Finite Group Databasebeta

← #201294

Subgroups — #201294

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic, complemented
3C2×#1821C211yesmaximal, characteristic, complemented
4#17577C212yesmaximal
5#9464C218yesmaximal, complemented
6#7942C212yesmaximal, complemented
7C22×#123—22no—
8C22×#121C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2501—21no—
12#2501—21no—
13#2501—22no—
14#2501C414yescomplemented
15#2501C2214yes—
16C2×#399—22no—
17#2345—24no—
18#2344—24no—
19#2316—21no—
20#2313C2214yescomplemented
21C2×#390C412yescomplemented
22C2×#384C412yescomplemented
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×#123—28no—
40C2×#123—24no—
41C2×#123—24no—
42C2×#123—416no—
43C2×#121—21no—
44C2×#121—21no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121C2×C412yes—
50C2×#121C2312yes—
51C2×#117—22no—
52C2×#117—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—21no—
69C22×C42C2×C412yes—
70#407—22no—
71#407—22no—
72#407—22no—
73#407C2314yes—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76#399—24no—
77#399—24no—
78#399—28no—
79#390—22no—
80#390—22no—
81#390C2×C418yes—
82#389—28no—
83#387—216no—
84#384—22no—
85#384—22no—
86#384C2×C418yes—
87#381—24no—
88#381C2314yes—
89#380—24no—
90#380—24no—
91#380—22no—
92#380D414yes—
93#379—22no—
94C4×C4⋊C4—28no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4C2×C412yes—
101C4×C22⋊C4D414yes—
102C4×C22⋊C4C2312yes—
103C2×#95—21no—
104#343—28no—
105#327—216no—
106C25—21no—
107C25C2×D411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—44no—
114C22×D4—48no—
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4—41no—
119C22×D4C22×C412yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—44no—
127C23×C4—48no—
128C23×C4C22⋊C414yes—
129C23×C4C22⋊C414yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—48no—
145C22×C8—42no—
146#123—432no—
147#123—416no—
148#123—416no—
149#121—24no—
150#121—24no—
151#121—24no—
152#121—24no—
153#121—42no—
154#121—42no—
155#121—42no—
156#121—42no—
157#117—48no—
158#117—88no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—416no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C2×D414yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—416no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—28no—
204C2×C42—44no—
205C2×C42C22×C414yes—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210#95—28no—
211#95—28no—
212#95—22no—
213#95—22no—
214#95—24no—
215#95C2×D418yes—
216C24—24no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24—44no—
221C24—44no—
222C24—44no—
223C24—41no—
224C24—41no—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C411yescharacteristic
227C24#12011yescharacteristic
228C24#12011yescharacteristic
229C24#12311yescharacteristic
230C24#12311yescharacteristic
231C24C22×D411yescharacteristic
232C2×D4—24no—
233C2×D4—44no—
234C2×D4—48no—
235C2×D4—48no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—84no—
243C2×D4—84no—
244C2×D4—84no—
245C2×D4—84no—
246C2×D4—84no—
247C2×D4—84no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4#12012yes—
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12312yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D412yes—
300C22×C4C2×#4112yes—
301M16—48no—
302M16—84no—
303C2×C8—44no—
304C2×C8—84no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—416no—
313C4⋊C4—416no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—416no—
320C22⋊C4—416no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C42—44no—
327C42—48no—
328C42—44no—
329C42—88no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—416no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—42no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—816no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23#38611yescharacteristic
363C23#38611yescharacteristic
364C23#38611yescharacteristic
365C23#38611yescharacteristic
366C23#45314yes—
367C23C22×C22⋊C411yescharacteristic
368C23C2×#12011yescharacteristic
369C23C2×#12311yescharacteristic
370D4—84no—
371D4—84no—
372D4—84no—
373D4—84no—
374D4—84no—
375D4—84no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—416no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#41014yes—
417C2×C4C2×C4×D412yes—
418C2×C4C2×#12112yes—
419C8—84no—
420C22—28no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—88no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—164no—
448C22—164no—
449C22#188712yes—
450C22#188712yes—
451C22#191212yes—
452C22C2×#38611yescharacteristic
453C22C2×#41012yes—
454C22C2×#45312yes—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—48no—
459C4—44no—
460C4—44no—
461C4—88no—
462C4—82no—
463C4#189914yes—
464C2—22no—
465C2—22no—
466C2—22no—
467C2—22no—
468C2—22no—
469C2—41no—
470C2—41no—
471C2—162no—
472C2C2×#188711yescharacteristic
473C2C2×#189911yescharacteristic
474C2C2×#191211yescharacteristic
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. #9464C2semidirect
6. #7942C2semidirect
14. #2501C4semidirect
20. #2313C22semidirect
21. C2×#390C4semidirect
22. C2×#384C4semidirect
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×#25012
#341782
C2×#18212
#175772
#94642
#79422