Finite Group Databasebeta

← #194849

Subgroups — #194849

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#21833C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9027C214yesmaximal, complemented
6#9025C214yesmaximal
7#8026C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C22×#117—21no—
11C22×#117C2211yescharacteristic
12C2×#431C2212yes—
13#2621—24no—
14#2621C418yescomplemented
15C2×#398—21no—
16C2×#398C2212yes—
17C2×#397—21no—
18C2×#397C2212yes—
19#2317—24no—
20#2316—21no—
21#2316C2212yes—
22#2315—21no—
23#2315C2212yes—
24#2296—24no—
25C2×#391—21no—
26#2285—21no—
27#2285C2212yes—
28C2×#382C2211yescharacteristic
29#1990—24no—
30#1988—24no—
31#1823C2218yes—
32#1458—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C24×C4D411yescharacteristic
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C414yescomplemented
43C2×C22⋊Q8—28no—
44C2×C22⋊Q8—44no—
45C2×C22⋊Q8—44no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—44no—
54C2×#117—44no—
55C2×#117C2×C414yes—
56C2×#117D414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71#431—24no—
72#431—24no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×C4×C8—22no—
76#398—28no—
77#398—24no—
78#398C2×C418yescomplemented
79#397—28no—
80#397—24no—
81#397C2×C418yes—
82#391—28no—
83#389—28no—
84#389—28no—
85#385—28no—
86#385—28no—
87#384—22no—
88#384D414yes—
89#382—22no—
90#382—22no—
91#382—22no—
92#382C2×C414yescomplemented
93#380—22no—
94#380—22no—
95#380—22no—
96#380C2314yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2312yes—
105#340—216no—
106#324—24no—
107#323—24no—
108C25C2×D411yescharacteristic
109C22×Q8—24no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4C22⋊C414yes—
126C23×C4D814yes—
127C23×C4SD1614yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C4⋊Q8—28no—
136C4⋊Q8—28no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C22⋊Q8—432no—
140#117—28no—
141#117—28no—
142#117—28no—
143#117—28no—
144#117—48no—
145#117—44no—
146#117—48no—
147#117—44no—
148#117—44no—
149#117—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4C22⋊C414yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D418yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C414yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—48no—
197C2×C42C22⋊C414yes—
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42#4114yes—
203C4⋊C8—44no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C4×C8—44no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—22no—
214#95—24no—
215#95C22×C414yes—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—42no—
221C24—42no—
222C24C2×C22⋊C412yes—
223C24C2×D812yes—
224C24C2×SD1612yes—
225C24C22×D411yescharacteristic
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—48no—
229C2×Q8—48no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—84no—
235C2×Q8—84no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4#102116yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4#12012yes—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4C2×D812yes—
285C22×C4C2×D812yes—
286C22×C4C2×SD1612yes—
287C22×C4C2×SD1612yes—
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4#13712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C2×C8—48no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—416no—
306C4⋊C4—48no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—416no—
317C42—24no—
318C42—24no—
319C42—48no—
320C42—48no—
321C42C4×D414yes—
322C42C4×D414yes—
323C42#12114yes—
324C42#12114yes—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23—81no—
342C23—81no—
343C23C2×#10214yes—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12011yescharacteristic
346C23C2×#12311yescharacteristic
347C23C22×D811yescharacteristic
348C23C22×SD1611yescharacteristic
349C23C2×AGL(1,ℤ/8)11yescharacteristic
350C23C2×#13711yescharacteristic
351Q8—88no—
352Q8—88no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—216no—
362C2×C4—216no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—816no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4#38614yes—
393C2×C4#38612yes—
394C2×C4#38612yes—
395C2×C4C2×#10214yes—
396C2×C4C2×#10214yes—
397C2×C4#41714yes—
398C2×C4#44712yes—
399C2×C4#44712yes—
400C2×C4#44812yes—
401C2×C4#44812yes—
402C2×C4#48014yes—
403C2×C4#48114yes—
404C2×C4C2×C4×D412yes—
405C2×C4C2×#12112yes—
406C8—84no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22C2×#38611yescharacteristic
425C22C22×#10211yescharacteristic
426C22C2×#41711yescharacteristic
427C22C2×#44711yescharacteristic
428C22C2×#44811yescharacteristic
429C22#302212yes—
430C22#302812yes—
431C22C2×#48011yescharacteristic
432C22C2×#48111yescharacteristic
433C4—24no—
434C4—24no—
435C4—22no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—84no—
440C4—88no—
441C4—84no—
442C4#187414yes—
443C4#187414yes—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—44no—
449C2#1830812yes—
450C2#1831012yes—
451C2C2×#187411yescharacteristic
452C2#2095511yescharacteristic
453C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×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
3. #21833C2semidirect
5. #9027C2semidirect
7. #8026C2semidirect
14. #2621C4semidirect
42. C2×C4⋊Q8C2×C4semidirect
78. #398C2×C4semidirect
92. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
#340522
#218332
C2×#14582
#90272
#90252
#80262