Finite Group Databasebeta

← #279404

Subgroups — #279404

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33066C211yesmaximal, characteristic
2#28515C211yesmaximal, characteristic, complemented
3#22650C211yesmaximal, characteristic
4C4×#474C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10592C211yesmaximal, characteristic, complemented
8#10588C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9603C211yesmaximal, characteristic, complemented
11#9227C211yesmaximal, characteristic
12#9227C211yesmaximal, characteristic
13#8652C211yesmaximal, characteristic
14#8650C211yesmaximal, characteristic
15#8589C211yesmaximal, characteristic, complemented
16C2×#479—21no—
17C2×#479—21no—
18C2×#474C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#423—21no—
22#2859C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#2097—21no—
26#2097—21no—
27#1988C2211yescharacteristic
28#1956C2211yescharacteristic
29#1956C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1918—21no—
36#1918—21no—
37#1918—21no—
38#1862—21no—
39#1862—21no—
40#1862—21no—
41#1851—21no—
42#1851—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1797—21no—
51C4×#107—21no—
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×#102—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#479—44no—
77#479—44no—
78#474—22no—
79#474Q814yes—
80#436—22no—
81#436—22no—
82#436—41no—
83#436—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92#423—44no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8D411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C43C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—41no—
154#334—41no—
155#332—41no—
156#332—41no—
157C22×Q8—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—81no—
167C2×M16—81no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C2×Q812yes—
182#125—42no—
183#125—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4C2×Q812yes—
193C4×D4#4112yes—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—41no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#107—24no—
228#107—42no—
229#107—42no—
230#107—42no—
231#107—42no—
232#107—42no—
233#106—42no—
234#106—42no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251C4⋊C8D814yes—
252C4⋊C8C2×Q812yes—
253#103—42no—
254#103—42no—
255#103—42no—
256#103—42no—
257#102—24no—
258#102—42no—
259#102—42no—
260#102—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—42no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4C22⋊Q812yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—81no—
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—84no—
315M16—84no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8C2×D812yes—
345C2×C8C2×D812yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4C22⋊Q812yes—
373C4⋊C4#13712yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—42no—
389C42—42no—
390C42—82no—
391C42C2×D812yes—
392C42AGL(1,ℤ/8)12yes—
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—22no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23C2×#12111yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—81no—
408Q8—81no—
409D4—22no—
410D4—22no—
411D4—41no—
412D4—41no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#45912yes—
453C2×C4#45912yes—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4#53711yescharacteristic
456C2×C4#54611yescharacteristic
457C2×C4D4×Q811yescharacteristic
458C2×C4#55211yescharacteristic
459C2×C4#55411yescharacteristic
460C2×C4C22×D811yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—22no—
469C8—22no—
470C8—41no—
471C8—41no—
472C8—42no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#241711yescharacteristic
486C22C2×#45911yescharacteristic
487C22#304611yescharacteristic
488C22#305411yescharacteristic
489C22#305611yescharacteristic
490C22#306711yescharacteristic
491C22#306911yescharacteristic
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—42no—
498C4—42no—
499C4—41no—
500C4—81no—
501C4—82no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4#307811yescharacteristic
506C4#308011yescharacteristic
507C4#325211yescharacteristic
508C4#325611yescharacteristic
509C4#329411yescharacteristic
510C4#333011yescharacteristic
511C4Q8×D811yescharacteristic
512C4#338411yescharacteristic
513C2—82no—
514C2#1913311yescharacteristic
515C2#1916311yescharacteristic
516C2#1945411yescharacteristic
517C2#1949911yescharacteristic
518C2#2481111yescharacteristic
519C2#2484211yescharacteristic
520C2#2486011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28515C2semidirect
5. #10653C2semidirect
6. #10650C2semidirect
7. #10592C2semidirect
8. #10588C2semidirect
9. #9900C2semidirect
10. #9603C2semidirect
15. #8589C2semidirect

Maximal subgroups

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

subgroupindex
#330662
#285152
#226502
C4×#4742
#106532
#106502
#105922
#105882
#99002
#96032
#92272
#86522
#86502
#85892