Finite Group Databasebeta

← #364174

Subgroups — #364174

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34199C211yesmaximal, characteristic
2C2×#1890C212yesmaximal
3#20633C211yesmaximal, characteristic
4#20354C211yesmaximal, characteristic
5#11205C212yesmaximal, complemented
6#11070C212yesmaximal, complemented
7#10829C212yesmaximal, complemented
8#10754C212yesmaximal
9#10726C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#99C2212yes—
15#2359—24no—
16#2344—22no—
17#2342—22no—
18#2322C414yescomplemented
19#2315C2211yescharacteristic
20#2315C2212yes—
21#2308C2211yescharacteristic
22C4×#123C414yescomplemented
23C2×#381C2211yescharacteristic
24C2×#381C2214yes—
25#2273C2211yescharacteristic
26#1890—28no—
27#1799—24no—
28#1740—22no—
29C2×#342—22no—
30C2×#342C2214yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2214yes—
35C2×#342C2214yes—
36C23×D4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C2×#123—22no—
39C2×#123—22no—
40C2×#123—22no—
41C2×#123C2×C412yes—
42C2×#123C2×C412yes—
43C2×#123D414yes—
44C2×#123Q814yes—
45#518—28no—
46#518—24no—
47#518—24no—
48#518—44no—
49C2×C4×D4—44no—
50C2×C4×D4C2×C414yes—
51C2×C4×D4D414yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—44no—
56C2×#117D412yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C412yescomplemented
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C22⋊C4C2314yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C22⋊C4C2311yescharacteristic
69C22×C42C2×C412yescomplemented
70C2×#99—24no—
71C2×#99—24no—
72C2×#99—24no—
73#389—24no—
74#388—24no—
75#388—24no—
76#387—24no—
77#387—24no—
78#386—22no—
79#386C2×C414yes—
80#385—24no—
81#385—22no—
82#385—44no—
83#385C2×C414yescomplemented
84#381—21no—
85#381—22no—
86#381—28no—
87#381—28no—
88#381C2×C412yescomplemented
89#381C2×C412yescomplemented
90#380—22no—
91#380—24no—
92#379—21no—
93C4×C4⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C414yescomplemented
98C2×#95—22no—
99C2×#95—44no—
100#342—24no—
101#342—28no—
102#342—24no—
103#342—24no—
104#342—28no—
105#342—44no—
106C25C2×D411yescharacteristic
107C25C2411yescharacteristic
108C22×D4—24no—
109C22×D4—24no—
110C22×D4C22×C412yes—
111C22×D4C2×D412yes—
112C22×D4C2×D412yes—
113C22×D4C2×Q812yes—
114C22×D4#4112yes—
115C22×D4#4112yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4C22⋊C414yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×Q812yes—
136#123—24no—
137#123—24no—
138#123—28no—
139#123—28no—
140#123C4⋊C418yescomplemented
141#123C4⋊C418yescomplemented
142C4×D4—48no—
143C4×D4—48no—
144C4×D4—416no—
145C4×D4—416no—
146#117—28no—
147#117—44no—
148#117—48no—
149#117—48no—
150#117—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C414yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4C22⋊C414yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C22⋊C4C2×Q812yes—
200C2×C22⋊C4#4118yes—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—44no—
204C2×C42—42no—
205C2×C42C22⋊C414yes—
206#99—48no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—416no—
213#95—44no—
214C24—22no—
215C24—22no—
216C24—82no—
217C24—82no—
218C24C2×C22⋊C412yes—
219C24#12112yes—
220C24C22⋊Q812yes—
221C24C23×C411yescharacteristic
222C24C22×D411yescharacteristic
223C24C22×D411yescharacteristic
224C24C22×D411yescharacteristic
225C24C22×Q811yescharacteristic
226C24C2×#4111yescharacteristic
227C24C2×#4111yescharacteristic
228C24C2×#4114yes—
229C24H(5,𝔽2)14yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—28no—
235C2×D4—28no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4C2×C4⋊C414yes—
239C2×D4#11714yes—
240C2×D4#12114yes—
241C2×D4C22⋊Q814yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—84no—
287C22×C4—88no—
288C22×C4—88no—
289C22×C4—164no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C414yes—
292C22×C4C2×C22⋊C414yes—
293C22×C4C2×C4⋊C412yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4#12112yes—
296C22×C4C22⋊Q812yes—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—44no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—84no—
303C4⋊C4—164no—
304C22⋊C4—22no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C22⋊C4—28no—
315C22⋊C4—28no—
316C22⋊C4—44no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—84no—
328C22⋊C4C2×C4⋊C414yes—
329C42—44no—
330C42—48no—
331C42—48no—
332C42—84no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—48no—
340C23—42no—
341C23—44no—
342C23—84no—
343C23—81no—
344C23—81no—
345C23—84no—
346C23—81no—
347C23—81no—
348C23—82no—
349C23—82no—
350C23—162no—
351C23#38014yes—
352C23C22×C22⋊C411yescharacteristic
353C23C22×C4⋊C411yescharacteristic
354C23C2×#11711yescharacteristic
355C23C4×#4112yes—
356C23#51812yes—
357C23C2×#12111yescharacteristic
358C23C2×#12111yescharacteristic
359C23C2×C22⋊Q811yescharacteristic
360C23C2×C22⋊Q811yescharacteristic
361C23#52512yes—
362C23#53212yes—
363C23#53412yes—
364C23#53412yes—
365C23#54614yes—
366C23#55114yes—
367C23#55314yes—
368C23#55414yes—
369D4—28no—
370D4—28no—
371D4—28no—
372D4—28no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—88no—
407C2×C4—84no—
408C2×C4—88no—
409C2×C4—88no—
410C2×C4—88no—
411C2×C4—84no—
412C2×C4—164no—
413C2×C4—162no—
414C2×C4—164no—
415C2×C4—168no—
416C2×C4—164no—
417C2×C4#38014yes—
418C22—22no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—44no—
424C22—41no—
425C22—41no—
426C22—42no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—84no—
439C22—82no—
440C22—84no—
441C22C2×#38011yescharacteristic
442C22#229711yescharacteristic
443C22#230811yescharacteristic
444C22#234912yes—
445C22#235312yes—
446C22#253412yes—
447C22#254312yes—
448C22#256412yes—
449C22#257912yes—
450C4—24no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—42no—
455C4—42no—
456C4—82no—
457C4—88no—
458C4—82no—
459C4—164no—
460C2—21no—
461C2—22no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—22no—
466C2—22no—
467C2—22no—
468C2—44no—
469C2#1396011yescharacteristic
470C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11205C2semidirect
6. #11070C2semidirect
7. #10829C2semidirect
9. #10726C2semidirect
18. #2322C4semidirect
22. C4×#123C4semidirect
58. C22×C4⋊C4C2×C4semidirect
69. C22×C42C2×C4semidirect
83. #385C2×C4semidirect
88. #381C2×C4semidirect
89. #381C2×C4semidirect
97. C4×C22⋊C4C2×C4semidirect
140. #123C4⋊C4semidirect
141. #123C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341992
C2×#18902
#206332
#203542
#112052
#110702
#108292
#107542
#107262