Finite Group Databasebeta

← #192437

Subgroups — #192437

Order 512.

1068 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
1#34213C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#8867C218yesmaximal
7#8014C211yesmaximal, characteristic
8#7908C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2370—22no—
13#2369—21no—
14#2365—21no—
15#2355—21no—
16#2355—22no—
17Q8×C4⋊C4—21no—
18#2329C2214yes—
19#2317—21no—
20C4×C22⋊Q8—22no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24#2274—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990—21no—
28#1956—21no—
29#1947—22no—
30#1872C2214yes—
31#1852C2214yes—
32#1845—21no—
33#1834—21no—
34#1803—22no—
35#1503C418yescomplemented
36C2×#328C2214yes—
37#1442C2214yes—
38C4⋊#95—22no—
39C4⋊#95—21no—
40#1294C2214yes—
41#1280C2214yes—
42#533—22no—
43#533—22no—
44#533C2×C414yes—
45#533Q814yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8D412yes—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42—21no—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2×C414yes—
69C2×#107—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×#103—24no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—22no—
78#391—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#384—22no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#382—22no—
95#382—21no—
96#382—44no—
97#380—21no—
98#380C2312yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112C4×C4⋊C4D412yes—
113C4×C22⋊C4—24no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95C2312yes—
117C2×#95C2312yes—
118#340—24no—
119#340—24no—
120#328C2×C4116yes—
121C22×Q8—21no—
122C22×Q8C2×D412yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8C4⋊C414yes—
135C4⋊Q8C4⋊C414yes—
136#125—24no—
137#125—24no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8C22×C414yes—
143C22⋊Q8C2×Q814yes—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—84no—
148C4×Q8C4⋊C418yes—
149#117—24no—
150#117—24no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4Q1614yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—44no—
200C2×C42C22×C412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208#107—84no—
209C4⋊C8—24no—
210C4⋊C8—42no—
211#103—84no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8C22×C414yes—
216C4×C8—24no—
217C4×C8—42no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—48no—
228#95—84no—
229#95C22×C414yes—
230C24C22×D411yescharacteristic
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8C2×C4⋊C414yes—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4C22⋊Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4C2×Q1614yes—
276C22×C4C2×Q1612yes—
277C22×C4#13512yes—
278C22×C4AGL(1,ℤ/8)12yes—
279C22×C4#13712yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×Q811yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4C2×C4⋊C414yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—28no—
324C42—48no—
325C42—44no—
326C42—44no—
327C42—82no—
328C42—84no—
329C42—84no—
330C42C2×C4⋊C412yes—
331C42C2×C4⋊C412yes—
332C42C4×D412yes—
333C42C4×D412yes—
334C42C22⋊Q812yes—
335C42C22⋊Q812yes—
336C42#12312yes—
337C42#12312yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23C2×C4×D411yescharacteristic
343C23C2×#12011yescharacteristic
344C23C2×C22⋊Q811yescharacteristic
345C23C22×Q1611yescharacteristic
346C23C2×#13511yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349Q8—44no—
350Q8—48no—
351Q8—82no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4#38512yes—
401C2×C4#38512yes—
402C2×C4C4×Q1614yes—
403C2×C4#44014yes—
404C2×C4#44912yes—
405C2×C4#44912yes—
406C2×C4#45112yes—
407C2×C4#45112yes—
408C2×C4#45412yes—
409C2×C4#45512yes—
410C2×C4#47714yes—
411C2×C4#47914yes—
412C2×C4C22×C4⋊C411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C8—44no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22C2×#38511yescharacteristic
431C22C2×C4×Q1611yescharacteristic
432C22C2×#44011yescharacteristic
433C22C2×#45111yescharacteristic
434C22C2×#44911yescharacteristic
435C22C2×#45511yescharacteristic
436C22C2×#45411yescharacteristic
437C22#302711yescharacteristic
438C22#302811yescharacteristic
439C22C2×#47711yescharacteristic
440C22C2×#47911yescharacteristic
441C4—22no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—44no—
447C4—44no—
448C4—44no—
449C4—84no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—162no—
454C4#186014yes—
455C4#186314yes—
456C4#186314yes—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2#1839011yescharacteristic
462C2#1839111yescharacteristic
463C2#1841111yescharacteristic
464C2#1841211yescharacteristic
465C2C2×#186011yescharacteristic
466C2C2×#186311yescharacteristic
467C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
35. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219822
#218602
C2×#15032
#139112
#88672
#80142
#79082