Finite Group Databasebeta

← #407264

Subgroups — #407264

Order 512.

1126 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#20845C211yesmaximal, characteristic
2#20844C211yesmaximal, characteristic
3#20835C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#11290C212yesmaximal, complemented
6#11290C212yesmaximal, complemented
7#11216C212yesmaximal, complemented
8#11216C212yesmaximal, complemented
9#10728C211yesmaximal, characteristic
10#10727C211yesmaximal, characteristic
11#10724C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#428C2212yes—
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2280C2211yescharacteristic
23#1854—22no—
24#1854—22no—
25#1853—22no—
26#1852—22no—
27#1851—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#341C2212yes—
33C2×#341C2212yes—
34#1736—24no—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1729C2211yescharacteristic
40#1729C2211yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—42no—
46C22×M16—42no—
47C22×M16D414yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C42—21no—
69#428—24no—
70#428—24no—
71#428—42no—
72#428—42no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77#424—42no—
78#424—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407—22no—
85#407Q814yes—
86#407Q814yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90#381—21no—
91#380D412yes—
92#380D412yes—
93#379—21no—
94C4×C22⋊C4—22no—
95#343—24no—
96#343—24no—
97#343—42no—
98#343—42no—
99#341—42no—
100#341—42no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108C25#4111yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—44no—
116C23×C4—42no—
117C23×C4—84no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×Q812yes—
121C23×C4C2×Q812yes—
122C23×C4#4111yescharacteristic
123C23×C4#4112yes—
124C23×C4#4112yes—
125C23×C4C2411yescharacteristic
126C2×M16—28no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—48no—
148C2×M16—48no—
149C2×M16C2×Q814yes—
150C2×M16C2×Q814yes—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165#117—28no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—48no—
171#117—44no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4#4114yes—
191C2×C4⋊C4#4114yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—44no—
201C2×C42—42no—
202#107—44no—
203#107—44no—
204#106—44no—
205#106—44no—
206C4⋊C8—44no—
207C4⋊C8—44no—
208C22⋊C8—28no—
209C22⋊C8—28no—
210C22⋊C8—44no—
211#95—22no—
212#95—22no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—42no—
217C24#12112yes—
218C24#12512yes—
219C24#12512yes—
220C24C2×#4111yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—48no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—816no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4#12112yes—
254C22×C4C22⋊Q814yes—
255C22×C4C22⋊Q814yes—
256C22×C4#12314yes—
257C22×C4#12512yes—
258C22×C4#12512yes—
259C22×C4C4⋊Q814yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)12yes—
269C22×C4H(5,𝔽2)12yes—
270M16—24no—
271M16—24no—
272M16—24no—
273M16—24no—
274M16—416no—
275M16—42no—
276M16—42no—
277M16—42no—
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C22⋊C4—48no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—48no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—24no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23#39814yes—
345C23C2×#12111yescharacteristic
346C23C2×#12511yescharacteristic
347C23C2×#12511yescharacteristic
348C23#55112yes—
349C23#55112yes—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—816no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4#39814yes—
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4#52511yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#52912yes—
384C2×C4#52912yes—
385C2×C4C2×C4⋊Q811yescharacteristic
386C2×C4#53411yescharacteristic
387C2×C4#53411yescharacteristic
388C2×C4#53812yes—
389C2×C4#53812yes—
390C2×C4#54612yes—
391C2×C4#54612yes—
392C2×C4#55412yes—
393C2×C4#55412yes—
394C2×C4#55412yes—
395C2×C4#55412yes—
396C8—42no—
397C8—42no—
398C8—44no—
399C8—44no—
400C8—81no—
401C8—81no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—48no—
412C22—48no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22C2×#39811yescharacteristic
418C22#250512yes—
419C22#250512yes—
420C22#253411yescharacteristic
421C22#253411yescharacteristic
422C22#306812yes—
423C22#306912yes—
424C4—24no—
425C4—81no—
426C4—84no—
427C4—81no—
428C4—82no—
429C4—84no—
430C4#245012yes—
431C4#245012yes—
432C4#258311yescharacteristic
433C4#258311yescharacteristic
434C4#262211yescharacteristic
435C4#285311yescharacteristic
436C2—22no—
437C2—22no—
438C2—44no—
439C2#1495511yescharacteristic
440C2#2367512yes—
441C2#2813711yescharacteristic
442C2#2813811yescharacteristic
443C2#2813912yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11290C2semidirect
6. #11290C2semidirect
7. #11216C2semidirect
8. #11216C2semidirect

Maximal subgroups

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

subgroupindex
#208452
#208442
#208352
C2×#17362
#112902
#112162
#107282
#107272
#107242