Finite Group Databasebeta

← #305977

Subgroups — #305977

Order 512.

661 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28809C211yesmaximal, characteristic, complemented
2#26044C211yesmaximal, characteristic
3#26040C211yesmaximal, characteristic
4#22703C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10575C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#9613C211yesmaximal, characteristic, complemented
10#9597C211yesmaximal, characteristic, complemented
11#8894C211yesmaximal, characteristic
12#8886C211yesmaximal, characteristic
13#8819C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic
15#8200C211yesmaximal, characteristic
16C2×#498—21no—
17C2×#476—21no—
18C2×#474—21no—
19C2×C8×D4C2211yescharacteristic
20#2551C2211yescharacteristic
21#2551C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#2088—21no—
25#2086—21no—
26#2058—21no—
27#2058—21no—
28#2058—21no—
29#2057—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1941—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1864—21no—
40#1862—21no—
41#1853—21no—
42#1853—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56C2×#334C2211yescharacteristic
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⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77#498—44no—
78#476—44no—
79#474—44no—
80C8×D4—22no—
81C8×D4Q814yescomplemented
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—41no—
95C2×#106D411yescharacteristic
96C2×#106D411yescharacteristic
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#395—41no—
113#395—41no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#385—41no—
121#385—41no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
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#340—41no—
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#334—22no—
150D4⋊C8Q814yescomplemented
151#322—22no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—41no—
157C23×C4—41no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—81no—
175C4⋊Q8—81no—
176C4⋊Q8—81no—
177C4⋊Q8—81no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4—81no—
184C4×D4C2×Q812yes—
185C4×D4C2×Q812yes—
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—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C2411yescharacteristic
222#107—42no—
223#107—42no—
224#107—81no—
225#106—22no—
226#106—22no—
227#106—42no—
228#106—42no—
229#106—42no—
230#106—42no—
231#106—42no—
232#106—81no—
233#106—81no—
234#106—81no—
235#106—81no—
236#106—82no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8C2×Q812yes—
246C4⋊C8C2×Q812yes—
247#102—42no—
248#102—42no—
249#102SD1614yes—
250#102SD1614yes—
251C22⋊C8—24no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—42no—
256C4×C8C2×Q812yes—
257C4×C8C2×Q812yes—
258C4×C8#4112yes—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24—21no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—161no—
272C2×Q8—161no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C4⋊Q812yes—
277C2×D4C2×SD1612yes—
278C2×D4C2×SD1612yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—161no—
301C22×C4—161no—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—84no—
329C2×C8C22⋊Q812yes—
330C2×C8C22⋊Q812yes—
331C2×C8C2×SD1612yes—
332C2×C8C2×SD1612yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4C4⋊Q812yes—
369C4⋊C4C2×SD1612yes—
370C4⋊C4C2×SD1612yes—
371C4⋊C4AGL(1,ℤ/8)12yes—
372C4⋊C4#13712yes—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42C22×Q811yescharacteristic
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C422−1+411yescharacteristic
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23D4211yescharacteristic
395Q8—164no—
396Q8—164no—
397D4—22no—
398D4—22no—
399D4—22no—
400D4#49814yescomplemented
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—164no—
441C2×C4—164no—
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×C4⋊Q811yescharacteristic
444C2×C4#54411yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C2×C4#55411yescharacteristic
448C2×C4C22×SD1611yescharacteristic
449C2×C4C22×SD1611yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57711yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—22no—
460C8—42no—
461C8—42no—
462C8—42no—
463C8—81no—
464C8—81no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22#244011yescharacteristic
473C22#326411yescharacteristic
474C22#326611yescharacteristic
475C22D4×SD1611yescharacteristic
476C22D4×SD1611yescharacteristic
477C22#328811yescharacteristic
478C22#328911yescharacteristic
479C4—21no—
480C4—21no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4#307911yescharacteristic
493C4#308111yescharacteristic
494C4C2×#49811yescharacteristic
495C4#316311yescharacteristic
496C4#333511yescharacteristic
497C4#333511yescharacteristic
498C4#338411yescharacteristic
499C4#338511yescharacteristic
500C2—44no—
501C2#1976311yescharacteristic
502C2#1976511yescharacteristic
503C2#1980311yescharacteristic
504C2#1980511yescharacteristic
505C2#2605811yescharacteristic
506C2#2606011yescharacteristic
507C2#2608511yescharacteristic
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. #28809C2semidirect
5. #10654C2semidirect
6. #10652C2semidirect
7. #10575C2semidirect
8. #10574C2semidirect
9. #9613C2semidirect
10. #9597C2semidirect
13. #8819C2semidirect
81. C8×D4Q8semidirect
150. D4⋊C8Q8semidirect
400. D4#498semidirect

Maximal subgroups

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

subgroupindex
#288092
#260442
#260402
#227032
#106542
#106522
#105752
#105742
#96132
#95972
#88942
#88862
#88192
#86422
#82002