Finite Group Databasebeta

← #226534

Subgroups — #226534

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22799C211yesmaximal, characteristic, complemented
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9560C212yesmaximal, complemented
5#9540C212yesmaximal, complemented
6#8947C212yesmaximal
7#8784C211yesmaximal, characteristic, complemented
8#8563C211yesmaximal, characteristic, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8261C211yesmaximal, characteristic, complemented
11#8204C212yesmaximal, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×C8×D4C2212yes—
16C2×#432—21no—
17C22×C4⋊C8—21no—
18C42×D4C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#1954C2212yes—
21#1939C2212yes—
22#1924—22no—
23#1850—22no—
24#1842C2211yescharacteristic
25#1842C2211yescharacteristic
26#1797—21no—
27#1797—21no—
28C4×#107C411yescharacteristic
29C4×#107C411yescharacteristic
30C4×#107C2211yescharacteristic
31#1770C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C411yescharacteristic
37C4×#102C411yescharacteristic
38C4×#102C2211yescharacteristic
39C4×#102C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×#322C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1277C2212yes—
47#1277C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C23×C8—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C4×D8—24no—
60C4×D8—24no—
61C4×D8—22no—
62C4×D8—22no—
63C8×D4—24no—
64C8×D4—42no—
65#432—44no—
66C2×#107C2×C411yescharacteristic
67C2×#107C2×C411yescharacteristic
68C2×#107C2×C411yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—44no—
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#102C2×C411yescharacteristic
81C2×#102C2×C411yescharacteristic
82C2×#102C2×C411yescharacteristic
83C2×#102C2×C411yescharacteristic
84C2×#102C2×C411yescharacteristic
85C2×#102C2×C411yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×C4×C8C2×C411yescharacteristic
91C2×C4×C8C2×C411yescharacteristic
92C2×C4×C8C2×C411yescharacteristic
93C2×C4×C8C2×C411yescharacteristic
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2312yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103C4×C4⋊C4C2311yescharacteristic
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C43C2311yescharacteristic
111#340C2×C411yescharacteristic
112#340C2×C411yescharacteristic
113#340C2×C411yescharacteristic
114#340C2×C411yescharacteristic
115#340C2×C411yescharacteristic
116#340C2×C411yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336D412yes—
121D4⋊C8—24no—
122D4⋊C8—42no—
123#322—42no—
124C22×D4C22×C411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8M1614yes—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—44no—
139C22×C8—44no—
140C22×C8—82no—
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C22×C8C22×C411yescharacteristic
144C22×C8C22×C411yescharacteristic
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4M1614yes—
161C4×D4#4112yes—
162C4×D4#4112yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2411yescharacteristic
192#107—22no—
193#107—22no—
194#107—22no—
195C4⋊C8—24no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—44no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—81no—
206C4⋊C8—81no—
207#102—22no—
208#102—22no—
209#102—22no—
210#102—22no—
211#102—22no—
212#102—22no—
213C22⋊C8—44no—
214C22⋊C8—44no—
215C22⋊C8—82no—
216#97—44no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235C24—21no—
236C24—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4#11712yes—
246C2×D4C2×M1612yes—
247C2×D4C2×M1612yes—
248C2×D4C2×M1612yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4111yescharacteristic
286D8—28no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—24no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—88no—
314C2×C8C2×M1612yes—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4#11712yes—
326C4⋊C4C2×M1612yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42C2×M1612yes—
350C42#13112yes—
351C42AGL(1,ℤ/8)12yes—
352C42AGL(1,ℤ/8)12yes—
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C42H(5,𝔽2)11yescharacteristic
356C422−1+411yescharacteristic
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C2×C4×D411yescharacteristic
372D4—22no—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—22no—
377D4—41no—
378D4—41no—
379D4#43214yescomplemented
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—84no—
418C2×C4—81no—
419C2×C4#43512yes—
420C2×C4#43512yes—
421C2×C4#44212yes—
422C2×C4#44212yes—
423C2×C4#44312yes—
424C2×C4C2×#11711yescharacteristic
425C2×C4#52011yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4#54811yescharacteristic
428C2×C4#55211yescharacteristic
429C2×C4#55311yescharacteristic
430C2×C4C22×M1611yescharacteristic
431C2×C4C22×M1611yescharacteristic
432C2×C4C2×#13111yescharacteristic
433C2×C4#56811yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57811yescharacteristic
438C8—22no—
439C8—44no—
440C8—42no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—42no—
445C8—82no—
446C8—82no—
447C22—24no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#235211yescharacteristic
462C22C2×#43511yescharacteristic
463C22D4×M1611yescharacteristic
464C22C2×#44211yescharacteristic
465C22#295111yescharacteristic
466C22C2×#44311yescharacteristic
467C22#295711yescharacteristic
468C4—22no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—41no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4#158614yes—
480C4C2×#43211yescharacteristic
481C4#292511yescharacteristic
482C4#298911yescharacteristic
483C4#299011yescharacteristic
484C4#332511yescharacteristic
485C4#333011yescharacteristic
486C4#336611yescharacteristic
487C4#336711yescharacteristic
488C2—44no—
489C2—82no—
490C2C2×#158611yescharacteristic
491C2#1788111yescharacteristic
492C2#1804411yescharacteristic
493C2#1805511yescharacteristic
494C2#2275011yescharacteristic
495C2#2275811yescharacteristic
496C2#2278311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×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. #22799C2semidirect
2. D4×C4⋊C8C2semidirect
4. #9560C2semidirect
5. #9540C2semidirect
7. #8784C2semidirect
8. #8563C2semidirect
9. #8563C2semidirect
10. #8261C2semidirect
11. #8204C2semidirect
379. D4#432semidirect

Maximal subgroups

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

subgroupindex
#227992
D4×C4⋊C82
C42×D82
#95602
#95402
#89472
#87842
#85632
#82612
#82042