Finite Group Databasebeta

← #305865

Subgroups — #305865

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28816C211yesmaximal, characteristic, complemented
2#26106C211yesmaximal, characteristic
3#26040C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10579C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#8894C211yesmaximal, characteristic
12#8882C211yesmaximal, characteristic
13#8820C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8202C211yesmaximal, characteristic
16C2×#501—21no—
17C2×#474—21no—
18C2×C8×D4C2211yescharacteristic
19C22×#106—21no—
20#2552C2211yescharacteristic
21#2551C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#2099—21no—
25#2086—21no—
26#2086—21no—
27#2059—21no—
28#2057—21no—
29#2057—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1944—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1862—21no—
40#1862—21no—
41#1853—21no—
42#1853—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802—21no—
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53C4×#106C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#102C2211yescharacteristic
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⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×#125—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76#501—44no—
77#474—44no—
78C8×D4—22no—
79C8×D4Q814yescomplemented
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—44no—
94C2×#106D411yescharacteristic
95C2×#106D411yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102D411yescharacteristic
105C2×#102D411yescharacteristic
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#398—41no—
112#395—41no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#385—41no—
120#385—41no—
121#384—21no—
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#335—22no—
150D4⋊C8Q814yescomplemented
151#322—22no—
152C22×Q8—41no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—41no—
156C23×C4—41no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—81no—
172C4⋊Q8—81no—
173#125—42no—
174#125—81no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—81no—
180C4×D4—81no—
181C4×D4C2×Q812yes—
182C4×D4C2×Q812yes—
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—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C2411yescharacteristic
220#107—42no—
221#107—42no—
222#107—42no—
223#107—42no—
224#107—81no—
225#107—81no—
226#107—81no—
227#107—82no—
228#106—22no—
229#106—42no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—81no—
234#106D814yes—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—42no—
242C4⋊C8—81no—
243C4⋊C8C2×Q812yes—
244C4⋊C8C2×Q812yes—
245#102—22no—
246#102—22no—
247#102—42no—
248#102—42no—
249C22⋊C8—24no—
250#97—42no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8C2×Q812yes—
255C4×C8C2×Q812yes—
256C4×C8#4112yes—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264C24—21no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—161no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4C4⋊Q812yes—
272C2×D4AGL(1,ℤ/8)12yes—
273C2×D4#13712yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—161no—
296C22×C4—161no—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—48no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8C22⋊Q812yes—
325C2×C8C22⋊Q812yes—
326C2×C8C2×D812yes—
327C2×C8#13712yes—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
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—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—82no—
365C4⋊C4—161no—
366C4⋊C4C4⋊Q812yes—
367C4⋊C4C2×D812yes—
368C4⋊C4C2×D812yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42C22×Q811yescharacteristic
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C23—22no—
384C23—22no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23D4211yescharacteristic
391Q8—164no—
392D4—22no—
393D4—22no—
394D4—22no—
395D4#50114yescomplemented
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—164no—
436C2×C4—164no—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C4⋊Q811yescharacteristic
439C2×C4#54411yescharacteristic
440C2×C4#54711yescharacteristic
441C2×C4D4×Q811yescharacteristic
442C2×C4#55411yescharacteristic
443C2×C4C22×D811yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57611yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—42no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—42no—
457C8—81no—
458C8—81no—
459C8#47414yescomplemented
460C22—24no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22#244011yescharacteristic
468C22#326511yescharacteristic
469C22#326611yescharacteristic
470C22#327611yescharacteristic
471C22D4×D811yescharacteristic
472C22#328811yescharacteristic
473C22#328911yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4C2×#47411yescharacteristic
488C4#308311yescharacteristic
489C4C2×#50111yescharacteristic
490C4#316311yescharacteristic
491C4#334411yescharacteristic
492C4#334511yescharacteristic
493C4Q8×D811yescharacteristic
494C4#338511yescharacteristic
495C2—44no—
496C2#1975611yescharacteristic
497C2#1975711yescharacteristic
498C2#2018111yescharacteristic
499C2#2019711yescharacteristic
500C2#2604011yescharacteristic
501C2#2606811yescharacteristic
502C2#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. #28816C2semidirect
5. #10655C2semidirect
6. #10651C2semidirect
7. #10579C2semidirect
8. #10575C2semidirect
9. #9617C2semidirect
10. #9596C2semidirect
13. #8820C2semidirect
79. C8×D4Q8semidirect
150. D4⋊C8Q8semidirect
395. D4#501semidirect
459. C8#474semidirect

Maximal subgroups

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

subgroupindex
#288162
#261062
#260402
D4×#1062
#106552
#106512
#105792
#105752
#96172
#95962
#88942
#88822
#88202
#86442
#82022