Finite Group Databasebeta

← #276708

Subgroups — #276708

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29998C211yesmaximal, characteristic
2#29207C211yesmaximal, characteristic, complemented
3#22796C211yesmaximal, characteristic
4#22146C211yesmaximal, characteristic
5#10619C211yesmaximal, characteristic
6#10616C211yesmaximal, characteristic
7#10476C211yesmaximal, characteristic
8#10476C211yesmaximal, characteristic
9#9900C211yesmaximal, characteristic, complemented
10#9618C211yesmaximal, characteristic, complemented
11#9615C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9573C211yesmaximal, characteristic, complemented
14#9565C211yesmaximal, characteristic, complemented
15#8586C211yesmaximal, characteristic, complemented
16C2×#461C2211yescharacteristic
17C2×#433—21no—
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2293C2211yescharacteristic
21#2088—21no—
22#2088—21no—
23#2065—21no—
24#2065—21no—
25#2059—21no—
26#2059—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1981C2211yescharacteristic
30#1968—21no—
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1949—21no—
34#1949—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1933—21no—
39#1924—21no—
40#1923—21no—
41#1880—21no—
42#1865—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1798—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#461—22no—
74#461—22no—
75#433—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—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⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#390D411yescharacteristic
112#390D411yescharacteristic
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389D411yescharacteristic
120#389D411yescharacteristic
121#388—21no—
122#388—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C43C2311yescharacteristic
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#335—22no—
149#332—22no—
150#332—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C2×SD16—24no—
155C2×SD16—81no—
156C2×SD16—81no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8#4112yes—
171#125—81no—
172#125—81no—
173C4×Q8—81no—
174C4×Q8—81no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—81no—
215#107—81no—
216#106—81no—
217#106—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221C4⋊C8—22no—
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—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237C4⋊C8SD1614yescomplemented
238C4⋊C8#4112yes—
239#103—24no—
240#103—81no—
241#103—81no—
242#102—81no—
243#102—81no—
244#102—81no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—81no—
248C22⋊C8—81no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—42no—
255C4×C8—81no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296SD16—84no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
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×C8C2×SD1612yes—
326C2×C8C2×SD1612yes—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—81no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—82no—
374C42C2×SD1612yes—
375C42#13712yes—
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C422−1+411yescharacteristic
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C2×#12111yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—81no—
391Q8—81no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4#46112yes—
434C2×C4#46112yes—
435C2×C4#52511yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4#54811yescharacteristic
439C2×C4#55211yescharacteristic
440C2×C4#55611yescharacteristic
441C2×C4C22×SD1611yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—41no—
452C8—41no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#241511yescharacteristic
467C22C2×#46111yescharacteristic
468C22#304511yescharacteristic
469C22#306111yescharacteristic
470C22#306211yescharacteristic
471C22#306711yescharacteristic
472C22#306911yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—41no—
478C4—41no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—81no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4#312011yescharacteristic
487C4#312211yescharacteristic
488C4#324111yescharacteristic
489C4#324311yescharacteristic
490C4#333511yescharacteristic
491C4#335711yescharacteristic
492C4#336211yescharacteristic
493C4#336911yescharacteristic
494C2—161no—
495C2#1919511yescharacteristic
496C2#1923211yescharacteristic
497C2#1954311yescharacteristic
498C2#1957611yescharacteristic
499C2#2468011yescharacteristic
500C2#2471311yescharacteristic
501C2#2472511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
2. #29207C2semidirect
9. #9900C2semidirect
10. #9618C2semidirect
11. #9615C2semidirect
12. #9601C2semidirect
13. #9573C2semidirect
14. #9565C2semidirect
15. #8586C2semidirect
237. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#299982
#292072
#227962
#221462
#106192
#106162
#104762
#99002
#96182
#96152
#96012
#95732
#95652
#85862