Finite Group Databasebeta

← #274106

Subgroups — #274106

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33146C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10590C212yesmaximal, complemented
6#10542C212yesmaximal, complemented
7#9736C212yesmaximal, complemented
8#9566C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C2×#431—21no—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103—21no—
17C22×#103—21no—
18#2857C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#2024—22no—
21#1990C2212yes—
22#1988C2211yescharacteristic
23#1988C2211yescharacteristic
24#1981—21no—
25#1981—21no—
26#1981C2212yes—
27#1981C2212yes—
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1939C2212yes—
31#1918—21no—
32#1918—21no—
33#1916—22no—
34#1872—21no—
35#1872—21no—
36#1852—21no—
37#1823—21no—
38#1817—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41C8×C4⋊C4C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43C2×#334C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#487—24no—
66#487—42no—
67#431—44no—
68C2×#107—21no—
69C2×#107—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—44no—
84C2×#103—44no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8D411yescharacteristic
88C2×C22⋊C8D411yescharacteristic
89C2×C4×C8—22no—
90C2×C4×C8C2312yes—
91#391—21no—
92#389—21no—
93#389—21no—
94#384C2311yescharacteristic
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103C2×#95—21no—
104#340—22no—
105#340—22no—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—21no—
111#340—21no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336D412yes—
117#334—42no—
118#332—24no—
119#332—42no—
120#332—42no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—41no—
128C22×Q8—42no—
129C22×Q8—42no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C4⋊Q8—24no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—44no—
152C4⋊Q8—44no—
153C4⋊Q8—42no—
154C4⋊Q8#4114yes—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4SD1614yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2411yescharacteristic
192#107—42no—
193#107—42no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—42no—
201C4⋊C8—44no—
202#103—44no—
203#103—44no—
204#103—44no—
205#103—44no—
206#103—44no—
207#103—44no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—44no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—41no—
220#95—41no—
221C24#12111yescharacteristic
222C2×Q8—24no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—82no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4C2×SD1612yes—
268C22×C4#13512yes—
269C22×C4#13512yes—
270C22×C4AGL(1,ℤ/8)12yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—84no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—82no—
315C4⋊C4C2×SD1612yes—
316C4⋊C4C2×SD1612yes—
317C4⋊C4AGL(1,ℤ/8)12yes—
318C22⋊C4—82no—
319C42—21no—
320C42—21no—
321C42—22no—
322C42—22no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42C2×#4112yes—
330C42H(5,𝔽2)12yes—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#46312yes—
336C23#46512yes—
337C23#47312yes—
338C23C2×#12111yescharacteristic
339Q8—44no—
340Q8—44no—
341Q8—88no—
342Q8—82no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4#46312yes—
387C2×C4#46512yes—
388C2×C4#52511yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#54612yes—
391C2×C4#55112yes—
392C2×C4C22×SD1611yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
397C2×C4#57511yescharacteristic
398C2×C4#57812yes—
399C8—42no—
400C8—42no—
401C8—81no—
402C8—82no—
403C8—81no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#167914yes—
412C22#241111yescharacteristic
413C22C2×#46311yescharacteristic
414C22#304411yescharacteristic
415C22C2×#46511yescharacteristic
416C22#305511yescharacteristic
417C22C2×#47311yescharacteristic
418C22#306911yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—41no—
425C4—42no—
426C4—84no—
427C4—81no—
428C4—161no—
429C4—161no—
430C4#310111yescharacteristic
431C4#310511yescharacteristic
432C4#320811yescharacteristic
433C4#321111yescharacteristic
434C4#330012yes—
435C4#330812yes—
436C2—24no—
437C2C2×#167911yescharacteristic
438C2#1916911yescharacteristic
439C2#1940211yescharacteristic
440C2#1947611yescharacteristic
441C2#2448111yescharacteristic
442C2#2450111yescharacteristic
443C2#2452211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10590C2semidirect
6. #10542C2semidirect
7. #9736C2semidirect
8. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#331462
#259022
#225992
C2×#19812
#105902
#105422
#97362
#95662
#91292
#88302
#83402