Finite Group Databasebeta

← #276706

Subgroups — #276706

Order 512.

535 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#29995C211yesmaximal, characteristic
2#29193C211yesmaximal, characteristic, complemented
3#22836C211yesmaximal, characteristic
4#22110C211yesmaximal, characteristic
5#10625C211yesmaximal, characteristic
6#10615C211yesmaximal, characteristic
7#10485C211yesmaximal, characteristic
8#10485C211yesmaximal, characteristic
9#9901C211yesmaximal, characteristic, complemented
10#9619C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9569C211yesmaximal, characteristic, complemented
14#9562C211yesmaximal, characteristic, complemented
15#8591C211yesmaximal, characteristic, complemented
16C2×#474C2211yescharacteristic
17#2859C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2293C2211yescharacteristic
20#2088—21no—
21#2087—21no—
22#2068—21no—
23#2068—21no—
24#2067—21no—
25#2065—21no—
26#1988C2211yescharacteristic
27#1956C2211yescharacteristic
28#1956C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1942—21no—
35#1934—21no—
36#1925—21no—
37#1924—21no—
38#1923—21no—
39#1918—21no—
40#1863—21no—
41#1863—21no—
42#1845C2211yescharacteristic
43#1842C2211yescharacteristic
44#1834C2211yescharacteristic
45#1831C2211yescharacteristic
46#1818—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1750—21no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C4×Q8—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72#474—22no—
73#474—22no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390D411yescharacteristic
109#390D411yescharacteristic
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389D411yescharacteristic
117#389D411yescharacteristic
118#388—21no—
119#388—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
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#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#335—22no—
148#332—22no—
149#332—22no—
150C22×Q8—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8#4112yes—
167#125—82no—
168C4×Q8—81no—
169C4×Q8—81no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174#117—81no—
175#117—81no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
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—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—24no—
210#107—81no—
211#107—81no—
212#107—81no—
213#107—81no—
214#106—81no—
215C4⋊C8—22no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—41no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8#4112yes—
231#103—42no—
232#103—42no—
233#103—81no—
234#103—81no—
235#102—24no—
236#102—81no—
237#102—81no—
238#102—81no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—81no—
242C22⋊C8—81no—
243#97—81no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—81no—
249C4×C8—81no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
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
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—21no—
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—41no—
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#13512yes—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
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—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C22⋊C4—81no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—82no—
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C422−1+411yescharacteristic
372C422−1+411yescharacteristic
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23C2×#12111yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—81no—
384Q8—81no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389C2×C4—21no—
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—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
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—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#47312yes—
428C2×C4#52511yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#54711yescharacteristic
431C2×C4#54811yescharacteristic
432C2×C4#55211yescharacteristic
433C2×C4#55611yescharacteristic
434C2×C4C2×#13511yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57611yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57811yescharacteristic
441C2×C4#57811yescharacteristic
442C8—22no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—42no—
447C8—81no—
448C8—81no—
449C8—81no—
450C8—81no—
451C8—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22#241511yescharacteristic
460C22#303511yescharacteristic
461C22#304111yescharacteristic
462C22#306111yescharacteristic
463C22#306311yescharacteristic
464C22C2×#47311yescharacteristic
465C22#306811yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—82no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4#311911yescharacteristic
480C4#312711yescharacteristic
481C4#324211yescharacteristic
482C4#324311yescharacteristic
483C4#333911yescharacteristic
484C4#335511yescharacteristic
485C4#336111yescharacteristic
486C4#337011yescharacteristic
487C2—161no—
488C2#1922311yescharacteristic
489C2#1925311yescharacteristic
490C2#1951011yescharacteristic
491C2#1955511yescharacteristic
492C2#2468411yescharacteristic
493C2#2471511yescharacteristic
494C2#2472211yescharacteristic
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. #29193C2semidirect
9. #9901C2semidirect
10. #9619C2semidirect
11. #9614C2semidirect
12. #9601C2semidirect
13. #9569C2semidirect
14. #9562C2semidirect
15. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#299952
#291932
#228362
#221102
#106252
#106152
#104852
#99012
#96192
#96142
#96012
#95692
#95622
#85912