Finite Group Databasebeta

← #124694

Subgroups — #124694

Order 512.

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

normal subgroups only

This group has 900 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×#1442C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#7909C211yesmaximal, characteristic
4#7122C214yesmaximal, complemented
5C23×C4⋊C4C411yescharacteristic
6C2×#407—21no—
7C22×#95—21no—
8C22×#95—21no—
9C22×#95C411yescharacteristic
10C22×#95C2211yescharacteristic
11#1779—21no—
12#1442—21no—
13#1442—21no—
14#1442C412yes—
15#1442C2212yes—
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—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C4⋊C4—44no—
30C22×C4⋊C4C2×C412yes—
31#407—24no—
32#407—42no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—41no—
38C2×#95—21no—
39C2×#95—24no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—41no—
45C2×#95—41no—
46C2×#95—44no—
47C2×#95—42no—
48C2×#95—42no—
49C2×#95C2×C412yes—
50C2×#95C2×C411yescharacteristic
51C2×#95D411yescharacteristic
52C2×#95D411yescharacteristic
53C2×#95C2311yescharacteristic
54C25C22⋊C411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—22no—
69C23×C4—41no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—42no—
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—41no—
85C23×C4—41no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4D812yes—
93C23×C4SD1612yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C2×M16—44no—
98C2×M16—82no—
99C2×M16—82no—
100C22×C8—42no—
101C22×C8—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—44no—
109C22⋊C8—24no—
110C22⋊C8—44no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—82no—
115C22⋊C8—82no—
116#95—24no—
117#95—22no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—44no—
123#95—44no—
124#95—44no—
125#95—42no—
126#95—42no—
127#95M1614yescomplemented
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24#9911yescharacteristic
137C24#9911yescharacteristic
138C24#10212yes—
139C24#10412yes—
140C24C2×C22⋊C411yescharacteristic
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—48no—
166C22×C4—48no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—48no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4#10112yes—
225C22×C4#10212yes—
226C22×C4C2×M1612yes—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×SD1611yescharacteristic
229C22×C4#13712yes—
230M16—88no—
231C2×C8—44no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—22no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23#32712yes—
269C23#32712yes—
270C23AΣL(1,𝔽16)12yes—
271C23#35212yes—
272C23#35412yes—
273C23#35512yes—
274C23#40711yescharacteristic
275C23C2×#9911yescharacteristic
276C23C2×#10111yescharacteristic
277C23C2×#10211yescharacteristic
278C23#41911yescharacteristic
279C23C2×#10411yescharacteristic
280C23#42111yescharacteristic
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—816no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C8—84no—
339C8—162no—
340C22—28no—
341C22—28no—
342C22—28no—
343C22—28no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—22no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—21no—
361C22—22no—
362C22—22no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—21no—
368C22#134414yes—
369C22#134514yes—
370C22#146311yescharacteristic
371C22#148611yescharacteristic
372C22#149111yescharacteristic
373C22C2×#32711yescharacteristic
374C22#151111yescharacteristic
375C22#152811yescharacteristic
376C22#154611yescharacteristic
377C22C2×AΣL(1,𝔽16)11yescharacteristic
378C22C2×#35211yescharacteristic
379C22C2×#35411yescharacteristic
380C22C2×#35511yescharacteristic
381C4—42no—
382C4—41no—
383C4—42no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—84no—
388C4—81no—
389C4—84no—
390C2—21no—
391C2—21no—
392C2—24no—
393C2—24no—
394C2—21no—
395C2—21no—
396C2#761011yescharacteristic
397C2#1151111yescharacteristic
398C2#1151711yescharacteristic
399C2C2×#134411yescharacteristic
400C2C2×#134511yescharacteristic
401C2#1180811yescharacteristic
402C2#1181411yescharacteristic
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. #7122C2semidirect
127. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#14422
#86982
#79092
#71222