Finite Group Databasebeta

← #279593

Subgroups — #279593

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29986C211yesmaximal, characteristic
2#29179C211yesmaximal, characteristic, complemented
3#22701C211yesmaximal, characteristic
4#22111C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10597C211yesmaximal, characteristic, complemented
8#10593C211yesmaximal, characteristic, complemented
9#9899C211yesmaximal, characteristic, complemented
10#9603C211yesmaximal, characteristic, complemented
11#9223C211yesmaximal, characteristic
12#9215C211yesmaximal, characteristic
13#8656C211yesmaximal, characteristic
14#8644C211yesmaximal, characteristic
15#8594C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#2293C2211yescharacteristic
22#2101—21no—
23#2092—21no—
24#2087—21no—
25#2087—21no—
26#2059—21no—
27#2025—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1941—21no—
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1934—21no—
38#1926—21no—
39#1921—21no—
40#1921—21no—
41#1919—21no—
42#1851—21no—
43#1850—21no—
44#1845C2211yescharacteristic
45#1834C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1798—21no—
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#476—22no—
75#476Q814yes—
76#436—22no—
77#436—41no—
78C8×D4—22no—
79C8×D4—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#391—21no—
106#390D411yescharacteristic
107#390D411yescharacteristic
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389D411yescharacteristic
115#389D411yescharacteristic
116#388—21no—
117#388—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C43C2311yescharacteristic
127#340—21no—
128#340—21no—
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#340—21no—
140#340—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—41no—
152#333—41no—
153#333—41no—
154C22×Q8—21no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C2×M16—42no—
158C2×M16—42no—
159C2×M16—81no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—81no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C2×Q812yes—
177#125—81no—
178#125—81no—
179C4×Q8—82no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C2×Q812yes—
183C4×D4#4112yes—
184#117—81no—
185#117—81no—
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—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—42no—
220#107—42no—
221#107—81no—
222#106—24no—
223#106—81no—
224#106—81no—
225C4⋊C8—22no—
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—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—41no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8C2×Q812yes—
241#102—24no—
242#102—81no—
243#102—81no—
244#102—81no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—81no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—21no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C22⋊Q812yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290M16—84no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
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—84no—
321C2×C8#13512yes—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4C22⋊Q812yes—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—42no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—82no—
371C42C22×Q811yescharacteristic
372C42C2×#4111yescharacteristic
373C42H(5,𝔽2)11yescharacteristic
374C422−1+411yescharacteristic
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23C2×#12111yescharacteristic
383Q8—82no—
384Q8—82no—
385D4—22no—
386D4—22no—
387D4—41no—
388D4—41no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
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—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—81no—
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×C4C2×C22⋊Q811yescharacteristic
429C2×C4#53711yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4D4×Q811yescharacteristic
432C2×C4#55211yescharacteristic
433C2×C4#55411yescharacteristic
434C2×C4C2×#13511yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57811yescharacteristic
441C2×C4#57811yescharacteristic
442C8—22no—
443C8—42no—
444C8—41no—
445C8—41no—
446C8—41no—
447C8—81no—
448C8—81no—
449C8—81no—
450C8—81no—
451C8—81no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22#241711yescharacteristic
460C22#303511yescharacteristic
461C22#304211yescharacteristic
462C22#305911yescharacteristic
463C22#306211yescharacteristic
464C22#306711yescharacteristic
465C22#306811yescharacteristic
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4#308211yescharacteristic
480C4#308311yescharacteristic
481C4#325811yescharacteristic
482C4#326211yescharacteristic
483C4#329611yescharacteristic
484C4#332311yescharacteristic
485C4#338111yescharacteristic
486C4#338811yescharacteristic
487C2—82no—
488C2#1922011yescharacteristic
489C2#1924811yescharacteristic
490C2#1951511yescharacteristic
491C2#1956311yescharacteristic
492C2#2482211yescharacteristic
493C2#2484911yescharacteristic
494C2#2485911yescharacteristic
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. #29179C2semidirect
5. #10655C2semidirect
6. #10652C2semidirect
7. #10597C2semidirect
8. #10593C2semidirect
9. #9899C2semidirect
10. #9603C2semidirect
15. #8594C2semidirect

Maximal subgroups

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

subgroupindex
#299862
#291792
#227012
#221112
#106552
#106522
#105972
#105932
#98992
#96032
#92232
#92152
#86562
#86442
#85942