Finite Group Databasebeta

← #188309

Subgroups — #188309

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#387C211yesmaximal, characteristic
2#20626C212yesmaximal
3C2×#1463C212yesmaximal
4#8562C218yesmaximal, complemented
5#7909C212yesmaximal
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#423—22no—
10C2×#407C2214yes—
11C2×#387—22no—
12C2×#387—22no—
13C2×#387—22no—
14C2×#387—22no—
15C2×#387C414yes—
16C2×#387C2214yes—
17C22×#95C411yescharacteristic
18C22×#95C411yescharacteristic
19C22×#95C411yescharacteristic
20C22×#95C2211yescharacteristic
21#1797—22no—
22#1797—22no—
23#1797C2214yes—
24#1797C2214yes—
25#1728—22no—
26#1463—28no—
27C2×#324C2218yes—
28#1280C2218yes—
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—24no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C414yes—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4Q814yes—
48C22×C4⋊C4C2312yes—
49C22×C42—44no—
50C22×C42C2×C412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2312yes—
56#423—48no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—24no—
59#407—28no—
60#407—28no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8C2318yes—
63#387—24no—
64#387—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95C2×C414yes—
83C2×#95C2×C414yes—
84C2×#95C2×C414yes—
85C2×#95C2×C412yes—
86C2×#95C2312yes—
87#336—24no—
88#336—24no—
89#324—216no—
90C25C22×C411yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C411yescharacteristic
125C23×C4C4⋊C414yes—
126C23×C4C4⋊C414yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×Q812yes—
138C23×C4#4114yes—
139C23×C4C2411yescharacteristic
140C2×M16—48no—
141C2×M16—48no—
142C2×M16—48no—
143C22×C8—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C4⋊C418yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×Q814yes—
161C2×C42—28no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C4⋊C8—48no—
176C4⋊C8—48no—
177C22⋊C8—28no—
178C22⋊C8—28no—
179C22⋊C8—48no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—22no—
194C24C2×C22⋊C411yescharacteristic
195C24C2×C22⋊C411yescharacteristic
196C24C2×C4⋊C412yes—
197C24#11712yes—
198C24C23×C411yescharacteristic
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—48no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—416no—
249C22×C4—416no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—88no—
279C22×C4—88no—
280C22×C4#9914yes—
281C22×C4#10114yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C4⋊C414yes—
288C22×C4C2×C4⋊C414yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4#11712yes—
291C22×C4#12114yes—
292C22×C4#12114yes—
293C22×C4C22⋊Q814yes—
294C22×C4C22⋊Q814yes—
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4112yes—
299M16—816no—
300C2×C8—48no—
301C2×C8—48no—
302C2×C8—48no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—216no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C42—24no—
309C42—24no—
310C42—24no—
311C42—24no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C23—28no—
317C23—28no—
318C23—28no—
319C23—28no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—24no—
338C23#38012yes—
339C23#38012yes—
340C23C2×#9912yes—
341C23#41012yes—
342C23C2×#10112yes—
343C23#41312yes—
344C23C22×C22⋊C411yescharacteristic
345C23C22×C4⋊C411yescharacteristic
346C23C2×#11711yescharacteristic
347C23#56812yes—
348C23#56812yes—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4#38014yes—
407C2×C4C2×#9914yes—
408C2×C4C2×#10114yes—
409C2×C4C2×#12112yes—
410C2×C4C2×C22⋊Q812yes—
411C8—88no—
412C22—216no—
413C22—216no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22#179914yes—
435C22#180114yes—
436C22C2×#38011yescharacteristic
437C22#287911yescharacteristic
438C22#287911yescharacteristic
439C22C22×#9911yescharacteristic
440C22C2×#41011yescharacteristic
441C22C22×#10111yescharacteristic
442C22C2×#41311yescharacteristic
443C22#290512yes—
444C22#292112yes—
445C4—22no—
446C4—22no—
447C4—24no—
448C4—24no—
449C4—24no—
450C4—44no—
451C4—44no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—82no—
456C2—24no—
457C2—24no—
458C2—22no—
459C2—22no—
460C2#1739211yescharacteristic
461C2#1739411yescharacteristic
462C2#1739711yescharacteristic
463C2#1739811yescharacteristic
464C2#2062711yescharacteristic
465C2C2×#179911yescharacteristic
466C2C2×#180111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8562C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3872
#206262
C2×#14632
#85622
#79092