Finite Group Databasebeta

← #380010

Subgroups — #380010

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×#1985C212yesmaximal, complemented
4C2×#1797C211yesmaximal, characteristic
5#11202C214yesmaximal, complemented
6#11147C214yesmaximal
7#10835C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—24no—
12C2×#432—28no—
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2212yes—
18C4×C22⋊Q8—24no—
19C4×C22⋊Q8—24no—
20C2×#391C411yescharacteristic
21C2×#389C411yescharacteristic
22C2×#387C412yes—
23C2×#385C412yes—
24C2×#380C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2212yes—
28#1985—28no—
29#1797—24no—
30C2×#336C2218yes—
31C2×#336C2212yes—
32C2×#336C2218yes—
33C23×Q8C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8C2314yes—
38C23×C8C2314yes—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—28no—
41C2×C4×Q8—24no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—24no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C42—24no—
54C22×C42—24no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42D414yes—
58C22×C42D414yes—
59C22×C42C2312yes—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#432—232no—
64C2×C4⋊C8—28no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—28no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—44no—
73C2×#97—24no—
74C2×#97—24no—
75C2×#97D418yes—
76#391—24no—
77#389—24no—
78#387—28no—
79#385—28no—
80#380—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92#336—216no—
93#336—24no—
94#336—48no—
95C25C22×C411yescharacteristic
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—24no—
99C22×Q8—42no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4C22⋊C414yes—
112C23×C4C22⋊C414yes—
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—24no—
130C22×C8—24no—
131C22×C8—24no—
132C22×C8—24no—
133C22×C8—24no—
134C22×C8—28no—
135C22×C8—28no—
136C22×C8C2×D418yes—
137C22⋊Q8—232no—
138C4×Q8—44no—
139C4×Q8—48no—
140C4×Q8—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4M1618yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—28no—
164C2×C42—216no—
165C2×C42—28no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42C22⋊C414yes—
175C2×C42C22⋊C414yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42#4114yes—
182C4⋊C8—216no—
183C4⋊C8—48no—
184C22⋊C8—216no—
185C22⋊C8—48no—
186#97—216no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192C24—22no—
193C24—22no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24C2×C22⋊C412yes—
199C24#13114yes—
200C24C23×C411yescharacteristic
201C2×Q8—28no—
202C2×Q8—416no—
203C2×Q8—44no—
204C2×Q8—44no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—416no—
248C22×C4—416no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C4×D414yes—
259C22×C4C4×D414yes—
260C22×C4C2×M1614yes—
261C22×C4C2×M1614yes—
262C22×C4C2×M1614yes—
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4H(5,𝔽2)12yes—
268C22×C42−1+412yes—
269C22×C42−1+414yes—
270C2×C8—28no—
271C2×C8—28no—
272C2×C8—28no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—48no—
281C2×C8—48no—
282C4⋊C4—28no—
283C4⋊C4—216no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C22⋊C4—216no—
289C22⋊C4—48no—
290C42—22no—
291C42—24no—
292C42—22no—
293C42—28no—
294C42—28no—
295C42—42no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—44no—
301C42#12014yes—
302C42#12118yes—
303C42#12314yes—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—28no—
310C23—28no—
311C23—28no—
312C23—22no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23#40814yes—
318C23C22×C22⋊C411yescharacteristic
319C23C2×C4×D412yes—
320C23#51912yes—
321C23#52012yes—
322C23C22×M1612yes—
323C23C2×#13112yes—
324C23#56812yes—
325C23#56812yes—
326Q8—416no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—416no—
347C2×C4—416no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4#38614yes—
369C2×C4#38614yes—
370C2×C4#40714yes—
371C2×C4C2×#12011yescharacteristic
372C2×C4C2×#12112yes—
373C2×C4C2×#12311yescharacteristic
374C2×C4#53511yescharacteristic
375C2×C4#53614yes—
376C2×C4#53612yes—
377C2×C4#53612yes—
378C2×C4#53611yescharacteristic
379C2×C4#53714yes—
380C2×C4#53712yes—
381C2×C4#54114yes—
382C2×C4#54214yes—
383C8—48no—
384C8—84no—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—42no—
389C22—416no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22C2×#38611yescharacteristic
394C22#231011yescharacteristic
395C22#231111yescharacteristic
396C22#232314yes—
397C22C2×#40711yescharacteristic
398C22C2×#40811yescharacteristic
399C22#287911yescharacteristic
400C22#287911yescharacteristic
401C22#293314yes—
402C22D4×M1614yes—
403C22#298414yes—
404C22#298714yes—
405C22#298914yes—
406C22#299214yes—
407C4—24no—
408C4—24no—
409C4—24no—
410C4—44no—
411C4—42no—
412C4—44no—
413C4—42no—
414C4—82no—
415C4—82no—
416C4#261612yes—
417C4#262012yes—
418C4#262112yes—
419C4#262512yes—
420C2—44no—
421C2#1425811yescharacteristic
422C2#2089711yescharacteristic
423C2#2089811yescharacteristic
424C2#2182011yescharacteristic
425C2#2182311yescharacteristic
426C2#2185211yescharacteristic
427C2#2185311yescharacteristic
428C2#2210618yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1985C2semidirect
5. #11202C2semidirect
7. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
C2×C4×C22⋊Q82
C2×#19852
C2×#17972
#112022
#111472
#108352