yes—
331C2×C8C22⋊Q812yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—82no—
375C42AGL(1,ℤ/8)12yes—
376C42#13712yes—
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C42H(5,𝔽2)11yescharacteristic
380C422−1+411yescharacteristic
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23C2×#12111yescharacteristic
389Q8—82no—
390Q8—82no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—82no—
396D4—82no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4#46812yes—
436C2×C4#46812yes—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4#53711yescharacteristic
439C2×C4D4×Q811yescharacteristic
440C2×C4#55011yescharacteristic
441C2×C4#55411yescharacteristic
442C2×C4#55611yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57811yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—22no—
453C8—41no—
454C8—41no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22#253811yescharacteristic
469C22#304511yescharacteristic
470C22#304611yescharacteristic
471C22C2×#46811yescharacteristic
472C22#306111yescharacteristic
473C22#306811yescharacteristic
474C22#306811yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4#308311yescharacteristic
489C4#308411yescharacteristic
490C4#325711yescharacteristic
491C4#325811yescharacteristic
492C4#338511yescharacteristic
493C4#338611yescharacteristic
494C4#339811yescharacteristic
495C4#339911yescharacteristic
496C2—162no—
497C2#1944511yescharacteristic
498C2#1945011yescharacteristic
499C2#1954211yescharacteristic
500C2#1955111yescharacteristic
501C2#2827211yescharacteristic
502C2#2829311yescharacteristic
503C2#2829911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×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. #29289C2semidirect
3. #22303C2semidirect
5. #10601C2semidirect
7. #10593C2semidirect
11. #9618C2semidirect
12. #9617C2semidirect
13. #8810C2semidirect
14. #8587C2semidirect

Maximal subgroups

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

subgroupindex
#292892
#282862
#223032
#222192
#106012
#105992
#105932
#105912
#97472
#96182
#96172
#88102
#85872
#85852