Finite Group Databasebeta

← #193252

Subgroups — #193252

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#8926C214yesmaximal, complemented
7#8923C214yesmaximal, complemented
8#7904C211yesmaximal, characteristic
9#7904C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8—21no—
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2365—22no—
16#2365—22no—
17#2365—22no—
18#2365C2214yes—
19C2×#391—21no—
20C2×#389—21no—
21C2×#387—22no—
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1993—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1948C2212yes—
30#1948C2212yes—
31#1947C2212yes—
32#1947C2212yes—
33#1853—21no—
34#1853—21no—
35#1852—21no—
36#1852—21no—
37#1776—22no—
38#1776—22no—
39#1736—22no—
40#1556C418yescomplemented
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—21no—
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×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—48no—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—22no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#432—22no—
70#432—22no—
71#432C2×C414yes—
72#432C2×C414yes—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×C4×C8C2312yes—
87#391—24no—
88#391—24no—
89#389—24no—
90#389—24no—
91#387—28no—
92#387—28no—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—21no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2312yes—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—24no—
120#332C2×C4116yescomplemented
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—44no—
126C22×Q8—44no—
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—
151C4×Q8—416no—
152C4×Q8—48no—
153C4×Q8—48no—
154#117—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—21no—
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—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4C4⋊C418yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—41no—
184C2×C42—28no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—48no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—41no—
193C2×C42C22×C412yes—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q812yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C2×C42C2411yescharacteristic
201#107—44no—
202#107—44no—
203#106—44no—
204#106—44no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8C22×C414yes—
208C4⋊C8C22×C414yes—
209C22⋊C8—24no—
210C22⋊C8—24no—
211#97—24no—
212#97—24no—
213#97—42no—
214C4×C8—22no—
215C4×C8—22no—
216#95—28no—
217#95—28no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224C24#12011yescharacteristic
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—28no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—416no—
231C2×Q8—416no—
232C2×Q8—44no—
233C2×Q8—44no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×C4⋊C412yes—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4#12312yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—48no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—416no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4C2×C4⋊C418yes—
307C4⋊C4AGL(1,ℤ/8)14yes—
308C4⋊C4#13714yes—
309C22⋊C4—82no—
310C42—44no—
311C42—44no—
312C42—416no—
313C42—44no—
314C42—44no—
315C42—82no—
316C42C2×C4⋊C412yes—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C22⋊Q812yes—
320C42C22⋊Q812yes—
321C42#12312yes—
322C42C23×C411yescharacteristic
323C42C22×Q811yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#38512yes—
331C23#45412yes—
332C23#45612yes—
333C23C2×#12011yescharacteristic
334Q8—416no—
335Q8—48no—
336Q8—48no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—416no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4#38512yes—
378C2×C4#38512yes—
379C2×C4#38512yes—
380C2×C4C22×C4⋊C411yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×C22⋊Q811yescharacteristic
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4C2×AGL(1,ℤ/8)12yes—
387C2×C4C2×#13712yes—
388C2×C4#57512yes—
389C2×C4#57512yes—
390C8—44no—
391C8—44no—
392C8—44no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22#185714yes—
400C22C2×#38511yescharacteristic
401C22#300311yescharacteristic
402C22#300411yescharacteristic
403C22#300511yescharacteristic
404C22#300511yescharacteristic
405C22C2×#45411yescharacteristic
406C22C2×#45611yescharacteristic
407C4—21no—
408C4—21no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—44no—
413C4—44no—
414C4—48no—
415C4—82no—
416C4—82no—
417C4—161no—
418C4—161no—
419C4C4×AGL(1,ℤ/8)12yes—
420C4C4×#13712yes—
421C4#307712yes—
422C4#307812yes—
423C2—24no—
424C2#1853411yescharacteristic
425C2#1853611yescharacteristic
426C2#1853911yescharacteristic
427C2#1854111yescharacteristic
428C2C2×#185711yescharacteristic
429C2#2087311yescharacteristic
430C2#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. #8926C2semidirect
7. #8923C2semidirect
40. #1556C4semidirect
120. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
#213652
#213242
C2×#15562
#89262
#89232
#79042