Finite Group Databasebeta

← #308119

Subgroups — #308119

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26511C211yesmaximal, characteristic, complemented
2#26091C211yesmaximal, characteristic
3#23061C211yesmaximal, characteristic
4#22707C211yesmaximal, characteristic
5#10681C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10589C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#9611C211yesmaximal, characteristic, complemented
10#9535C211yesmaximal, characteristic, complemented
11#9215C211yesmaximal, characteristic
12#8976C211yesmaximal, characteristic
13#8929C211yesmaximal, characteristic, complemented
14#8897C211yesmaximal, characteristic
15#8845C211yesmaximal, characteristic
16C2×#476C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2571C2211yescharacteristic
19#2369C2211yescharacteristic
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22#2101—21no—
23#2092—21no—
24#2088—21no—
25#2065—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1956C2211yescharacteristic
30#1949—21no—
31#1949—21no—
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
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#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1808—21no—
50#1802C2211yescharacteristic
51C4×#106—21no—
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#334C2211yescharacteristic
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⋊Q8C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72#476—22no—
73#476Q814yes—
74C8×D4—22no—
75C8×D4—22no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—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#400—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#382—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
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#334—22no—
149#333—22no—
150D4⋊C8—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—24no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—81no—
161C22×C8—81no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C2×Q812yes—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4C2×Q812yes—
173C4×D4#4112yes—
174#117—42no—
175#117—42no—
176#117—81no—
177#117—81no—
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⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2411yescharacteristic
210#107—42no—
211#107—81no—
212#107—81no—
213#107—81no—
214#106—24no—
215#106—22no—
216#106—41no—
217#106—42no—
218#106—42no—
219#106—81no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8C2×Q812yes—
229C4⋊C8#4112yes—
230#103—82no—
231#102—24no—
232#102—22no—
233#102—41no—
234#102—81no—
235C22⋊C8—24no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—42no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—21no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—81no—
257C2×Q8—81no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4C22⋊Q812yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—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×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
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—84no—
311C2×C8#13512yes—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
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—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—161no—
344C4⋊C4—161no—
345C4⋊C4C22⋊Q812yes—
346C4⋊C4#13512yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—82no—
360C42—82no—
361C42—161no—
362C42C22×Q811yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C422−1+411yescharacteristic
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23D4211yescharacteristic
374Q8—82no—
375Q8—161no—
376D4—22no—
377D4—22no—
378D4—41no—
379D4—41no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
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—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4C2×C22⋊Q811yescharacteristic
420C2×C4#52511yescharacteristic
421C2×C4#53911yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4D4×Q811yescharacteristic
424C2×C4#55411yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4#57511yescharacteristic
428C2×C4#57511yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57811yescharacteristic
433C8—22no—
434C8—42no—
435C8—41no—
436C8—41no—
437C8—41no—
438C8—42no—
439C8—81no—
440C8—81no—
441C8—81no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#244911yescharacteristic
450C22#326711yescharacteristic
451C22#326911yescharacteristic
452C22#328211yescharacteristic
453C22#328411yescharacteristic
454C22#328811yescharacteristic
455C22#329011yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—42no—
460C4—82no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4#307611yescharacteristic
470C4#308211yescharacteristic
471C4#310211yescharacteristic
472C4#312311yescharacteristic
473C4#333611yescharacteristic
474C4#335011yescharacteristic
475C4#338111yescharacteristic
476C4#338811yescharacteristic
477C2—82no—
478C2#1980011yescharacteristic
479C2#1985411yescharacteristic
480C2#1999911yescharacteristic
481C2#2014511yescharacteristic
482C2#2635111yescharacteristic
483C2#2637811yescharacteristic
484C2#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. #26511C2semidirect
5. #10681C2semidirect
6. #10657C2semidirect
7. #10589C2semidirect
8. #10578C2semidirect
9. #9611C2semidirect
10. #9535C2semidirect
13. #8929C2semidirect

Maximal subgroups

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

subgroupindex
#265112
#260912
#230612
#227072
#106812
#106572
#105892
#105782
#96112
#95352
#92152
#89762
#89292
#88972
#88452