Finite Group Databasebeta

← #189862

Subgroups — #189862

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#21738C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#8716C214yesmaximal, complemented
5#8712C214yesmaximal, complemented
6#8023C212yesmaximal, complemented
7C2×#1294C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2349—24no—
12D4×C4⋊C4C418yescomplemented
13#2316—21no—
14#2315—21no—
15#2315—21no—
16C2×#386—21no—
17#2285C2212yes—
18C2×#380C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C22×#95C2211yescharacteristic
22#1869—24no—
23#1817C2212yes—
24#1779C2212yes—
25#1441—21no—
26#1439—21no—
27#1434C2212yes—
28C22⋊#95—21no—
29C22⋊#95C2212yes—
30#1294C2218yes—
31#1294C2218yes—
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C23×C8—22no—
37#518—44no—
38#518—44no—
39C2×C4×D4—24no—
40C2×C4×D4—24no—
41C2×C4×D4—28no—
42C2×C4×D4C2×C414yes—
43C2×C4×D4D414yes—
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4C2×C414yescomplemented
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8C2314yes—
61#389—28no—
62#386—44no—
63#385—28no—
64#385—28no—
65#384C2×C414yescomplemented
66#382—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—24no—
73#380C2×C414yes—
74#380C2×C414yescomplemented
75#380C2314yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2×C414yescomplemented
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—24no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95C2312yes—
86C2×#95C2312yes—
87#336—24no—
88C25C2×D411yescharacteristic
89C25C2×D411yescharacteristic
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—22no—
93C22×D4—22no—
94C22×D4C22⋊C414yes—
95C22×D4D814yes—
96C22×D4SD1614yes—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—44no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4—82no—
110C23×C4C22×C412yes—
111C23×C4C22×C412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4#4114yes—
119C23×C4#4114yes—
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C22×C8—44no—
123C22×C8—84no—
124C4×D4—216no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—432no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—84no—
138C2×C4⋊C4—84no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22×C418yes—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—82no—
159C2×C22⋊C4C22⋊C414yes—
160C2×C22⋊C4C22⋊C414yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42C22×C412yes—
165C2×C42C2×D412yes—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—44no—
169C22⋊C8—84no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—44no—
177#95—42no—
178#95—44no—
179#95—82no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—42no—
185C24—42no—
186C24—42no—
187C24—42no—
188C24C2×C22⋊C412yes—
189C24C2×C22⋊C412yes—
190C24#12312yes—
191C24#12312yes—
192C24C2×D812yes—
193C24C2×D812yes—
194C24C2×SD1612yes—
195C24C2×SD1612yes—
196C24#13712yes—
197C24#13712yes—
198C24C22×D411yescharacteristic
199C2×D4—28no—
200C2×D4—28no—
201C2×D4—28no—
202C2×D4—28no—
203C2×D4—28no—
204C2×D4—28no—
205C2×D4#102116yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—28no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—88no—
251C22×C4—88no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4#11718yes—
259C22×C4#12314yes—
260C22×C4#12314yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4C2×D812yes—
264C22×C4C2×SD1612yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C2×C8—48no—
270C2×C8—44no—
271C2×C8—88no—
272C4⋊C4—28no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—88no—
286C22⋊C4—84no—
287C42—24no—
288C42—24no—
289C42—48no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—44no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—81no—
313C23—81no—
314C23—82no—
315C23—82no—
316C23—81no—
317C23—81no—
318C23#38114yes—
319C23C2×#10214yes—
320C23C2×#10214yes—
321C23#41914yes—
322C23#48014yes—
323C23#48014yes—
324C23#48314yes—
325C23#48314yes—
326C23C22×C22⋊C411yescharacteristic
327C23C2×#12311yescharacteristic
328C23C2×#12311yescharacteristic
329C23C22×D811yescharacteristic
330C23C22×SD1611yescharacteristic
331C23C2×#13711yescharacteristic
332C23C2×#13711yescharacteristic
333D4—232no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—84no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—816no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—168no—
379C2×C4—168no—
380C2×C4#38114yes—
381C2×C4C2×#10214yes—
382C2×C4C2×#11712yes—
383C2×C4C2×#12312yes—
384C8—84no—
385C22—22no—
386C22—28no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—24no—
397C22—28no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—42no—
404C22—44no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22#182318yes—
411C22C2×#38111yescharacteristic
412C22#288312yes—
413C22C22×#10211yescharacteristic
414C22C2×#41911yescharacteristic
415C22#290112yes—
416C22C2×#48011yescharacteristic
417C22C2×#48011yescharacteristic
418C22C2×#48311yescharacteristic
419C22C2×#48311yescharacteristic
420C4—22no—
421C4—22no—
422C4—24no—
423C4—24no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—88no—
428C4—162no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—28no—
434C2—44no—
435C2#1761312yes—
436C2#1762112yes—
437C2#2068911yescharacteristic
438C2C2×#182311yescharacteristic
439C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8716C2semidirect
5. #8712C2semidirect
6. #8023C2semidirect
7. C2×#1294C2semidirect
12. D4×C4⋊C4C4semidirect
47. C22×C4⋊C4C2×C4semidirect
65. #384C2×C4semidirect
74. #380C2×C4semidirect
77. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
#217382
#139602
#87162
#87122
#80232
C2×#12942