Finite Group Databasebeta

← #305991

Subgroups — #305991

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic, complemented
2#26091C211yesmaximal, characteristic
3#26087C211yesmaximal, characteristic
4#22708C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10578C211yesmaximal, characteristic, complemented
8#10576C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9598C211yesmaximal, characteristic, complemented
11#8893C211yesmaximal, characteristic
12#8885C211yesmaximal, characteristic
13#8820C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8200C211yesmaximal, characteristic
16C2×C8×D4C2211yescharacteristic
17#2571C2211yescharacteristic
18#2571C2211yescharacteristic
19#2354C2211yescharacteristic
20#2354C2211yescharacteristic
21#2092—21no—
22#2092—21no—
23#2085—21no—
24#2085—21no—
25#2067—21no—
26#2065—21no—
27#1949—21no—
28#1949—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1942—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1921—21no—
39#1919—21no—
40#1854—21no—
41#1854—21no—
42#1846—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C8×D4—22no—
72C8×D4Q814yes—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—41no—
79C2×#107D411yescharacteristic
80C2×#107D411yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102D411yescharacteristic
97C2×#102D411yescharacteristic
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#400—41no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#388—41no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#382—21no—
118#382—21no—
119#382—21no—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
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—41no—
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147D4⋊C8Q814yes—
148#322—22no—
149C22×D4C2×D411yescharacteristic
150C23×C4—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—81no—
158C22×C8—81no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C4×D4—21no—
164C4×D4—21no—
165C4×D4C2×Q812yes—
166C4×D4C2×Q812yes—
167#117—42no—
168#117—42no—
169#117—81no—
170#117—81no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
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—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2411yescharacteristic
203#107—41no—
204#107—41no—
205#107—42no—
206#107—42no—
207#107—81no—
208#107—81no—
209#107—81no—
210#107—82no—
211#106—42no—
212#106—81no—
213#106—81no—
214#106—81no—
215#106—81no—
216#106—81no—
217#106—81no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8C2×Q812yes—
227C4⋊C8C2×Q812yes—
228#102—22no—
229#102—22no—
230#102—81no—
231#102—81no—
232C22⋊C8—24no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—42no—
237C4×C8C2×Q812yes—
238C4×C8C2×Q812yes—
239C4×C8#4112yes—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4C4⋊Q812yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—84no—
302C2×C8C22⋊Q812yes—
303C2×C8C22⋊Q812yes—
304C2×C8#13512yes—
305C2×C8#13512yes—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
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—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—161no—
340C4⋊C4—161no—
341C4⋊C4C4⋊Q812yes—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—82no—
355C42—82no—
356C42C22×Q811yescharacteristic
357C42C22×Q811yescharacteristic
358C42C2×#4111yescharacteristic
359C422−1+411yescharacteristic
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23D4211yescharacteristic
368D4—22no—
369D4—22no—
370D4—41no—
371D4—41no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×C4⋊Q811yescharacteristic
413C2×C4#54411yescharacteristic
414C2×C4#54711yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C2×C4#55411yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4#57511yescharacteristic
420C2×C4#57511yescharacteristic
421C2×C4#57611yescharacteristic
422C2×C4#57611yescharacteristic
423C2×C4#57711yescharacteristic
424C2×C4#57811yescharacteristic
425C8—22no—
426C8—22no—
427C8—22no—
428C8—22no—
429C8—42no—
430C8—42no—
431C8—42no—
432C8—81no—
433C8—81no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#244011yescharacteristic
442C22#326711yescharacteristic
443C22#326811yescharacteristic
444C22#328311yescharacteristic
445C22#328711yescharacteristic
446C22#328811yescharacteristic
447C22#328911yescharacteristic
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4#308011yescharacteristic
462C4#308111yescharacteristic
463C4#316011yescharacteristic
464C4#316411yescharacteristic
465C4#333611yescharacteristic
466C4#333911yescharacteristic
467C4#338811yescharacteristic
468C4#338911yescharacteristic
469C2—82no—
470C2#1986211yescharacteristic
471C2#1986311yescharacteristic
472C2#1987011yescharacteristic
473C2#1987111yescharacteristic
474C2#2605911yescharacteristic
475C2#2606611yescharacteristic
476C2#2608411yescharacteristic
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. #28828C2semidirect
5. #10654C2semidirect
6. #10651C2semidirect
7. #10578C2semidirect
8. #10576C2semidirect
9. #9616C2semidirect
10. #9598C2semidirect
13. #8820C2semidirect

Maximal subgroups

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

subgroupindex
#288282
#260912
#260872
#227082
#106542
#106512
#105782
#105762
#96162
#95982
#88932
#88852
#88202
#86442
#82002