Finite Group Databasebeta

← #319637

Subgroups — #319637

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29844C211yesmaximal, characteristic
2#27949C211yesmaximal, characteristic
3#27878C211yesmaximal, characteristic
4#25902C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic
6#21804C211yesmaximal, characteristic
7#10684C211yesmaximal, characteristic, complemented
8#10674C211yesmaximal, characteristic, complemented
9#10313C211yesmaximal, characteristic, complemented
10#10313C211yesmaximal, characteristic, complemented
11#9836C211yesmaximal, characteristic, complemented
12#9833C211yesmaximal, characteristic, complemented
13#9823C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2285C2211yescharacteristic
26#2085—22no—
27#2076—22no—
28#2068—22no—
29#2057—22no—
30#2036—22no—
31#2024—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1981—22no—
42#1956C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1921—22no—
48#1872—22no—
49#1854—22no—
50#1852—22no—
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1845C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1787—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×Q8—21no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—41no—
70C2×C4⋊Q8—41no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42C2311yescharacteristic
81#500—22no—
82#500—22no—
83#500—41no—
84#500—41no—
85#487—22no—
86#487—22no—
87#487—41no—
88#487—41no—
89#487—41no—
90#487—41no—
91#431—41no—
92#431—41no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#106—22no—
99C2×C4⋊C8D412yes—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×C22⋊C8D412yes—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8D412yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
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#340—22no—
145#340—22no—
146#336D412yes—
147#336D412yes—
148#322—41no—
149C82—41no—
150C22×Q8—24no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—42no—
156C22×Q8—41no—
157C22×Q8—82no—
158C22×Q8—82no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C4⋊Q8—24no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
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⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189C4⋊Q8#4112yes—
190C4⋊Q8#4112yes—
191C22⋊Q8—44no—
192C22⋊Q8—82no—
193#117—82no—
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—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—21no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—41no—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234#107—44no—
235#107—44no—
236#107—44no—
237#107—82no—
238#107—82no—
239#106—82no—
240C4⋊C8—42no—
241#103—44no—
242#103—44no—
243#103—44no—
244#103—82no—
245#103—82no—
246#103—82no—
247#103—82no—
248#103—82no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8Q1618yes—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24#12711yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—44no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8—88no—
291C2×Q8—88no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—44no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4#12711yescharacteristic
315C22×C4#12711yescharacteristic
316C22×C4#12711yescharacteristic
317C22×C4C2×Q1614yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8C2×Q1614yes—
338C2×C8C2×Q1614yes—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—41no—
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—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—82no—
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C42H(5,𝔽2)11yescharacteristic
384C42H(5,𝔽2)11yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23#49412yes—
392C23C2×#12711yescharacteristic
393Q8—44no—
394Q8—44no—
395Q8—88no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—88no—
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—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4#49412yes—
442C2×C4#52511yescharacteristic
443C2×C4#53411yescharacteristic
444C2×C4#54611yescharacteristic
445C2×C4#54611yescharacteristic
446C2×C4#54611yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4C22×Q1612yes—
449C2×C4#57512yes—
450C2×C4#57612yes—
451C2×C4#57712yes—
452C2×C4#57712yes—
453C2×C4#57812yes—
454C8—24no—
455C8—24no—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22#253911yescharacteristic
468C22C2×#49411yescharacteristic
469C22#315311yescharacteristic
470C22#315411yescharacteristic
471C22#315411yescharacteristic
472C22#315511yescharacteristic
473C22#315611yescharacteristic
474C4—22no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—82no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#311812yes—
486C4#319812yes—
487C4#329912yes—
488C4#329912yes—
489C4#331312yes—
490C4#332712yes—
491C2—42no—
492C2#1963411yescharacteristic
493C2#1966411yescharacteristic
494C2#1972211yescharacteristic
495C2#1973811yescharacteristic
496C2#2832111yescharacteristic
497C2#2833311yescharacteristic
498C2#2835911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10684C2semidirect
8. #10674C2semidirect
9. #10313C2semidirect
10. #10313C2semidirect
11. #9836C2semidirect
12. #9833C2semidirect
13. #9823C2semidirect
14. #9821C2semidirect

Maximal subgroups

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

subgroupindex
#298442
#279492
#278782
#259022
#258132
#218042
#106842
#106742
#103132
#98362
#98332
#98232
#98212
#81482