Finite Group Databasebeta

← #274244

Subgroups — #274244

Order 512.

753 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#33108C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10600C212yesmaximal, complemented
6#10542C212yesmaximal, complemented
7#9748C212yesmaximal, complemented
8#9570C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C2×#483—21no—
15C2×#432—21no—
16C22×C4⋊C8C2211yescharacteristic
17#2857C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2043—22no—
20#1988—21no—
21#1988—21no—
22#1988C2212yes—
23#1988C2212yes—
24#1981C2211yescharacteristic
25#1981C2211yescharacteristic
26#1956C2212yes—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1939C2212yes—
30#1927—21no—
31#1927—21no—
32#1917—22no—
33#1872—21no—
34#1817—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1787—21no—
38#1787—21no—
39#1770C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#335C2212yes—
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#483—44no—
68#432—44no—
69C2×#106—21no—
70C2×#106—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—42no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—22no—
85C2×#103—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8D411yescharacteristic
89C2×C22⋊C8D411yescharacteristic
90C2×#97—22no—
91C2×C4×C8C2312yes—
92#391—21no—
93#389—21no—
94#389—21no—
95#384C2311yescharacteristic
96#380—21no—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103C4×C4⋊C4C2311yescharacteristic
104C2×#95—21no—
105#340—21no—
106#340—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#336D412yes—
116#335—42no—
117#332—24no—
118#332—42no—
119#332—42no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—44no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—82no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C4⋊Q8—24no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—42no—
151C4⋊Q8#4114yes—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42C2411yescharacteristic
190#106—42no—
191#106—42no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—42no—
199C4⋊C8—44no—
200#103—42no—
201#103—42no—
202#103—42no—
203#103—44no—
204#103—44no—
205#103—42no—
206#103—82no—
207#103—82no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212#97—44no—
213C4×C8—22no—
214C4×C8—22no—
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—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×Q8—82no—
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—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
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×C4#13512yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4AGL(1,ℤ/8)12yes—
270C22×C4#13712yes—
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—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—84no—
293C4⋊C4—21no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—81no—
314C4⋊C4—82no—
315C4⋊C4#13712yes—
316C22⋊C4—82no—
317C42—21no—
318C42—21no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42C2×#4112yes—
328C42H(5,𝔽2)12yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#46312yes—
334C23#46812yes—
335C23#47312yes—
336C23C2×#12111yescharacteristic
337Q8—44no—
338Q8—44no—
339Q8—88no—
340Q8—82no—
341Q8—82no—
342Q8—81no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
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—21no—
361C2×C4—21no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—84no—
376C2×C4—88no—
377C2×C4—84no—
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#46812yes—
388C2×C4#52511yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#54612yes—
391C2×C4#55112yes—
392C2×C4C2×#13511yescharacteristic
393C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4#57511yescharacteristic
398C2×C4#57712yes—
399C8—42no—
400C8—42no—
401C8—81no—
402C8—81no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#169914yes—
412C22#241111yescharacteristic
413C22C2×#46311yescharacteristic
414C22#304311yescharacteristic
415C22C2×#46811yescharacteristic
416C22#305611yescharacteristic
417C22C2×#47311yescharacteristic
418C22#306911yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—41no—
423C4—42no—
424C4—84no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—161no—
429C4—161no—
430C4#311211yescharacteristic
431C4#311711yescharacteristic
432C4#321311yescharacteristic
433C4#321811yescharacteristic
434C4#329612yes—
435C4#330612yes—
436C2—24no—
437C2C2×#169911yescharacteristic
438C2#1916711yescharacteristic
439C2#1943311yescharacteristic
440C2#1947111yescharacteristic
441C2#2447711yescharacteristic
442C2#2450911yescharacteristic
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. #10600C2semidirect
6. #10542C2semidirect
7. #9748C2semidirect
8. #9570C2semidirect

Maximal subgroups

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

subgroupindex
#331082
#258562
#226032
C2×#19882
#106002
#105422
#97482
#95702
#91312
#88292
#83402