Finite Group Databasebeta

← #194756

Subgroups — #194756

Order 512.

2174 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3C2×#1787C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9016C214yesmaximal, complemented
6#9014C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C24×Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C22×C4×D4C2211yescharacteristic
11C22×#103—24no—
12C22×C22⋊C8C2212yes—
13#2409—24no—
14#2409C418yescomplemented
15C2×#395C2212yes—
16C2×#393C2212yes—
17#2353—24no—
18#2352—24no—
19#2313C2212yes—
20#2312C2212yes—
21#2308—21no—
22C2×#391—21no—
23C2×#386—21no—
24C2×#385—21no—
25#2283—21no—
26#2280—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29#2273—21no—
30#1787—24no—
31#1787C2218yes—
32#1459—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C23×Q8—22no—
38C23×Q8—22no—
39C23×Q8—22no—
40C23×Q8—24no—
41C23×Q8—48no—
42C23×Q8C2×C414yes—
43C23×Q8C2312yes—
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—48no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C42D411yescharacteristic
67C2×#103—216no—
68C2×#103—416no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2314yes—
73#395—24no—
74#395C2×C418yescomplemented
75#393—24no—
76#393C2×C418yescomplemented
77#391—28no—
78#388—28no—
79#387—28no—
80#386—28no—
81#386—28no—
82#385—28no—
83#383—22no—
84#382—22no—
85#381—22no—
86#381—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380C2×C414yes—
92#380D414yes—
93#380C2314yes—
94#379—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—28no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100#376—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2312yes—
105#324—24no—
106#323—24no—
107C25C2×D411yescharacteristic
108C25C2×D411yescharacteristic
109C22×Q8—28no—
110C22×Q8—24no—
111C22×Q8—22no—
112C22×Q8—216no—
113C22×Q8—28no—
114C22×Q8—28no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—48no—
118C22×Q8—48no—
119C22×Q8—44no—
120C22×Q8—416no—
121C22×Q8—416no—
122C22×Q8—432no—
123C22×Q8—416no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8C22×C414yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4C22⋊C414yes—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—44no—
143C23×C4—48no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C414yes—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4#4114yes—
157C23×C4C2411yescharacteristic
158C22×C8—24no—
159C22×C8—48no—
160C22×C8—44no—
161C22⋊Q8—432no—
162C4×D4—28no—
163C4×D4—28no—
164C4×D4—28no—
165C4×D4—28no—
166#117—84no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D418yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4C22⋊C414yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C2×D412yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42C22⋊C414yes—
201#103—432no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—44no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95C22×C414yes—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24C2×C22⋊C412yes—
224C24C2×C22⋊C412yes—
225C24#12012yes—
226C24#12312yes—
227C24#13514yes—
228C24#13714yes—
229C24C22×D411yescharacteristic
230C2×Q8—216no—
231C2×Q8—216no—
232C2×Q8—232no—
233C2×Q8—28no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—432no—
239C2×Q8—416no—
240C2×Q8—416no—
241C2×Q8—48no—
242C2×Q8—432no—
243C2×Q8—416no—
244C2×Q8—432no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—48no—
249C2×D4—28no—
250C2×D4—28no—
251C2×D4—28no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—48no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—416no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—416no—
280C22×C4—416no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—84no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C4×D414yes—
300C22×C4C4×D414yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12114yes—
304C22×C4#12114yes—
305C22×C4#12312yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D412yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4112yes—
310C2×C8—48no—
311C2×C8—44no—
312C2×C8—88no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—416no—
318C4⋊C4—48no—
319C22⋊C4—28no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—416no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—84no—
328C22⋊C4—84no—
329C42—24no—
330C42—24no—
331C42—84no—
332C42—84no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—24no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—48no—
345C23—48no—
346C23—42no—
347C23—42no—
348C23—48no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—84no—
356C23#38614yes—
357C23#41614yes—
358C23#41914yes—
359C23#45214yes—
360C23#45214yes—
361C23#48518yes—
362C23C22×C22⋊C411yescharacteristic
363C23C2×#12011yescharacteristic
364C23C2×#12311yescharacteristic
365C23C2×#13512yes—
366C23C2×#13712yes—
367Q8—216no—
368Q8—416no—
369Q8—416no—
370Q8—48no—
371Q8—416no—
372D4—416no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—24no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—432no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—84no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—88no—
412C2×C4—168no—
413C2×C4#38612yes—
414C2×C4#38612yes—
415C2×C4B(4,𝔽2)14yes—
416C2×C4C2×C4×D412yes—
417C2×C4C2×#12112yes—
418C8—84no—
419C22—24no—
420C22—28no—
421C22—24no—
422C22—28no—
423C22—21no—
424C22—21no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—44no—
430C22—44no—
431C22—48no—
432C22—44no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—82no—
437C22—84no—
438C22—84no—
439C22#187318yes—
440C22C2×#38611yescharacteristic
441C22C2×#41611yescharacteristic
442C22C2×#41911yescharacteristic
443C22C2×#45212yes—
444C22#302212yes—
445C22C2×B(4,𝔽2)12yes—
446C22C2×#48512yes—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—48no—
453C4—42no—
454C4—84no—
455C4—88no—
456C4#188914yes—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—28no—
461C2—82no—
462C2#1819914yes—
463C2C2×#187311yescharacteristic
464C2#2095511yescharacteristic
465C2C2×#188911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1787C2semidirect
5. #9016C2semidirect
6. #9014C2semidirect
7. #8023C2semidirect
14. #2409C4semidirect
74. #395C2×C4semidirect
76. #393C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#341852
C2×#17872
C2×#14592
#90162
#90142
#80232