Finite Group Databasebeta

← #194842

Subgroups — #194842

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#21804C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9015C214yesmaximal, complemented
6#9013C214yesmaximal
7#8026C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×#117—21no—
10C22×#117C2211yescharacteristic
11C2×#431C2212yes—
12#2628—24no—
13#2628C418yescomplemented
14C2×#400—21no—
15C2×#399—21no—
16C2×#395C2212yes—
17C2×#393C2212yes—
18#2317—24no—
19#2316—21no—
20#2316C2212yes—
21#2315—21no—
22#2315C2212yes—
23#2296—24no—
24C2×#389—21no—
25#2285—21no—
26#2285C2212yes—
27C2×#382C2211yescharacteristic
28C2×#380—21no—
29#1981—24no—
30#1956—24no—
31#1787C2218yes—
32#1458—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8C2312yes—
37C24×C4D411yescharacteristic
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C414yescomplemented
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—24no—
50C2×#117—44no—
51C2×#117—44no—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42—22no—
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—24no—
69#431—24no—
70C2×C4⋊C8—22no—
71C2×#103—216no—
72C2×C22⋊C8C2314yes—
73C2×C4×C8—22no—
74#400—28no—
75#399—28no—
76#395—24no—
77#395C2×C418yescomplemented
78#393—24no—
79#393C2×C418yes—
80#389—28no—
81#389—28no—
82#389—28no—
83#385—28no—
84#385—28no—
85#384—22no—
86#384D414yes—
87#382—22no—
88#382—22no—
89#382—22no—
90#382C2×C414yescomplemented
91#380—28no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380C2314yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95C2312yes—
104#324—24no—
105#323—24no—
106C25C2×D411yescharacteristic
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8—84no—
111C22×Q8C22×C414yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4C22⋊C414yes—
121C23×C4SD1614yes—
122C23×C4Q1614yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—24no—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134#117—28no—
135#117—28no—
136#117—28no—
137#117—28no—
138#117—48no—
139#117—44no—
140#117—48no—
141#117—44no—
142#117—44no—
143#117—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D418yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—84no—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C22×C412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—48no—
189C2×C42C22⋊C414yes—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42#4114yes—
195C4⋊C8—44no—
196#103—816no—
197C22⋊C8—24no—
198C22⋊C8—24no—
199C4×C8—44no—
200#95—24no—
201#95—24no—
202#95—28no—
203#95—24no—
204#95—28no—
205#95—24no—
206#95—22no—
207#95—24no—
208#95C22×C414yes—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—42no—
214C24—42no—
215C24C2×C22⋊C412yes—
216C24C2×SD1612yes—
217C24C2×Q1612yes—
218C24C22×D411yescharacteristic
219C2×Q8—28no—
220C2×Q8—416no—
221C2×Q8—44no—
222C2×Q8—84no—
223C2×Q8—84no—
224C2×Q8—88no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4#103116yescomplemented
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4#12012yes—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4C2×SD1612yes—
272C22×C4C2×SD1612yes—
273C22×C4C2×Q1612yes—
274C22×C4C2×Q1612yes—
275C22×C4AGL(1,ℤ/8)12yes—
276C22×C4#13712yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4112yes—
281C2×C8—48no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—28no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—88no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—28no—
300C22⋊C4—48no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—88no—
304C42—24no—
305C42—24no—
306C42—48no—
307C42—48no—
308C42C4×D414yes—
309C42C4×D414yes—
310C42#12114yes—
311C42#12114yes—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—44no—
323C23—44no—
324C23—42no—
325C23—42no—
326C23—82no—
327C23—82no—
328C23—81no—
329C23—81no—
330C23C2×#10314yes—
331C23C22×C22⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12311yescharacteristic
334C23C22×SD1611yescharacteristic
335C23C22×Q1611yescharacteristic
336C23C2×AGL(1,ℤ/8)11yescharacteristic
337C23C2×#13711yescharacteristic
338Q8—416no—
339Q8—88no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—216no—
349C2×C4—216no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—816no—
376C2×C4—84no—
377C2×C4#38614yes—
378C2×C4#38612yes—
379C2×C4#38612yes—
380C2×C4C2×#10314yes—
381C2×C4C2×#10314yes—
382C2×C4#41714yes—
383C2×C4#45012yes—
384C2×C4#45012yes—
385C2×C4#45112yes—
386C2×C4#45112yes—
387C2×C4#48314yes—
388C2×C4#48414yes—
389C2×C4C2×C4×D412yes—
390C2×C4C2×#12112yes—
391C8—84no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22C2×#38611yescharacteristic
410C22C22×#10311yescharacteristic
411C22C2×#41711yescharacteristic
412C22C2×#45011yescharacteristic
413C22C2×#45111yescharacteristic
414C22#302212yes—
415C22#302712yes—
416C22C2×#48311yescharacteristic
417C22C2×#48411yescharacteristic
418C4—24no—
419C4—24no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—84no—
425C4—88no—
426C4—84no—
427C4#187214yes—
428C4#187214yes—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—44no—
434C2#1830112yes—
435C2#1830312yes—
436C2C2×#187211yescharacteristic
437C2#2095511yescharacteristic
438C2#2096111yescharacteristic
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. #21804C2semidirect
5. #9015C2semidirect
7. #8026C2semidirect
13. #2628C4semidirect
42. C2×C4⋊Q8C2×C4semidirect
77. #395C2×C4semidirect
90. #382C2×C4semidirect
260. C22×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#340522
#218042
C2×#14582
#90152
#90132
#80262