Finite Group Databasebeta

← #192139

Subgroups — #192139

Order 512.

1606 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2#21942C212yesmaximal, complemented
3C2×#1462C211yesmaximal, characteristic
4#13903C211yesmaximal, characteristic
5#8858C218yesmaximal, complemented
6#8019C212yesmaximal, complemented
7C23×C22⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9#2312—24no—
10#2312—22no—
11#2312C2214yes—
12C2×#388—22no—
13C2×#388—22no—
14C2×#381—21no—
15C2×#380C2212yes—
16C2×#380C2211yescharacteristic
17C2×#379—21no—
18C2×#379—21no—
19C2×C4×C22⋊C4—21no—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1853—22no—
22#1853C2214yescomplemented
23#1852—22no—
24#1852C2214yescomplemented
25#1809—24no—
26#1462C418yescomplemented
27C2×#323C2214yes—
28#1439C2214yes—
29#1437—22no—
30#1294C2218yescomplemented
31C26D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16—22no—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4C2312yes—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—24no—
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—24no—
48C22×C22⋊C4C2×C414yes—
49C22×C22⋊C4D412yes—
50C22×C22⋊C4Q814yes—
51C22×C22⋊C4C2312yes—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—21no—
55C2×#107—24no—
56C2×#106—24no—
57#407—24no—
58#407C2×C418yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2314yes—
61#388—28no—
62#388—28no—
63#388—28no—
64#388—28no—
65#381—44no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380D414yes—
71#380C2314yes—
72#379—24no—
73#379—24no—
74#379—44no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4—24no—
77C4×C22⋊C4—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—24no—
82C2×#95C2312yes—
83#340—24no—
84#340—24no—
85#323C2×C4116yescomplemented
86C25—21no—
87C25—21no—
88C25—22no—
89C25—44no—
90C25—42no—
91C25—42no—
92C25—82no—
93C25C4⋊C414yes—
94C25C2×D411yescharacteristic
95C25C2×D412yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—82no—
106C23×C4C4⋊C414yes—
107C23×C4C22×C412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4#4114yes—
114C23×C4C2411yescharacteristic
115C2×M16—28no—
116C2×M16—82no—
117C2×M16—82no—
118C22×C8—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×D414yes—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—48no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—84no—
152C2×C22⋊C4—82no—
153C2×C22⋊C4—82no—
154C2×C22⋊C4—84no—
155C2×C22⋊C4C4⋊C418yes—
156C2×C22⋊C4C22×C414yes—
157C2×C22⋊C4C2×Q814yes—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42—82no—
162C2×C42—82no—
163C2×C42—82no—
164#107—84no—
165#106—84no—
166C22⋊C8—28no—
167C22⋊C8—24no—
168C22⋊C8C22×C418yes—
169#95—28no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—48no—
174#95—42no—
175C24—24no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—24no—
180C24—24no—
181C24—22no—
182C24—24no—
183C24—24no—
184C24—42no—
185C24—48no—
186C24—42no—
187C24—44no—
188C24—42no—
189C24—44no—
190C24—44no—
191C24—48no—
192C24—44no—
193C24—48no—
194C24—42no—
195C24—42no—
196C24—84no—
197C24—81no—
198C24—81no—
199C24—82no—
200C24—84no—
201C24—84no—
202C24—816no—
203C24—816no—
204C24C2×C4⋊C412yes—
205C24C2×C4⋊C414yes—
206C24#12011yescharacteristic
207C24#12011yescharacteristic
208C24#12312yes—
209C24C22×D411yescharacteristic
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—48no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4C2×C4⋊C414yes—
250C22×C4C2×C4⋊C412yes—
251C22×C4C4×D414yes—
252C22×C4C4×D414yes—
253C22×C4#12012yes—
254C22×C4C22⋊Q814yes—
255C22×C4C22⋊Q814yes—
256C22×C4#12312yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×Q811yescharacteristic
260C22×C4C2×#4112yes—
261M16—88no—
262C2×C8—28no—
263C2×C8—44no—
264C2×C8—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—28no—
272C22⋊C4—28no—
273C22⋊C4—216no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—48no—
277C22⋊C4—84no—
278C22⋊C4—84no—
279C22⋊C4—88no—
280C22⋊C4—816no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C42—84no—
284C42—84no—
285C42—84no—
286C23—22no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—22no—
291C23—24no—
292C23—24no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—42no—
303C23—42no—
304C23—48no—
305C23—48no—
306C23—44no—
307C23—44no—
308C23—48no—
309C23—48no—
310C23—48no—
311C23—48no—
312C23—48no—
313C23—44no—
314C23—48no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—82no—
320C23—82no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—88no—
326C23—88no—
327C23—832no—
328C23—832no—
329C23—88no—
330C23—88no—
331C23#38512yes—
332C23#38512yes—
333C23#45312yes—
334C23#45512yes—
335C23C22×C4⋊C411yescharacteristic
336C23C2×#12011yescharacteristic
337C23C2×#12311yescharacteristic
338C23#57514yes—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38514yes—
378C2×C4C2×C4×D412yes—
379C2×C4C2×C22⋊Q812yes—
380C8—48no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—48no—
395C22—48no—
396C22—48no—
397C22—42no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—84no—
403C22—88no—
404C22—816no—
405C22—816no—
406C22—84no—
407C22—84no—
408C22#185614yes—
409C22C2×#38511yescharacteristic
410C22#291112yes—
411C22#300512yes—
412C22C2×#45311yescharacteristic
413C22C2×#45511yescharacteristic
414C22#302612yes—
415C22#309312yes—
416C4—22no—
417C4—22no—
418C4—24no—
419C4—44no—
420C4—84no—
421C4—84no—
422C4—84no—
423C4—162no—
424C2—22no—
425C2—22no—
426C2—24no—
427C2—42no—
428C2—84no—
429C2#1832812yes—
430C2#1832912yes—
431C2#2084511yescharacteristic
432C2C2×#185611yescharacteristic
433C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21942C2semidirect
5. #8858C2semidirect
6. #8019C2semidirect
22. #1853C22semidirect
24. #1852C22semidirect
26. #1462C4semidirect
30. #1294C22semidirect
85. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
#219422
C2×#14622
#139032
#88582
#80192