Finite Group Databasebeta

← #393528

Subgroups — #393528

Order 512.

1592 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
1C2×#2873C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3#20972C212yesmaximal
4#20688C211yesmaximal, characteristic
5#10947C214yesmaximal, complemented
6#10889C214yesmaximal, complemented
7#10717C212yesmaximal
8C23×M16C2212yes—
9C22×C22⋊Q8C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C2×#419C2214yes—
12#2873—24no—
13#2873—22no—
14#2873C2214yes—
15C2×#405—24no—
16C2×C4×M16—22no—
17#2855—22no—
18#2625—22no—
19#2503C412yescomplemented
20#2502C412yescomplemented
21#2499C414yescomplemented
22#2484—22no—
23#2483—22no—
24#2317C2212yes—
25#2310C2212yes—
26#2280C2212yes—
27#2280C2211yescharacteristic
28#1894—24no—
29#1892—24no—
30#1820—22no—
31#1819—22no—
32#1740C2214yes—
33#1736C2214yes—
34C2×H(3,ℤ/4)C2218yes—
35C23×Q8C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C22×M16—22no—
39C22×M16—22no—
40C22×M16—22no—
41C22×M16—216no—
42C22×M16D414yes—
43C22×M16C2314yes—
44C23×C8D414yes—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2×C412yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117C2×C412yes—
58C2×#117D412yes—
59C2×#117D412yes—
60C2×#117C2314yes—
61C2×#117C2312yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4D412yes—
69C22×C42D412yes—
70#419—216no—
71C2×#103—28no—
72#405—216no—
73#405—216no—
74C4×M16—28no—
75C4×M16—28no—
76C2×#97—24no—
77C2×C4×C8—24no—
78#400—24no—
79#398—24no—
80#397—28no—
81#391—24no—
82#391C2×C414yes—
83#391C2×C414yes—
84#389—24no—
85#387—28no—
86#385—24no—
87#385—24no—
88#384C2×C414yes—
89#382C2×C414yes—
90#381—24no—
91#381—21no—
92#381—24no—
93#381—22no—
94#380—21no—
95#380—21no—
96#380—22no—
97#380—22no—
98#379—21no—
99#379—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2×C412yes—
104#342—28no—
105#340—28no—
106H(3,ℤ/4)—216no—
107C25C2×D411yescharacteristic
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—28no—
112C22×Q8C22×C412yes—
113C23×C4—28no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4#4112yes—
126C23×C4#4112yes—
127C23×C4C2411yescharacteristic
128C2×M16—216no—
129C2×M16—216no—
130C2×M16—216no—
131C2×M16—216no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—28no—
137C2×M16—216no—
138C2×M16#41116yes—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—216no—
142C22×C8C2×D418yes—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—42no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—42no—
153#117—24no—
154#117—24no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—28no—
162#117—28no—
163#117—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4C22⋊C414yes—
196C2×C22⋊C4C2×D414yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42C22⋊C414yes—
203C2×C42C22×C414yes—
204C2×C42C2×D414yes—
205#103—416no—
206#97—216no—
207C4×C8—216no—
208#95—22no—
209#95—22no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—42no—
214#95—42no—
215C24—24no—
216C24—42no—
217C24—81no—
218C24C2×C22⋊C411yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C24#12112yes—
221C24#12112yes—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—28no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—48no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C414yes—
259C22×C4C2×C22⋊C414yes—
260C22×C4C4×D412yes—
261C22×C4C4×D412yes—
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12012yes—
267C22×C4#12112yes—
268C22×C4#12112yes—
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12312yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4118yes—
277M16—216no—
278M16—216no—
279M16—216no—
280M16—216no—
281C2×C8—216no—
282C2×C8—216no—
283C2×C8—28no—
284C2×C8—28no—
285C2×C8#123116yes—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—88no—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—48no—
308C22⋊C4—82no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—28no—
316C42—48no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—24no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—88no—
332C23#38612yes—
333C23#38611yescharacteristic
334C23#38611yescharacteristic
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12112yes—
337C23#53214yes—
338Q8—48no—
339Q8—82no—
340Q8—82no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4#38611yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38612yes—
379C2×C4C2×C4×D412yes—
380C2×C4C4×#4114yes—
381C2×C4C2×#12011yescharacteristic
382C2×C4C2×#12314yes—
383C2×C4C2×#12311yescharacteristic
384C2×C4#52514yes—
385C2×C4#52514yes—
386C2×C4#52918yes—
387C8—216no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—21no—
400C22—44no—
401C22—44no—
402C22—48no—
403C22—82no—
404C22—84no—
405C22#190912yes—
406C22C2×#38611yescharacteristic
407C22#229712yes—
408C22#285414yes—
409C4—21no—
410C4—22no—
411C4—21no—
412C4—44no—
413C4—44no—
414C4—84no—
415C4—82no—
416C4—82no—
417C4—164no—
418C4C4×#12314yes—
419C4#284912yes—
420C4#285312yes—
421C2—22no—
422C2—24no—
423C2—22no—
424C2—41no—
425C2—42no—
426C2#1470611yescharacteristic
427C2C2×#190911yescharacteristic
428C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10947C2semidirect
6. #10889C2semidirect
19. #2503C4semidirect
20. #2502C4semidirect
21. #2499C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28732
#353112
#209722
#206882
#109472
#108892
#107172