Finite Group Databasebeta

← #119610

Subgroups — #119610

Order 512.

1078 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 142 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×#1463C211yesmaximal, characteristic
2#13870C211yesmaximal, characteristic
3#13776C211yesmaximal, characteristic
4#10733C212yesmaximal
5C2×#1284C212yesmaximal
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9C2×#381—21no—
10C2×#381—21no—
11C2×#381C2211yescharacteristic
12C2×C4×C22⋊C4C2212yes—
13C2×#339—24no—
14#1463—24no—
15C2×#324—22no—
16#1441C412yescomplemented
17#1441C412yescomplemented
18#1436—24no—
19#1436—22no—
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1284—24no—
23C24×C4D411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—44no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D412yes—
40C22×C22⋊C4—21no—
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C42—21no—
44C22×C42D414yes—
45#407—22no—
46#407—22no—
47#407—22no—
48#407D414yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#381—28no—
52#381—22no—
53#381—22no—
54#381—44no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59C2×#95—24no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—21no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95C2×C412yes—
69#339—48no—
70#324—28no—
71C25C2×D411yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—24no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—24no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4#4112yes—
97C23×C4#4112yes—
98C23×C4#4112yes—
99C23×C4#4112yes—
100C2×M16—24no—
101C2×M16—42no—
102C2×M16—44no—
103C2×M16—44no—
104C22×C8—42no—
105#117—28no—
106#117—44no—
107#117—44no—
108#117—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—44no—
146C2×C42—44no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151#95—216no—
152#95—24no—
153#95—44no—
154#95—44no—
155#95—44no—
156#95—44no—
157#95—41no—
158#95—42no—
159#95—41no—
160#95—42no—
161#95—82no—
162#95—82no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—41no—
168C24—41no—
169C24C2×C22⋊C411yescharacteristic
170C24#12011yescharacteristic
171C24#12112yes—
172C24#12311yescharacteristic
173C24#12312yes—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—48no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—84no—
237C22×C4#10414yes—
238C22×C4#11711yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4C4×D412yes—
241C22×C4C4×D412yes—
242C22×C4#12011yescharacteristic
243C22×C4#12112yes—
244C22×C4#12112yes—
245C22×C4#12112yes—
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#12311yescharacteristic
250C22×C4#12412yes—
251C22×C4#12412yes—
252C22×C4#12612yes—
253C22×C4#12612yes—
254M16—48no—
255M16—84no—
256C2×C8—44no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C22⋊C4—48no—
268C22⋊C4—48no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—84no—
272C42—48no—
273C42—48no—
274C42—84no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—24no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—24no—
292C23—24no—
293C23—42no—
294C23—42no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23#38111yescharacteristic
300C23#38611yescharacteristic
301C23#38611yescharacteristic
302C23#39412yes—
303C23#39712yes—
304C23C2×#10412yes—
305C23#42112yes—
306C23B(4,𝔽2)12yes—
307C23#45812yes—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#38812yes—
367C2×C4#38812yes—
368C2×C4#39412yes—
369C2×C4#39712yes—
370C8—84no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—48no—
389C22—48no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#144011yescharacteristic
395C22#182612yes—
396C22#187112yes—
397C22#188911yescharacteristic
398C22#189011yescharacteristic
399C22#189611yescharacteristic
400C22#189711yescharacteristic
401C22#191211yescharacteristic
402C22#191211yescharacteristic
403C22#204414yes—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—44no—
409C4—88no—
410C4—82no—
411C4—82no—
412C4—82no—
413C4—162no—
414C2—24no—
415C2—22no—
416C2—22no—
417C2—42no—
418C2#947611yescharacteristic
419C2#947811yescharacteristic
420C2#948011yescharacteristic
421C2#948211yescharacteristic
422C2#1115611yescharacteristic
423C2#1117411yescharacteristic
424C2#1120111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1441C4semidirect
17. #1441C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14632
#138702
#137762
#107332
C2×#12842