Finite Group Databasebeta

← #307899

Subgroups — #307899

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26442C211yesmaximal, characteristic, complemented
2#26020C211yesmaximal, characteristic
3#23061C211yesmaximal, characteristic
4#22707C211yesmaximal, characteristic
5#10682C211yesmaximal, characteristic, complemented
6#10658C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#8976C211yesmaximal, characteristic
13#8929C211yesmaximal, characteristic, complemented
14#8893C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#476C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2551C2211yescharacteristic
19#2369C2211yescharacteristic
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22#2088—21no—
23#2067—21no—
24#2058—21no—
25#2023—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1988—21no—
29#1988—21no—
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1934—21no—
40#1926—21no—
41#1920—21no—
42#1919—21no—
43#1918—21no—
44#1854—21no—
45#1854—21no—
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1808—21no—
50#1802C2211yescharacteristic
51C4×#106—21no—
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74#476—22no—
75#476Q814yescomplemented
76C8×D4—22no—
77C8×D4—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106D411yescharacteristic
85C2×#106D411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#103—41no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102D411yescharacteristic
96C2×#102D411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#395—41no—
103#391—41no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—41no—
111#388—41no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124#340—21no—
125#340—21no—
126#340—21no—
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—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#333—22no—
150D4⋊C8—22no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—24no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—81no—
162C22×C8—81no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C4⋊Q8C2×Q812yes—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4C2×Q812yes—
178C4×D4#4112yes—
179#117—42no—
180#117—42no—
181#117—81no—
182#117—81no—
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×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—81no—
218#107—81no—
219#106—24no—
220#106—22no—
221#106—22no—
222#106—42no—
223#106—81no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232C4⋊C8C2×Q812yes—
233C4⋊C8#4112yes—
234#103—82no—
235#102—24no—
236#102—22no—
237#102—41no—
238#102—81no—
239C22⋊C8—24no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×Q8—161no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C22⋊Q812yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C2×C8—22no—
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—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—84no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
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—82no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—161no—
349C4⋊C4C22⋊Q812yes—
350C4⋊C4#13512yes—
351C4⋊C4#13512yes—
352C4⋊C4#13712yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—161no—
358C22⋊C4—161no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—82no—
366C42—82no—
367C42—161no—
368C42C22×Q811yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C422−1+411yescharacteristic
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23D4211yescharacteristic
380Q8—82no—
381Q8—161no—
382Q8—164no—
383D4—22no—
384D4—22no—
385D4—41no—
386D4—41no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
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×C4C2×C22⋊Q811yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4#53911yescharacteristic
429C2×C4#54711yescharacteristic
430C2×C4D4×Q811yescharacteristic
431C2×C4#55411yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57811yescharacteristic
440C8—42no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—42no—
446C8—81no—
447C8—81no—
448C8—81no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#244911yescharacteristic
457C22#326611yescharacteristic
458C22#326911yescharacteristic
459C22#328011yescharacteristic
460C22#328211yescharacteristic
461C22#328811yescharacteristic
462C22#329011yescharacteristic
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4#307611yescharacteristic
477C4#308211yescharacteristic
478C4#310711yescharacteristic
479C4#311811yescharacteristic
480C4#334211yescharacteristic
481C4#334811yescharacteristic
482C4#338111yescharacteristic
483C4#338511yescharacteristic
484C2—82no—
485C2#1978411yescharacteristic
486C2#1982811yescharacteristic
487C2#1998811yescharacteristic
488C2#2004411yescharacteristic
489C2#2633711yescharacteristic
490C2#2637211yescharacteristic
491C2#2640511yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26442C2semidirect
5. #10682C2semidirect
6. #10658C2semidirect
7. #10593C2semidirect
8. #10575C2semidirect
9. #9616C2semidirect
10. #9534C2semidirect
13. #8929C2semidirect
75. #476Q8semidirect

Maximal subgroups

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

subgroupindex
#264422
#260202
#230612
#227072
#106822
#106582
#105932
#105752
#96162
#95342
#92152
#89762
#89292
#88932
#88432