Finite Group Databasebeta

← #225942

Subgroups — #225942

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22700C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4C2×#1586C212yesmaximal
5#9599C212yesmaximal, complemented
6#8847C212yesmaximal, complemented
7#8843C212yesmaximal, complemented
8#8807C211yesmaximal, characteristic, complemented
9#8565C211yesmaximal, characteristic, complemented
10#8565C211yesmaximal, characteristic, complemented
11#8261C211yesmaximal, characteristic, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×#436C2212yes—
16C2×C8×D4C2212yes—
17C42×D4C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1939C2212yes—
20#1939C2212yes—
21#1851—22no—
22#1850—22no—
23#1831C2211yescharacteristic
24#1831C2211yescharacteristic
25#1797—21no—
26#1797—21no—
27C4×#107C411yescharacteristic
28C4×#107C411yescharacteristic
29C4×#107C2211yescharacteristic
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C4×#102C411yescharacteristic
33C4×#102C411yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C2211yescharacteristic
37C4×#102C2211yescharacteristic
38#1728—21no—
39C2×#336—21no—
40C42×C8C2211yescharacteristic
41#1586—28no—
42C2×#335C2212yes—
43C2×#334C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×#322C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1273C2212yes—
50C22×D8C2×C411yescharacteristic
51C22×M16—22no—
52C23×C8—22no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C4×D8—24no—
62C4×D8—24no—
63C4×D8D414yes—
64C4×D8Q814yes—
65#436—24no—
66#436—42no—
67C8×D4—24no—
68C8×D4—42no—
69C2×#107C2×C411yescharacteristic
70C2×#107C2×C411yescharacteristic
71C2×#107C2×C411yescharacteristic
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×#102C2×C411yescharacteristic
79C2×#102C2×C411yescharacteristic
80C2×#102C2×C411yescharacteristic
81C2×#102C2×C411yescharacteristic
82C2×#102C2×C411yescharacteristic
83C2×#102C2×C411yescharacteristic
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8C2×C411yescharacteristic
87C2×C4×C8C2×C411yescharacteristic
88C2×C4×C8C2×C411yescharacteristic
89C2×C4×C8C2×C411yescharacteristic
90C2×C4×C8C2312yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C43C2311yescharacteristic
107#340C2×C411yescharacteristic
108#340C2×C411yescharacteristic
109#340C2×C411yescharacteristic
110#340C2×C411yescharacteristic
111#340C2×C411yescharacteristic
112#340C2×C411yescharacteristic
113#336—22no—
114#336—22no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336D411yescharacteristic
121#336D411yescharacteristic
122#335—42no—
123#334—24no—
124D4⋊C8—24no—
125D4⋊C8—24no—
126#322—24no—
127C22×D4C22×C411yescharacteristic
128C22×D4C22×C411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C2×D8—22no—
136C2×D8—22no—
137C2×D8C4⋊C414yes—
138C2×M16—44no—
139C2×M16—44no—
140C2×M16—82no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—44no—
145C22×C8—44no—
146C22×C8—82no—
147C22×C8C22×C411yescharacteristic
148C22×C8C22×C411yescharacteristic
149C22×C8C22×C411yescharacteristic
150C22×C8C22×C411yescharacteristic
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—41no—
163C4×D4—41no—
164C4×D4M1614yescomplemented
165C4×D4C2×D412yes—
166C4×D4C2×D412yes—
167C4×D4C2×Q812yes—
168C4×D4C2×Q812yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#107—22no—
197#107—22no—
198#107C4⋊C414yes—
199C4⋊C8—24no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205C4⋊C8—82no—
206C4⋊C8—82no—
207#102—22no—
208#102—22no—
209#102—22no—
210#102C4⋊C414yes—
211#102C4⋊C414yes—
212#102M1614yescomplemented
213C22⋊C8—44no—
214C22⋊C8—44no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—21no—
221C4×C8—22no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—41no—
225C4×C8C2×D412yes—
226C4×C8C2×Q812yes—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C24—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4C2×C4⋊C412yes—
244C2×D4C2×C4⋊C412yes—
245C2×D4C2×M1612yes—
246C2×D4C2×M1612yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4111yescharacteristic
280D8—28no—
281M16—88no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—24no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—88no—
307C2×C8C2×C4⋊C412yes—
308C2×C8C2×C4⋊C412yes—
309C2×C8C2×M1612yes—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4C2×C4⋊C412yes—
319C4⋊C4C2×C4⋊C412yes—
320C4⋊C4C2×M1612yes—
321C4⋊C4C2×M1612yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—41no—
336C42—42no—
337C42—42no—
338C42—81no—
339C42—81no—
340C42—81no—
341C42—81no—
342C42C2×M1612yes—
343C42#13112yes—
344C42C22×D411yescharacteristic
345C42C22×Q811yescharacteristic
346C42H(5,𝔽2)11yescharacteristic
347C422−1+411yescharacteristic
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23C2×C4×D411yescharacteristic
363D4—22no—
364D4—22no—
365D4—22no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4#42314yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
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—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4#43512yes—
409C2×C4#43512yes—
410C2×C4#44312yes—
411C2×C4C22×C4⋊C411yescharacteristic
412C2×C4#52011yescharacteristic
413C2×C4D4211yescharacteristic
414C2×C4#54711yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C2×C4#55411yescharacteristic
417C2×C4C22×M1611yescharacteristic
418C2×C4C22×M1611yescharacteristic
419C2×C4C2×#13111yescharacteristic
420C2×C4#56811yescharacteristic
421C2×C4#57512yes—
422C2×C4#57612yes—
423C8—22no—
424C8—22no—
425C8—22no—
426C8—44no—
427C8—41no—
428C8—42no—
429C8—42no—
430C8—82no—
431C8—82no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22D4×C4⋊C411yescharacteristic
447C22C2×#43511yescharacteristic
448C22D4×M1611yescharacteristic
449C22#294311yescharacteristic
450C22#294811yescharacteristic
451C22C2×#44311yescharacteristic
452C22#295711yescharacteristic
453C4—22no—
454C4—21no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4C2×#42311yescharacteristic
466C4#290611yescharacteristic
467C4#298911yescharacteristic
468C4#299011yescharacteristic
469C4#327312yes—
470C4#338912yes—
471C2—44no—
472C2—82no—
473C2#1786811yescharacteristic
474C2#1787711yescharacteristic
475C2#1804111yescharacteristic
476C2#1805411yescharacteristic
477C2#2264811yescharacteristic
478C2#2267211yescharacteristic
479C2#2268811yescharacteristic
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. #22700C2semidirect
2. #22696C2semidirect
5. #9599C2semidirect
6. #8847C2semidirect
7. #8843C2semidirect
8. #8807C2semidirect
9. #8565C2semidirect
10. #8565C2semidirect
11. #8261C2semidirect
164. C4×D4M16semidirect
212. #102M16semidirect

Maximal subgroups

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

subgroupindex
#227002
#226962
C42×D82
C2×#15862
#95992
#88472
#88432
#88072
#85652
#82612