Finite Group Databasebeta

← #406778

Subgroups — #406778

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20844C211yesmaximal, characteristic
2#20835C211yesmaximal, characteristic
3#20701C211yesmaximal, characteristic
4C2×#1822C211yesmaximal, characteristic
5C2×#1811C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7#11303C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11242C212yesmaximal, complemented
10#11216C212yesmaximal, complemented
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2212yes—
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2280C2211yescharacteristic
23#1853—22no—
24#1852—22no—
25#1851—24no—
26#1825—22no—
27#1825—22no—
28#1822—24no—
29#1811—24no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2212yes—
35#1736—24no—
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1729C2211yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—42no—
49C22×M16D414yes—
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117Q814yes—
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C2×#108—24no—
74C2×#108—24no—
75C2×#108—42no—
76#428—24no—
77#428—42no—
78C2×#107—22no—
79C2×#106—22no—
80#424—24no—
81#424—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84#407—22no—
85#407—22no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407D414yes—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93#381—21no—
94#380D412yes—
95#380D412yes—
96#379—21no—
97C4×C22⋊C4—22no—
98#343—24no—
99#343—24no—
100#343—24no—
101#343—42no—
102#341—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—24no—
109#336—22no—
110#336—22no—
111C25#4111yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—44no—
119C23×C4—42no—
120C23×C4—84no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×Q812yes—
125C23×C4#4111yescharacteristic
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4C2411yescharacteristic
129C2×M16—28no—
130C2×M16—22no—
131C2×M16—28no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—44no—
151C2×M16—48no—
152C2×M16—48no—
153C2×M16C2×D414yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8#4114yes—
168#117—28no—
169#117—28no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—28no—
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—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×Q814yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C42—24no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42—42no—
208#108—48no—
209#107—44no—
210#106—44no—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C22⋊C8—28no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216#95—22no—
217#95—22no—
218C24—22no—
219C24—22no—
220C24—42no—
221C24—42no—
222C24#12112yes—
223C24C22⋊Q812yes—
224C24#12612yes—
225C24C2×#4111yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—48no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—816no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—84no—
258C22×C4#12114yes—
259C22×C4#12114yes—
260C22×C4#12112yes—
261C22×C4C22⋊Q814yes—
262C22×C4C22⋊Q814yes—
263C22×C4C22⋊Q812yes—
264C22×C4#12612yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4H(5,𝔽2)12yes—
274C22×C42−1+412yes—
275M16—24no—
276M16—24no—
277M16—416no—
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8#12314yes—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—48no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—48no—
332C23—22no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—24no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—48no—
346C23—44no—
347C23#39614yes—
348C23C2×#12111yescharacteristic
349C23C2×C22⋊Q811yescharacteristic
350C23C2×#12611yescharacteristic
351C23#55112yes—
352C23#55212yes—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—816no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4#39614yes—
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×C22⋊Q811yescharacteristic
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4#52511yescharacteristic
387C2×C4#52912yes—
388C2×C4#53312yes—
389C2×C4#53411yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4#53912yes—
392C2×C4#54412yes—
393C2×C4#54612yes—
394C2×C4#54612yes—
395C2×C4#54712yes—
396C2×C4#54712yes—
397C2×C4#55112yes—
398C2×C4#55212yes—
399C8—24no—
400C8—42no—
401C8—44no—
402C8—81no—
403C8—81no—
404C8—82no—
405C22—24no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—24no—
411C22—24no—
412C22—24no—
413C22—48no—
414C22—48no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22C2×#39611yescharacteristic
420C22#243812yes—
421C22#248611yescharacteristic
422C22#249312yes—
423C22#253411yescharacteristic
424C22#306812yes—
425C22#306912yes—
426C4—24no—
427C4—82no—
428C4—84no—
429C4—81no—
430C4—84no—
431C4—81no—
432C4#241211yescharacteristic
433C4#241311yescharacteristic
434C4#242512yes—
435C4#243812yes—
436C4#262711yescharacteristic
437C4#285511yescharacteristic
438C2—22no—
439C2—22no—
440C2—44no—
441C2#1493511yescharacteristic
442C2#2360412yes—
443C2#2767612yes—
444C2#2813711yescharacteristic
445C2#2813811yescharacteristic
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
7. #11303C2semidirect
8. #11249C2semidirect
9. #11242C2semidirect
10. #11216C2semidirect

Maximal subgroups

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

subgroupindex
#208442
#208352
#207012
C2×#18222
C2×#18112
C2×#17362
#113032
#112492
#112422
#112162
#107272