Finite Group Databasebeta

← #226540

Subgroups — #226540

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22839C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9560C212yesmaximal, complemented
5#9537C212yesmaximal, complemented
6#8947C212yesmaximal
7#8785C211yesmaximal, characteristic, complemented
8#8565C211yesmaximal, characteristic, complemented
9#8565C211yesmaximal, characteristic, complemented
10#8261C211yesmaximal, characteristic, complemented
11#8204C212yesmaximal, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×C8×D4C2212yes—
16C42×D4C2211yescharacteristic
17C42×D4C2211yescharacteristic
18#1954C2212yes—
19#1939C2212yes—
20#1924—22no—
21#1850—22no—
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1817—21no—
25#1797—21no—
26C4×#107C411yescharacteristic
27C4×#107C411yescharacteristic
28C4×#107C2211yescharacteristic
29#1770C2212yes—
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C4×#102C411yescharacteristic
33C4×#102C411yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C2211yescharacteristic
37C4×#102C2211yescharacteristic
38#1749—21no—
39C2×#336—21no—
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—
65C2×#107C2×C411yescharacteristic
66C2×#107C2×C411yescharacteristic
67C2×#107C2×C411yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#102C2×C411yescharacteristic
74C2×#102C2×C411yescharacteristic
75C2×#102C2×C411yescharacteristic
76C2×#102C2×C411yescharacteristic
77C2×#102C2×C411yescharacteristic
78C2×#102C2×C411yescharacteristic
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2×C411yescharacteristic
86C2×C4×C8C2×C411yescharacteristic
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2312yes—
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C4×C4⋊C4C2311yescharacteristic
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C43C2311yescharacteristic
104#340C2×C411yescharacteristic
105#340C2×C411yescharacteristic
106#340C2×C411yescharacteristic
107#340C2×C411yescharacteristic
108#340C2×C411yescharacteristic
109#340C2×C411yescharacteristic
110#336—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336D411yescharacteristic
118#336D411yescharacteristic
119D4⋊C8—24no—
120D4⋊C8—42no—
121#322—42no—
122C22×D4C22×C411yescharacteristic
123C22×D4C22×C411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8M1614yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—44no—
138C22×C8—82no—
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4M1614yes—
159C4×D4#4112yes—
160C4×D4#4112yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—82no—
180C2×C42—82no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—22no—
189#107—22no—
190#107—22no—
191C4⋊C8—24no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—42no—
197C4⋊C8—82no—
198#102—22no—
199#102—22no—
200#102—22no—
201#102—22no—
202#102—22no—
203#102—22no—
204C22⋊C8—44no—
205C22⋊C8—44no—
206C22⋊C8—82no—
207#97—82no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—41no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24—21no—
227C24—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4#11712yes—
237C2×D4C2×M1612yes—
238C2×D4C2×M1612yes—
239C2×D4C2×M1612yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273D8—28no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—24no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—88no—
301C2×C8C2×M1612yes—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4#11712yes—
313C4⋊C4C2×M1612yes—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C42—21no—
323C42—21no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—42no—
329C42—42no—
330C42—81no—
331C42—81no—
332C42—81no—
333C42—81no—
334C42C2×M1612yes—
335C42#13112yes—
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C42H(5,𝔽2)11yescharacteristic
339C422−1+411yescharacteristic
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C2×C4×D411yescharacteristic
355D4—22no—
356D4—22no—
357D4—22no—
358D4—22no—
359D4—22no—
360D4—41no—
361D4—41no—
362D4#43214yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#43512yes—
401C2×C4#43512yes—
402C2×C4#44312yes—
403C2×C4C2×#11711yescharacteristic
404C2×C4#52011yescharacteristic
405C2×C4#54611yescharacteristic
406C2×C4#54811yescharacteristic
407C2×C4#55211yescharacteristic
408C2×C4#55311yescharacteristic
409C2×C4C22×M1611yescharacteristic
410C2×C4C22×M1611yescharacteristic
411C2×C4C2×#13111yescharacteristic
412C2×C4#56811yescharacteristic
413C2×C4#57512yes—
414C2×C4#57612yes—
415C8—22no—
416C8—44no—
417C8—41no—
418C8—41no—
419C8—41no—
420C8—42no—
421C8—42no—
422C8—82no—
423C8—82no—
424C22—24no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22#235211yescharacteristic
439C22C2×#43511yescharacteristic
440C22D4×M1611yescharacteristic
441C22#294311yescharacteristic
442C22#294811yescharacteristic
443C22C2×#44311yescharacteristic
444C22#295711yescharacteristic
445C4—22no—
446C4—22no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4C2×#43211yescharacteristic
458C4#292511yescharacteristic
459C4#298911yescharacteristic
460C4#299011yescharacteristic
461C4#332212yes—
462C4#337012yes—
463C2—44no—
464C2—82no—
465C2#1786811yescharacteristic
466C2#1787711yescharacteristic
467C2#1804511yescharacteristic
468C2#1805411yescharacteristic
469C2#2275011yescharacteristic
470C2#2276411yescharacteristic
471C2#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. #22839C2semidirect
2. #22696C2semidirect
4. #9560C2semidirect
5. #9537C2semidirect
7. #8785C2semidirect
8. #8565C2semidirect
9. #8565C2semidirect
10. #8261C2semidirect
11. #8204C2semidirect

Maximal subgroups

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

subgroupindex
#228392
#226962
C42×D82
#95602
#95372
#89472
#87852
#85652
#82612
#82042