Finite Group Databasebeta

← #414738

Subgroups — #414738

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C212yesmaximal
2#35011C211yesmaximal, characteristic
3C2×#1996C212yesmaximal
4C4×#381C211yesmaximal, characteristic
5#11344C214yesmaximal
6#11070C214yesmaximal
7#10846C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C23×C42C2211yescharacteristic
10#2857—22no—
11#2855—28no—
12#2855—28no—
13#2659—22no—
14#2583—24no—
15#2502—22no—
16#2434—22no—
17#2315C2214yes—
18C2×#391—22no—
19C2×#389—22no—
20C2×#387—24no—
21C2×#385C2214yes—
22C2×#381C2214yes—
23C2×#381C2212yes—
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1996—28no—
28C2×#342C2218yes—
29C2×#342C2214yes—
30#1434—22no—
31C22⋊#95C2212yes—
32C4×#95—22no—
33C23×Q8D412yes—
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C2×C4⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4D414yes—
49C22×C4⋊C4D412yes—
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2312yes—
54C22×C42—24no—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42—21no—
61C22×C42D414yes—
62C22×C42D412yes—
63#398—28no—
64#395—28no—
65#391—216no—
66#391—24no—
67#389—28no—
68#389—216no—
69#387—232no—
70#385—28no—
71#385—28no—
72#385—28no—
73#385—28no—
74#384—22no—
75#381—28no—
76#381—28no—
77#381—24no—
78#381Q818yescomplemented
79#380—28no—
80#380—24no—
81#380D418yes—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—24no—
85C2×#95—24no—
86C2×#95—22no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95D412yes—
93#342—216no—
94#342—28no—
95C25C2411yescharacteristic
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—22no—
99C22×Q8—24no—
100C22×Q8—48no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D414yes—
119C23×C4C2×D414yes—
120C23×C4C2×D412yes—
121C23×C4C2×D414yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D414yes—
126C23×C4C2×Q814yes—
127C23×C4#4112yes—
128C4⋊Q8—84no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—416no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4C2×D418yes—
166C2×C22⋊C4C2×Q814yes—
167C2×C22⋊C4C2×Q814yes—
168C2×C22⋊C4#4114yes—
169C2×C42—22no—
170C2×C42—28no—
171C2×C42—28no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—42no—
185#95—216no—
186#95—216no—
187#95—216no—
188#95—216no—
189#95—28no—
190#95—24no—
191#95—216no—
192#95—48no—
193#95—42no—
194C24—42no—
195C24—42no—
196C24C22×D412yes—
197C24C22×D411yescharacteristic
198C24C22×D411yescharacteristic
199C24C22×D412yes—
200C24C22×Q812yes—
201C24C2×#4111yescharacteristic
202C24C2×#4112yes—
203C242−1+414yes—
204C2×Q8—416no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—48no—
208C2×Q8—84no—
209C2×Q8—88no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4#12014yes—
266C22×C4#12012yes—
267C22×C4#12011yescharacteristic
268C22×C4#12014yes—
269C22×C4#12011yescharacteristic
270C22×C4#12112yes—
271C22×C4#12114yes—
272C22×C4#12112yes—
273C22×C4#12114yes—
274C22×C4C22⋊Q818yes—
275C22×C4#12712yes—
276C22×C4#12712yes—
277C22×C4C4⋊Q814yescomplemented
278C4⋊C4—416no—
279C4⋊C4—432no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—88no—
285C4⋊C4—88no—
286C22⋊C4—28no—
287C22⋊C4—28no—
288C22⋊C4—216no—
289C22⋊C4—28no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C42—44no—
298C42—48no—
299C42—44no—
300C42—44no—
301C42—82no—
302C42—84no—
303C23—22no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23—41no—
308C23—42no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—44no—
314C23—44no—
315C23—48no—
316C23C2×#12012yes—
317C23C2×#12011yescharacteristic
318C23C2×#12111yescharacteristic
319C23C2×#12112yes—
320C23C2×C22⋊Q812yes—
321C23C2×#12711yescharacteristic
322C23C2×C4⋊Q811yescharacteristic
323C23#53211yescharacteristic
324C23#53312yes—
325C23#53612yes—
326C23#53612yes—
327C23#54112yes—
328C23#54714yes—
329C23#54814yes—
330C23D4×Q814yes—
331C23D4×Q814yes—
332Q8—88no—
333Q8—88no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—816no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#39212yes—
383C2×C4#39214yes—
384C2×C4#39212yes—
385C2×C4#45814yes—
386C22—24no—
387C22—24no—
388C22—28no—
389C22—28no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—44no—
404C22—44no—
405C22C2×#39211yescharacteristic
406C22#240912yes—
407C22#242914yes—
408C22#243212yes—
409C22#243412yes—
410C22#254212yes—
411C22#262111yescharacteristic
412C22#285711yescharacteristic
413C22C2×#45812yes—
414C22#302512yes—
415C4—22no—
416C4—24no—
417C4—48no—
418C4—44no—
419C4—44no—
420C4—88no—
421C4—84no—
422C4—84no—
423C4—84no—
424C4#202214yes—
425C2—22no—
426C2—22no—
427C2—21no—
428C2—21no—
429C2—24no—
430C2—22no—
431C2#1517811yescharacteristic
432C2C2×#202211yescharacteristic
433C2#2344311yescharacteristic
434C2#2435114yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
78. #381Q8semidirect
277. C22×C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#350112
C2×#19962
C4×#3812
#113442
#110702
#108462