Finite Group Databasebeta

← #120158

Subgroups — #120158

Order 512.

1022 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 150 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
1#14128C211yesmaximal, characteristic
2#13778C211yesmaximal, characteristic
3#10733C212yesmaximal
4C2×#1284C212yesmaximal
5#7909C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×#385—21no—
9C2×#381C2211yescharacteristic
10C2×#380—21no—
11C2×C4×C22⋊C4C2212yes—
12C22×#95C2211yescharacteristic
13C2×#339—24no—
14#1444C412yescomplemented
15#1442C412yescomplemented
16#1436—24no—
17#1436—22no—
18C22⋊#95—21no—
19C22⋊#95—21no—
20#1284—24no—
21#1280—22no—
22C24×C4D411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—44no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—22no—
35C22×C22⋊C4—21no—
36C22×C42—21no—
37C22×C42D414yes—
38#407—22no—
39#407—22no—
40#407—22no—
41#407Q814yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44#385—44no—
45#381—21no—
46#381—21no—
47#381D412yes—
48#381Q812yes—
49#380—28no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66C2×#95D412yes—
67#339—48no—
68C25C2×D411yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—24no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—24no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×Q811yescharacteristic
92C23×C4C2×Q812yes—
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4112yes—
96C23×C4#4112yes—
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—44no—
100C2×M16—44no—
101C22×C8—42no—
102#117—28no—
103#117—44no—
104#117—44no—
105#117—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4—82no—
115C2×C4⋊C4—82no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—28no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—42no—
128C2×C42—24no—
129C2×C42—24no—
130C2×C42—42no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—44no—
134C2×C42—44no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#95—28no—
140#95—22no—
141#95—24no—
142#95—24no—
143#95—22no—
144#95—22no—
145#95—44no—
146#95—44no—
147#95—42no—
148#95—42no—
149#95—82no—
150#95—82no—
151#95—82no—
152#95C4⋊C414yescomplemented
153C24—22no—
154C24—22no—
155C24—22no—
156C24—22no—
157C24—41no—
158C24—41no—
159C24C2×C22⋊C411yescharacteristic
160C24#12011yescharacteristic
161C24#12111yescharacteristic
162C24C22⋊Q811yescharacteristic
163C24#12311yescharacteristic
164C24#12411yescharacteristic
165C24C4⋊Q811yescharacteristic
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—88no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—88no—
228C22×C4#10414yes—
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4C4×D412yes—
232C22×C4C4×Q812yes—
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4#12111yescharacteristic
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12312yes—
241C22×C4#12512yes—
242C22×C4#12512yes—
243C22×C4#12612yes—
244C22×C4C4⋊Q812yes—
245M16—48no—
246M16—84no—
247C2×C8—44no—
248C2×C8—82no—
249C2×C8—82no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—88no—
255C4⋊C4—88no—
256C22⋊C4—48no—
257C22⋊C4—44no—
258C22⋊C4—48no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C42—48no—
263C42—48no—
264C42—84no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23#38011yescharacteristic
296C23#38611yescharacteristic
297C23#39111yescharacteristic
298C23#39411yescharacteristic
299C23#39511yescharacteristic
300C23#39811yescharacteristic
301C23#40011yescharacteristic
302C23C2×#10412yes—
303C23#42112yes—
304C23B(4,𝔽2)12yes—
305C23#45812yes—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4#38712yes—
364C2×C4#38912yes—
365C2×C4#39612yes—
366C2×C4#39812yes—
367C8—84no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22#144111yescharacteristic
399C22#180612yes—
400C22#188911yescharacteristic
401C22#189011yescharacteristic
402C22#189611yescharacteristic
403C22#189711yescharacteristic
404C22#191111yescharacteristic
405C22#191211yescharacteristic
406C22#198612yes—
407C22#204412yes—
408C22#206212yes—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—44no—
414C4—88no—
415C4—84no—
416C4—82no—
417C4—82no—
418C4—82no—
419C2—22no—
420C2—22no—
421C2—22no—
422C2—22no—
423C2—41no—
424C2—41no—
425C2#947511yescharacteristic
426C2#947811yescharacteristic
427C2#948011yescharacteristic
428C2#948111yescharacteristic
429C2#1122211yescharacteristic
430C2#1122411yescharacteristic
431C2#1122611yescharacteristic
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
14. #1444C4semidirect
15. #1442C4semidirect
152. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#141282
#137782
#107332
C2×#12842
#79092