Finite Group Databasebeta

← #124671

Subgroups — #124671

Order 512.

1052 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 108 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×#1444C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#7909C211yesmaximal, characteristic
4#7123C214yesmaximal, complemented
5C23×C4⋊C4—21no—
6C23×C4⋊C4C411yescharacteristic
7C2×#407—21no—
8C22×#95—21no—
9C22×#95C411yescharacteristic
10C22×#95C2211yescharacteristic
11#1779—21no—
12#1444—21no—
13#1444—21no—
14#1444C412yes—
15#1444C2212yes—
16#1280—21no—
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4—21no—
21C24×C4—21no—
22C24×C4C2×C411yescharacteristic
23C24×C4D411yescharacteristic
24C24×C4D411yescharacteristic
25C22×M16—41no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4—44no—
32C22×C4⋊C4—44no—
33C22×C4⋊C4—44no—
34C22×C4⋊C4—44no—
35C22×C4⋊C4C2×C412yes—
36#407—24no—
37#407—42no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—41no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43C2×#95—21no—
44C2×#95—24no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—41no—
48C2×#95—41no—
49C2×#95—44no—
50C2×#95C2×C412yes—
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95C2311yescharacteristic
55C25C22⋊C411yescharacteristic
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—41no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4SD1612yes—
94C23×C4Q1612yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—44no—
99C2×M16—82no—
100C2×M16—82no—
101C22×C8—42no—
102C22×C8—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—432no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C22⋊C8—24no—
118C22⋊C8—44no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#95—24no—
125#95—22no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—44no—
131#95—44no—
132#95M1614yescomplemented
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24#9911yescharacteristic
142C24#9911yescharacteristic
143C24#10312yes—
144C24#10412yes—
145C24C2×C22⋊C411yescharacteristic
146C22×C4—24no—
147C22×C4—216no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—48no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—42no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—42no—
183C22×C4—48no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—84no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4#10112yes—
230C22×C4#10312yes—
231C22×C4C2×M1612yes—
232C22×C4C2×SD1611yescharacteristic
233C22×C4C2×Q1611yescharacteristic
234C22×C4AGL(1,ℤ/8)12yes—
235M16—88no—
236C2×C8—44no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C4⋊C4—44no—
243C4⋊C4—416no—
244C4⋊C4—416no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—24no—
254C23—24no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23#32812yes—
276C23#32812yes—
277C23AΣL(1,𝔽16)12yes—
278C23#35212yes—
279C23#35312yes—
280C23#35612yes—
281C23#40711yescharacteristic
282C23C2×#9911yescharacteristic
283C23C2×#10111yescharacteristic
284C23C2×#10311yescharacteristic
285C23#41811yescharacteristic
286C23C2×#10411yescharacteristic
287C23#42111yescharacteristic
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—432no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—88no—
341C2×C4—88no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C8—84no—
346C8—162no—
347C22—28no—
348C22—28no—
349C22—28no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22#135214yes—
376C22#135414yes—
377C22#146311yescharacteristic
378C22#148711yescharacteristic
379C22#149011yescharacteristic
380C22C2×#32811yescharacteristic
381C22#151011yescharacteristic
382C22#152811yescharacteristic
383C22#154911yescharacteristic
384C22C2×AΣL(1,𝔽16)11yescharacteristic
385C22C2×#35211yescharacteristic
386C22C2×#35311yescharacteristic
387C22C2×#35611yescharacteristic
388C4—48no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—81no—
396C4—84no—
397C2—21no—
398C2—21no—
399C2—24no—
400C2—24no—
401C2—21no—
402C2—21no—
403C2#760711yescharacteristic
404C2#1151211yescharacteristic
405C2#1151711yescharacteristic
406C2C2×#135211yescharacteristic
407C2C2×#135411yescharacteristic
408C2#1179611yescharacteristic
409C2#1180311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
132. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#14442
#86982
#79092
#71232