Finite Group Databasebeta

← #228871

Subgroups — #228871

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23151C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic
3C4×#476C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9597C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#9539C211yesmaximal, characteristic, complemented
9#9534C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic
11#8843C211yesmaximal, characteristic
12#8263C211yesmaximal, characteristic, complemented
13#8202C211yesmaximal, characteristic
14#8200C211yesmaximal, characteristic
15#8172C211yesmaximal, characteristic
16C2×#476C411yescharacteristic
17C2×#446—21no—
18C2×C8×D4C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1987C411yescharacteristic
23#1985—21no—
24#1948C411yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1920C411yescharacteristic
30#1920C411yescharacteristic
31#1917—21no—
32#1916—21no—
33#1864C411yescharacteristic
34#1864C411yescharacteristic
35#1850—21no—
36#1850—21no—
37#1831C2211yescharacteristic
38#1802C411yescharacteristic
39#1797—21no—
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C2×#336—21no—
50C4×#97C2211yescharacteristic
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
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#476—24no—
76#446—44no—
77C8×D4—22no—
78C8×D4—22no—
79C8×D4—41no—
80C8×D4—41no—
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×#102C2×C411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#389—21no—
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
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#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—41no—
152D4⋊C8—22no—
153D4⋊C8Q814yescomplemented
154#322—41no—
155#322—41no—
156C22×Q8—21no—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—81no—
172C22×C8—81no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—22no—
182C4⋊Q8—41no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—41no—
191C4×D4C2×Q812yes—
192C4×D4#4112yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#106—22no—
223#106—22no—
224#106—22no—
225#106—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8C2×Q812yes—
242#102—22no—
243#102—22no—
244#102—22no—
245#102M1614yes—
246C22⋊C8—42no—
247C22⋊C8—42no—
248#97—42no—
249#97—42no—
250#97—42no—
251#97—42no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8C2×Q812yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24—21no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—81no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4C4×Q812yes—
280C2×D4C2×M1612yes—
281C2×D4#13112yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
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—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
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—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—84no—
349C2×C8—84no—
350C2×C8C2×M1612yes—
351C2×C8AGL(1,ℤ/8)12yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4C4×Q812yes—
369C4⋊C4C2×M1612yes—
370C4⋊C4#13112yes—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—81no—
392C42—81no—
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—22no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23C2×C4×D411yescharacteristic
405Q8—84no—
406D4—22no—
407D4—22no—
408D4—22no—
409D4#44614yescomplemented
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#44412yes—
451C2×C4C2×C4×Q811yescharacteristic
452C2×C4#52011yescharacteristic
453C2×C4#54611yescharacteristic
454C2×C4D4×Q811yescharacteristic
455C2×C4#55211yescharacteristic
456C2×C4#55411yescharacteristic
457C2×C4C22×M1611yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C8—22no—
466C8—22no—
467C8—42no—
468C8—42no—
469C8—42no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22#236011yescharacteristic
487C22#292911yescharacteristic
488C22D4×M1611yescharacteristic
489C22C4×AGL(1,ℤ/8)11yescharacteristic
490C22#294911yescharacteristic
491C22C2×#44411yescharacteristic
492C22#295611yescharacteristic
493C4—22no—
494C4—21no—
495C4—21no—
496C4—41no—
497C4—42no—
498C4—41no—
499C4—42no—
500C4—42no—
501C4—41no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4C2×#44611yescharacteristic
507C4#296211yescharacteristic
508C4#299011yescharacteristic
509C4#299211yescharacteristic
510C4#330611yescharacteristic
511C4#331111yescharacteristic
512C4#338411yescharacteristic
513C4#338811yescharacteristic
514C2—44no—
515C2#1791711yescharacteristic
516C2#1792411yescharacteristic
517C2#1804611yescharacteristic
518C2#1806011yescharacteristic
519C2#2295411yescharacteristic
520C2#2297511yescharacteristic
521C2#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. #23151C2semidirect
4. #9877C2semidirect
5. #9606C2semidirect
6. #9597C2semidirect
7. #9596C2semidirect
8. #9539C2semidirect
9. #9534C2semidirect
12. #8263C2semidirect
153. D4⋊C8Q8semidirect
409. D4#446semidirect

Maximal subgroups

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

subgroupindex
#231512
#226962
C4×#4762
#98772
#96062
#95972
#95962
#95392
#95342
#88452
#88432
#82632
#82022
#82002
#81722