Finite Group Databasebeta

← #274256

Subgroups — #274256

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33109C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3#22629C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10592C212yesmaximal, complemented
6#10542C212yesmaximal, complemented
7#9748C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C2×#484—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2857C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2043—22no—
19#1988—21no—
20#1988—21no—
21#1988C2212yes—
22#1988C2212yes—
23#1981C2211yescharacteristic
24#1981C2211yescharacteristic
25#1956C2212yes—
26#1947C2211yescharacteristic
27#1947C2211yescharacteristic
28#1939C2212yes—
29#1928—21no—
30#1928—21no—
31#1917—22no—
32#1872—21no—
33#1797—21no—
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1787—21no—
37#1787—21no—
38#1770C2212yes—
39#1749—21no—
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#484—44no—
68C2×#107—21no—
69C2×#107—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—21no—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8D411yescharacteristic
88C2×C22⋊C8D411yescharacteristic
89C2×#97—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—21no—
105#340—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#336D412yes—
115#335—42no—
116#332—24no—
117#332—42no—
118#332—42no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—44no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C4⋊Q8—24no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C4⋊Q8—42no—
150C4⋊Q8#4114yes—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
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—41no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—41no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—82no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#107—42no—
189#107—42no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—42no—
197C4⋊C8—82no—
198#103—42no—
199#103—42no—
200#103—42no—
201#103—44no—
202#103—44no—
203#103—42no—
204#103—82no—
205#103—82no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210#97—82no—
211C4×C8—22no—
212C4×C8—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—41no—
218#95—41no—
219C24#12111yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—88no—
234C2×Q8—88no—
235C2×Q8—82no—
236C2×Q8—82no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#13512yes—
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4#13712yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—84no—
291C4⋊C4—21no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C4⋊C4#13712yes—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—81no—
322C42—81no—
323C42C2×#4112yes—
324C42H(5,𝔽2)12yes—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#46412yes—
330C23#46812yes—
331C23#47312yes—
332C23C2×#12111yescharacteristic
333Q8—44no—
334Q8—44no—
335Q8—88no—
336Q8—82no—
337Q8—82no—
338Q8—81no—
339Q8—81no—
340Q8—81no—
341Q8—81no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—88no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#46412yes—
383C2×C4#46812yes—
384C2×C4#52511yescharacteristic
385C2×C4#53511yescharacteristic
386C2×C4#54612yes—
387C2×C4#55112yes—
388C2×C4C2×#13511yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C2×C4#57612yes—
395C8—42no—
396C8—42no—
397C8—81no—
398C8—81no—
399C8—82no—
400C8—82no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#170014yes—
408C22#241111yescharacteristic
409C22C2×#46411yescharacteristic
410C22#304111yescharacteristic
411C22C2×#46811yescharacteristic
412C22#305611yescharacteristic
413C22C2×#47311yescharacteristic
414C22#306911yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—42no—
419C4—41no—
420C4—81no—
421C4—84no—
422C4—81no—
423C4—81no—
424C4—161no—
425C4—161no—
426C4#311211yescharacteristic
427C4#311711yescharacteristic
428C4#321311yescharacteristic
429C4#321811yescharacteristic
430C4#329712yes—
431C4#331412yes—
432C2—24no—
433C2C2×#170011yescharacteristic
434C2#1917511yescharacteristic
435C2#1942111yescharacteristic
436C2#1945911yescharacteristic
437C2#2447911yescharacteristic
438C2#2450911yescharacteristic
439C2#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. #10592C2semidirect
6. #10542C2semidirect
7. #9748C2semidirect
8. #9569C2semidirect

Maximal subgroups

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

subgroupindex
#331092
#258562
#226292
C2×#19882
#105922
#105422
#97482
#95692
#91312
#88292
#83402