Finite Group Databasebeta

← #228798

Subgroups — #228798

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23253C211yesmaximal, characteristic, complemented
2#22700C211yesmaximal, characteristic
3C4×#478C211yesmaximal, characteristic
4#9871C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9598C211yesmaximal, characteristic, complemented
7#9597C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#8849C211yesmaximal, characteristic
11#8847C211yesmaximal, characteristic
12#8261C211yesmaximal, characteristic, complemented
13#8210C211yesmaximal, characteristic
14#8210C211yesmaximal, characteristic
15C4×D4⋊C8C211yesmaximal, characteristic
16C2×#478C411yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C4×#125C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1989C411yescharacteristic
22#1949C411yescharacteristic
23#1939—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1921C411yescharacteristic
29#1919C411yescharacteristic
30#1916—21no—
31#1916—21no—
32#1864C411yescharacteristic
33#1862C411yescharacteristic
34#1851—21no—
35#1851—21no—
36#1831C2211yescharacteristic
37#1804C411yescharacteristic
38#1797—21no—
39C4×#107C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C8×C4⋊C4—21no—
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1728—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C2×#125C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C22×C42—21no—
74#478—24no—
75#436—22no—
76#436—22no—
77#436—41no—
78#436—41no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#387—21no—
115#387—21no—
116#382C2×C411yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—41no—
150#334—41no—
151D4⋊C8—22no—
152D4⋊C8Q814yes—
153#322—41no—
154#322—41no—
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—81no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179#125—22no—
180#125M1614yescomplemented
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4C2×Q812yes—
188C4×D4#4112yes—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#107—22no—
218#107—22no—
219#106—22no—
220#106—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8C2×Q812yes—
237#102—22no—
238#102—22no—
239#102—41no—
240#102—41no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8—81no—
258C4×C8—81no—
259C4×C8C2×Q812yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4C4×Q812yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302M16—84no—
303M16—84no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
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×C8#13512yes—
345C2×C8#13512yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—42no—
359C4⋊C4—81no—
360C4⋊C4C4×Q812yes—
361C4⋊C4C2×M1612yes—
362C4⋊C4C2×M1612yes—
363C4⋊C4#13112yes—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—22no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—81no—
383C42—81no—
384C42C2×M1612yes—
385C42#13112yes—
386C42C22×Q811yescharacteristic
387C42C2×#4111yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C422−1+411yescharacteristic
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C2×C4×D411yescharacteristic
398D4—22no—
399D4—22no—
400D4—41no—
401D4—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4#43512yes—
441C2×C4#43512yes—
442C2×C4#44412yes—
443C2×C4C2×C4×Q811yescharacteristic
444C2×C4#52011yescharacteristic
445C2×C4#54611yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C2×C4#55211yescharacteristic
448C2×C4#55411yescharacteristic
449C2×C4C22×M1611yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C8—22no—
458C8—22no—
459C8—42no—
460C8—42no—
461C8—42no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—81no—
469C8—81no—
470C8—81no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#236011yescharacteristic
479C22C2×#43511yescharacteristic
480C22#293311yescharacteristic
481C22C4×#13511yescharacteristic
482C22#294611yescharacteristic
483C22C2×#44411yescharacteristic
484C22#295611yescharacteristic
485C4—22no—
486C4—22no—
487C4—21no—
488C4—21no—
489C4—22no—
490C4—41no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4#296011yescharacteristic
499C4Q8×M1611yescharacteristic
500C4#299111yescharacteristic
501C4#299311yescharacteristic
502C4#329611yescharacteristic
503C4#329711yescharacteristic
504C4#338011yescharacteristic
505C4#338111yescharacteristic
506C2—82no—
507C2#1786311yescharacteristic
508C2#1787511yescharacteristic
509C2#1799911yescharacteristic
510C2#1802011yescharacteristic
511C2#2295311yescharacteristic
512C2#2296611yescharacteristic
513C2#2298311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23253C2semidirect
4. #9871C2semidirect
5. #9603C2semidirect
6. #9598C2semidirect
7. #9597C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
12. #8261C2semidirect
180. #125M16semidirect

Maximal subgroups

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

subgroupindex
#232532
#227002
C4×#4782
#98712
#96032
#95982
#95972
#95362
#95352
#88492
#88472
#82612
#82102
C4×D4⋊C82