Finite Group Databasebeta

← #316360

Subgroups — #316360

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27681C211yesmaximal, characteristic
2#24301C211yesmaximal, characteristic
3#21597C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic
5#10588C212yesmaximal, complemented
6#10486C212yesmaximal, complemented
7#9569C212yesmaximal, complemented
8#9228C212yesmaximal, complemented
9#8725C211yesmaximal, characteristic
10#8595C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#475C2212yes—
14C2×#446C2211yescharacteristic
15C22×#103—21no—
16C22×#103—21no—
17C22×C4×C8C2211yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2100—21no—
21#2095—21no—
22#2058—21no—
23#2024—21no—
24#1988C2212yes—
25#1984C2211yescharacteristic
26#1956—21no—
27#1956—21no—
28#1956C2212yes—
29#1956C2212yes—
30#1947C2212yes—
31#1939C2211yescharacteristic
32#1916—21no—
33#1872—21no—
34#1872—21no—
35#1853—21no—
36#1852—21no—
37#1850—21no—
38#1803C2212yes—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1771—21no—
43#1770C2211yescharacteristic
44#1752—21no—
45C2×#334C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8C2312yes—
60C2×C22⋊Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#475—24no—
69#475—42no—
70#446—42no—
71C2×#107—22no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—44no—
86C2×#103—44no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—42no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391D411yescharacteristic
98#389D411yescharacteristic
99#387—21no—
100#385—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2312yes—
105#380D411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#334—42no—
121#332—24no—
122#332—42no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C2×D411yescharacteristic
132C23×C4—41no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—42no—
158C4⋊Q8#4114yes—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42C2411yescharacteristic
195#107—82no—
196#106—44no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205#103—44no—
206#103—44no—
207#103—44no—
208#103—44no—
209#103—44no—
210#103—82no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214#97—42no—
215#97—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225C24#12111yescharacteristic
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—22no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8—84no—
241C2×Q8—84no—
242C2×Q8—82no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4#13712yes—
270C22×C4#13712yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8AGL(1,ℤ/8)12yes—
299C2×C8AGL(1,ℤ/8)12yes—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42C2×#4112yes—
332C42H(5,𝔽2)12yes—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#46412yes—
338C23#47012yes—
339C23#47112yes—
340C23C2×#12111yescharacteristic
341Q8—44no—
342Q8—44no—
343Q8—42no—
344Q8—42no—
345Q8—88no—
346Q8—82no—
347Q8—82no—
348C2×C4—28no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—88no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—164no—
383C2×C4#46412yes—
384C2×C4#47012yes—
385C2×C4#47312yes—
386C2×C4#52511yescharacteristic
387C2×C4#53411yescharacteristic
388C2×C4#54612yes—
389C2×C4#55112yes—
390C2×C4C2×#13511yescharacteristic
391C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57712yes—
397C8—22no—
398C8—22no—
399C8—41no—
400C8—41no—
401C8—81no—
402C8—81no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#170414yes—
412C22#253411yescharacteristic
413C22C2×#46411yescharacteristic
414C22#303811yescharacteristic
415C22C2×#47011yescharacteristic
416C22#305711yescharacteristic
417C22C2×#47111yescharacteristic
418C22C2×#47311yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—84no—
427C4—81no—
428C4—162no—
429C4#310511yescharacteristic
430C4#311011yescharacteristic
431C4#318811yescharacteristic
432C4#319311yescharacteristic
433C4#331612yes—
434C4#332012yes—
435C2—24no—
436C2C2×#170411yescharacteristic
437C2#1921311yescharacteristic
438C2#1929311yescharacteristic
439C2#1935211yescharacteristic
440C2#2810811yescharacteristic
441C2#2812311yescharacteristic
442C2#2813511yescharacteristic
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. #10588C2semidirect
6. #10486C2semidirect
7. #9569C2semidirect
8. #9228C2semidirect

Maximal subgroups

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

subgroupindex
#276812
#243012
#215972
C2×#19562
#105882
#104862
#95692
#92282
#87252
#85952
#83402