Finite Group Databasebeta

← #193202

Subgroups — #193202

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#8929C214yesmaximal, complemented
7#8927C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15#2369—22no—
16#2369—22no—
17#2369—22no—
18#2369C2214yes—
19C2×#391—22no—
20C2×#389—21no—
21C2×#389—21no—
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1981—22no—
28#1956C2212yes—
29#1956C2212yes—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1947C2212yes—
34#1947C2212yes—
35#1852—21no—
36#1852—21no—
37#1787—22no—
38C4×#107—22no—
39#1736—21no—
40#1548C418yescomplemented
41C2×#333C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8D412yes—
45C22×M16—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C414yes—
51C2×C4⋊Q8D412yes—
52C2×C4⋊Q8Q814yes—
53C2×C4⋊Q8C2312yes—
54C2×#117—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2×C414yes—
70#428—44no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—22no—
75#423—22no—
76#423C2×C414yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—24no—
82C2×#103—24no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8C2312yes—
87#391—28no—
88#391—28no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389—24no—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—21no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#333C2×C4116yescomplemented
121C22×Q8—22no—
122C22×Q8—24no—
123C22×Q8—82no—
124C22×Q8—82no—
125C22×Q8C4⋊C418yes—
126C22×Q8C2×D414yes—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—24no—
136C2×M16—42no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8C4⋊C418yes—
149C4⋊Q8C22×C414yes—
150C4⋊Q8C2×Q814yes—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4Q1614yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—41no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—28no—
189C2×C42—44no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—41no—
193C2×C42—84no—
194C2×C42C22×C412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42C2411yescharacteristic
202#107—44no—
203#107—44no—
204#107—44no—
205#106—44no—
206C4⋊C8—24no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—42no—
210C4⋊C8C22×C414yes—
211C4⋊C8C22×C414yes—
212#103—84no—
213#103—84no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216C4×C8—24no—
217C4×C8—22no—
218C4×C8—22no—
219#95—28no—
220#95—28no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—41no—
226#95—41no—
227C24#12011yescharacteristic
228C2×Q8—24no—
229C2×Q8—28no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8C2×C4⋊C418yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4C2×C4⋊C412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4#12312yes—
272C22×C4C2×Q1612yes—
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—48no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—88no—
314C4⋊C4C2×Q1614yes—
315C4⋊C4#13714yes—
316C22⋊C4—82no—
317C42—44no—
318C42—42no—
319C42—42no—
320C42—44no—
321C42—44no—
322C42—88no—
323C42—82no—
324C42C2×C4⋊C412yes—
325C42C4×D412yes—
326C42C4×D412yes—
327C42C22⋊Q812yes—
328C42C22⋊Q812yes—
329C42#12312yes—
330C42C23×C411yescharacteristic
331C42C22×Q811yescharacteristic
332C42C2×#4111yescharacteristic
333C42C2×#4111yescharacteristic
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#38512yes—
339C23#44912yes—
340C23#45112yes—
341C23C2×#12011yescharacteristic
342Q8—48no—
343Q8—84no—
344Q8—88no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—416no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#38512yes—
382C2×C4#38512yes—
383C2×C4#38512yes—
384C2×C4C4×Q1614yes—
385C2×C4#44014yes—
386C2×C4#44912yes—
387C2×C4#45112yes—
388C2×C4#47714yes—
389C2×C4#47914yes—
390C2×C4C22×C4⋊C411yescharacteristic
391C2×C4C2×C4×D411yescharacteristic
392C2×C4C2×C4×D411yescharacteristic
393C2×C4C2×C22⋊Q811yescharacteristic
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×#12311yescharacteristic
396C2×C4C22×Q1611yescharacteristic
397C2×C4C2×#13511yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C2×C4C2×#13712yes—
401C2×C4#57512yes—
402C8—44no—
403C8—44no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#186314yes—
412C22C2×#38511yescharacteristic
413C22C2×#45111yescharacteristic
414C22C2×#44911yescharacteristic
415C22#300411yescharacteristic
416C22#300511yescharacteristic
417C22#301911yescharacteristic
418C22#301911yescharacteristic
419C4—21no—
420C4—21no—
421C4—22no—
422C4—22no—
423C4—44no—
424C4—44no—
425C4—48no—
426C4—82no—
427C4—82no—
428C4—161no—
429C4—161no—
430C4#186314yes—
431C4C2×C4×Q1611yescharacteristic
432C4C2×#44011yescharacteristic
433C4C4×#13712yes—
434C4C2×#47711yescharacteristic
435C4C2×#47911yescharacteristic
436C4#307812yes—
437C2—24no—
438C2#1874911yescharacteristic
439C2#1875211yescharacteristic
440C2#1876911yescharacteristic
441C2#1877211yescharacteristic
442C2#2085611yescharacteristic
443C2C2×#186311yescharacteristic
444C2#2087411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8929C2semidirect
7. #8927C2semidirect
40. #1548C4semidirect
120. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#218042
#213582
#211702
C2×#15482
#89292
#89272
#79012
#78922