Finite Group Databasebeta

← #192308

Subgroups — #192308

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34074C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13848C211yesmaximal, characteristic
6#8868C214yesmaximal
7#8867C214yesmaximal
8#8014C211yesmaximal, characteristic
9#7906C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2331—21no—
14#2331—21no—
15#2330—22no—
16#2329—22no—
17#2329C2214yes—
18#2296—21no—
19#2295—22no—
20#2295—22no—
21#2285C2211yescharacteristic
22C4×#117—21no—
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1987—21no—
28#1981—21no—
29#1947—22no—
30#1872C2214yes—
31#1852C2214yes—
32C4×#107—22no—
33C4×#103—22no—
34#1503C418yescomplemented
35C2×#328C2212yes—
36C2×#328C2212yes—
37#1444C2212yes—
38#1442C2212yes—
39#1441—22no—
40C4×#95—21no—
41#1294C2212yes—
42#1294C2212yes—
43#1280C2214yes—
44#533—22no—
45#533—22no—
46#533C2×C414yes—
47#533Q814yes—
48C2×C4⋊Q8D411yescharacteristic
49C2×C4⋊Q8D411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8C2312yes—
53C2×C4×Q8—21no—
54C2×C4×Q8D412yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C42—21no—
61C22×C42—21no—
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
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—24no—
78#391—24no—
79#389—24no—
80#389—22no—
81#389—22no—
82#389—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#385—24no—
88#384D412yes—
89#382—21no—
90#380—21no—
91#380C2312yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—44no—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—24no—
106#376—22no—
107#376—21no—
108#376—21no—
109#376—21no—
110#376—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95D412yes—
116C2×#95C2312yes—
117C2×#95C2311yescharacteristic
118C2×#95C2311yescharacteristic
119C43—22no—
120#340—24no—
121#340—24no—
122#328C2×C418yes—
123#328C2×C418yes—
124C22×Q8—21no—
125C22×Q8C2×D412yes—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8C4⋊C414yescomplemented
139C4⋊Q8C4⋊C414yescomplemented
140#125—24no—
141#125—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8C22×C414yes—
147C22⋊Q8C2×Q814yes—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8C4⋊C418yescomplemented
152#117—24no—
153#117—24no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D414yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42Q1614yes—
196C2×C42C22×C412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×Q812yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204#107—48no—
205C4⋊C8—24no—
206C4⋊C8—42no—
207#103—48no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8C22×C414yes—
212C4×C8—24no—
213C4×C8—42no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—41no—
230#95—41no—
231#95C22×C412yes—
232#95C22×C412yes—
233C24C22×D411yescharacteristic
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—82no—
238C2×Q8C2×C4⋊C414yes—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C2×Q1614yes—
279C22×C4C2×Q1612yes—
280C22×C4#13512yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—24no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—48no—
306C4⋊C4—416no—
307C4⋊C4—44no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4C2×C4⋊C414yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—28no—
317C42—24no—
318C42—24no—
319C42—48no—
320C42—42no—
321C42—42no—
322C42—44no—
323C42—84no—
324C42—84no—
325C42—82no—
326C42—82no—
327C42C2×C4⋊C412yes—
328C42C2×C4⋊C412yes—
329C42C4×D412yes—
330C42C4×D412yes—
331C42C22⋊Q812yes—
332C42C22⋊Q812yes—
333C42#12312yes—
334C42#12312yes—
335C23—22no—
336C23—22no—
337C23C2×C4×D411yescharacteristic
338C23C2×#12011yescharacteristic
339C23C2×C22⋊Q811yescharacteristic
340C23C22×Q1611yescharacteristic
341C23C2×#13511yescharacteristic
342C23C2×AGL(1,ℤ/8)11yescharacteristic
343C23C2×#13711yescharacteristic
344Q8—44no—
345Q8—82no—
346Q8—84no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#38512yes—
392C2×C4#38512yes—
393C2×C4#38512yes—
394C2×C4#38512yes—
395C2×C4C4×Q1614yes—
396C2×C4#44014yes—
397C2×C4#44912yes—
398C2×C4#44912yes—
399C2×C4#45112yes—
400C2×C4#45112yes—
401C2×C4#47714yes—
402C2×C4#47914yes—
403C2×C4C22×C4⋊C411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×#12311yescharacteristic
407C8—44no—
408C8—82no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22C2×#38511yescharacteristic
418C22C2×C4×Q1611yescharacteristic
419C22C2×#44011yescharacteristic
420C22C2×#45111yescharacteristic
421C22C2×#44911yescharacteristic
422C22#301812yes—
423C22#302512yes—
424C22C2×#47711yescharacteristic
425C22C2×#47911yescharacteristic
426C4—22no—
427C4—22no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—44no—
433C4—44no—
434C4—44no—
435C4—44no—
436C4—82no—
437C4—84no—
438C4—82no—
439C4—84no—
440C4#186314yes—
441C4#186314yes—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#1812512yes—
447C2#1812612yes—
448C2#2086311yescharacteristic
449C2C2×#186311yescharacteristic
450C2#2088811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #1503C4semidirect
138. C4⋊Q8C4⋊C4semidirect
139. C4⋊Q8C4⋊C4semidirect
151. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340742
#213852
#213582
C2×#15032
#138482
#88682
#88672
#80142
#79062