Finite Group Databasebeta

← #219215

Subgroups — #219215

Order 512.

629 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23819C211yesmaximal, characteristic
2#21682C211yesmaximal, characteristic
3#21675C211yesmaximal, characteristic, complemented
4#9872C211yesmaximal, characteristic, complemented
5#9642C211yesmaximal, characteristic
6#9546C211yesmaximal, characteristic, complemented
7#9546C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8849C211yesmaximal, characteristic, complemented
11#8847C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8812C211yesmaximal, characteristic, complemented
15#8134C211yesmaximal, characteristic, complemented
16C2×#436C2211yescharacteristic
17C2×#435—21no—
18C2×#435—21no—
19C2×C8×D4C2211yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#125C2211yescharacteristic
22#2100C411yescharacteristic
23#2100C411yescharacteristic
24#2073C411yescharacteristic
25#2073C411yescharacteristic
26#2041C411yescharacteristic
27#2041C411yescharacteristic
28#2025C411yescharacteristic
29#2025C411yescharacteristic
30#1955—21no—
31#1955—21no—
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1917—21no—
37#1917—21no—
38#1842C2211yescharacteristic
39#1831C2211yescharacteristic
40#1771—21no—
41#1770—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#127C2×C411yescharacteristic
70C2×#125C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75#436—41no—
76#436—41no—
77#435—44no—
78#435—44no—
79C8×D4—41no—
80C8×D4—41no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97D411yescharacteristic
105C2×#97D411yescharacteristic
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390—21no—
116#390—21no—
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#387—21no—
122#387—21no—
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C43C2311yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#335—41no—
151#335—41no—
152#334—41no—
153#334—41no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156D4⋊C8—41no—
157D4⋊C8—41no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184#127—22no—
185#127—41no—
186#125—22no—
187#125—41no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239#102—22no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—41no—
244#102—41no—
245#102—41no—
246#102—41no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253#97—22no—
254#97—22no—
255#97—41no—
256#97—41no—
257#97—42no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—42no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277C24—21no—
278C24—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4#13112yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—84no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—84no—
364C2×C8AGL(1,ℤ/8)12yes—
365C2×C8AGL(1,ℤ/8)12yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4#13112yes—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42H(5,𝔽2)11yescharacteristic
407C42H(5,𝔽2)11yescharacteristic
408C422−1+411yescharacteristic
409C422−1+411yescharacteristic
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C2×C4×D411yescharacteristic
425D4—22no—
426D4—22no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—84no—
434D4—84no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4#52011yescharacteristic
476C2×C4#52011yescharacteristic
477C2×C4#53511yescharacteristic
478C2×C4#53611yescharacteristic
479C2×C4#54211yescharacteristic
480C2×C4#54211yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4C2×#13111yescharacteristic
483C2×C4#56811yescharacteristic
484C2×C4#56811yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
487C2×C4#57511yescharacteristic
488C2×C4#57511yescharacteristic
489C8—22no—
490C8—22no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#232611yescharacteristic
518C22#293311yescharacteristic
519C22#293311yescharacteristic
520C22C4×AGL(1,ℤ/8)11yescharacteristic
521C22C4×AGL(1,ℤ/8)11yescharacteristic
522C22#295711yescharacteristic
523C22#295711yescharacteristic
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4#299211yescharacteristic
537C4#299311yescharacteristic
538C4#299411yescharacteristic
539C4#299511yescharacteristic
540C4#321711yescharacteristic
541C4#321911yescharacteristic
542C4#323411yescharacteristic
543C4#323511yescharacteristic
544C2—82no—
545C2—161no—
546C2#1800611yescharacteristic
547C2#1800611yescharacteristic
548C2#1800711yescharacteristic
549C2#1801111yescharacteristic
550C2#2221711yescharacteristic
551C2#2222611yescharacteristic
552C2#2223611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21675C2semidirect
4. #9872C2semidirect
6. #9546C2semidirect
7. #9546C2semidirect
8. #9541C2semidirect
9. #9541C2semidirect
10. #8849C2semidirect
11. #8847C2semidirect
12. #8845C2semidirect
13. #8843C2semidirect
14. #8812C2semidirect
15. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#238192
#216822
#216752
#98722
#96422
#95462
#95412
#88492
#88472
#88452
#88432
#88122
#81342