Finite Group Databasebeta

← #201299

Subgroups — #201299

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic, complemented
3C2×#1821C211yesmaximal, characteristic, complemented
4#17577C212yesmaximal
5#9465C218yesmaximal, complemented
6#7942C212yesmaximal, complemented
7C22×#121C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407C2212yes—
10#2504—21no—
11#2504—21no—
12#2504—22no—
13#2504C414yescomplemented
14#2504C2214yes—
15C2×#397—22no—
16#2345—24no—
17#2344—24no—
18#2316—21no—
19#2313C2214yescomplemented
20C2×#390C412yescomplemented
21C2×#386—22no—
22C2×#382C412yescomplemented
23#2280—22no—
24#2280C2212yes—
25#2274—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1821—22no—
28#1821C2214yescomplemented
29C2×#343—22no—
30#1519—24no—
31#1510C2218yes—
32#1507—24no—
33#1284C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4D412yes—
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C2×#121—21no—
40C2×#121—21no—
41C2×#121—22no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121C2×C412yes—
46C2×#121C2312yes—
47C2×#117—22no—
48C2×#117—24no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—24no—
64C22×C42C2×C412yes—
65#407—22no—
66#407—22no—
67#407—22no—
68#407C2314yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71#397—24no—
72#397—24no—
73#397—28no—
74#390—22no—
75#390—22no—
76#390C2×C418yes—
77#389—28no—
78#387—216no—
79#386—28no—
80#386—24no—
81#386—24no—
82#382—22no—
83#382—22no—
84#382C2×C418yes—
85#381—24no—
86#381C2314yes—
87#380—24no—
88#380—24no—
89#380—22no—
90#380D414yes—
91#379—22no—
92C4×C4⋊C4—28no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D414yes—
100C4×C22⋊C4C2312yes—
101C2×#95—24no—
102C2×#95—21no—
103#343—28no—
104#327—216no—
105C25—21no—
106C25C2×D411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—44no—
113C22×D4—48no—
114C22×D4—41no—
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4C22×C412yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—44no—
126C23×C4—84no—
127C23×C4C22⋊C414yes—
128C23×C4C22⋊C414yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—48no—
144C22×C8—42no—
145#121—24no—
146#121—24no—
147#121—24no—
148#121—24no—
149#121—42no—
150#121—42no—
151#121—42no—
152#121—42no—
153#117—48no—
154#117—88no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—88no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—28no—
197C2×C42—44no—
198C2×C42C22×C414yes—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—42no—
202C22⋊C8—42no—
203#95—28no—
204#95—28no—
205#95—22no—
206#95—28no—
207#95—28no—
208#95—22no—
209#95—24no—
210#95C2×D418yes—
211C24—24no—
212C24—22no—
213C24—42no—
214C24—42no—
215C24—44no—
216C24—44no—
217C24—44no—
218C24—41no—
219C24—41no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24#12011yescharacteristic
223C24#12011yescharacteristic
224C24#12311yescharacteristic
225C24#12311yescharacteristic
226C24C22×D411yescharacteristic
227C2×D4—24no—
228C2×D4—44no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—84no—
238C2×D4—84no—
239C2×D4—84no—
240C2×D4—84no—
241C2×D4—84no—
242C2×D4—84no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12012yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D412yes—
295C22×C4C2×#4112yes—
296M16—48no—
297M16—84no—
298C2×C8—44no—
299C2×C8—84no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—48no—
308C4⋊C4—42no—
309C4⋊C4—88no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—88no—
318C22⋊C4—88no—
319C42—44no—
320C42—48no—
321C42—44no—
322C42—88no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—416no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—816no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23#38611yescharacteristic
356C23#38611yescharacteristic
357C23#38611yescharacteristic
358C23#38611yescharacteristic
359C23#45414yes—
360C23C22×C22⋊C411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12311yescharacteristic
363D4—84no—
364D4—84no—
365D4—84no—
366D4—84no—
367D4—84no—
368D4—84no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—88no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4#41014yes—
410C2×C4C2×C4×D412yes—
411C2×C4C2×#12112yes—
412C8—84no—
413C22—28no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—88no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—164no—
441C22—164no—
442C22#188812yes—
443C22#188812yes—
444C22#191112yes—
445C22C2×#38611yescharacteristic
446C22C2×#41012yes—
447C22C2×#45412yes—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—88no—
454C4—84no—
455C4—82no—
456C4#189914yes—
457C2—22no—
458C2—22no—
459C2—22no—
460C2—22no—
461C2—22no—
462C2—41no—
463C2—41no—
464C2—162no—
465C2C2×#188811yescharacteristic
466C2C2×#189911yescharacteristic
467C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34178C2semidirect
3. C2×#1821C2semidirect
5. #9465C2semidirect
6. #7942C2semidirect
13. #2504C4semidirect
19. #2313C22semidirect
20. C2×#390C4semidirect
22. C2×#382C4semidirect
28. #1821C22semidirect
33. #1284C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
#341782
C2×#18212
#175772
#94652
#79422