Finite Group Databasebeta

← #110404

Subgroups — #110404

Order 512.

1006 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 166 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#20377C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#13870C211yesmaximal, characteristic
4#10728C211yesmaximal, characteristic
5C2×#1306C212yesmaximal
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2212yes—
10C2×C4×M16—21no—
11C2×#381C412yes—
12C2×#381C2211yescharacteristic
13#1750—24no—
14C2×#343—22no—
15#1728—21no—
16#1463—24no—
17#1463C418yescomplemented
18C2×#324—24no—
19#1441—22no—
20#1436—21no—
21#1436—21no—
22C22⋊#95C412yes—
23#1306—24no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C22×M16—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—44no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C414yes—
40C22×C4⋊C4D412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C412yes—
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45#407—22no—
46#407—22no—
47#407—22no—
48#407—22no—
49#407—22no—
50#407C2×C414yes—
51#407C2×C414yes—
52#407D414yes—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8D414yes—
57C4×M16—44no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#381—22no—
61#381—22no—
62#381—44no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—21no—
69#343—44no—
70#336—24no—
71#324—216no—
72C25C22×C411yescharacteristic
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—44no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4C4212yes—
88C23×C4C4212yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22×C412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4#4112yes—
97C23×C4#4112yes—
98C23×C4#4112yes—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—44no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16—44no—
106C2×M16—44no—
107C22×C8—24no—
108C22×C8—42no—
109#117—44no—
110#117—44no—
111#117—48no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—24no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—42no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8C22⋊C418yes—
151#97—44no—
152C4×C8—44no—
153#95—44no—
154#95—44no—
155#95—44no—
156#95—44no—
157#95—42no—
158#95—41no—
159#95—41no—
160C24—22no—
161C24—22no—
162C24—42no—
163C24—42no—
164C24C2×C4211yescharacteristic
165C24C2×C22⋊C411yescharacteristic
166C24#11711yescharacteristic
167C24C4×D412yes—
168C24C4×D412yes—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—84no—
221C22×C4#9914yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4#11712yes—
224C22×C4C4×D412yes—
225C22×C4C4×D412yes—
226C22×C4C4×D412yes—
227C22×C4C4×D412yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4#12311yescharacteristic
233C22×C4#12312yes—
234C22×C4#12411yescharacteristic
235C22×C4#12412yes—
236C22×C4#12612yes—
237C22×C4#12711yescharacteristic
238M16—48no—
239M16—48no—
240M16—48no—
241M16—84no—
242C2×C8—28no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C22⋊C4—48no—
254C22⋊C4—84no—
255C42—22no—
256C42—22no—
257C42—24no—
258C42—42no—
259C42—42no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—22no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—44no—
279C23C4×C22⋊C412yes—
280C23C4×C22⋊C411yescharacteristic
281C23#38611yescharacteristic
282C23#38812yes—
283C23#39011yescharacteristic
284C23C2×#9912yes—
285C23#41012yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#38612yes—
338C2×C4#38812yes—
339C2×C4#38812yes—
340C2×C4#39012yes—
341C8—48no—
342C8—84no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—48no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22#143711yescharacteristic
363C22C4×#9912yes—
364C22#189112yes—
365C22#189811yescharacteristic
366C22#189911yescharacteristic
367C22#191512yes—
368C22#196412yes—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—44no—
374C4—44no—
375C4—82no—
376C4—82no—
377C4—162no—
378C2—22no—
379C2—22no—
380C2—42no—
381C2—42no—
382C2#951312yes—
383C2#951412yes—
384C2#1093711yescharacteristic
385C2#1094011yescharacteristic
386C2#1096011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ 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
17. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#203772
C2×#14632
#138702
#107282
C2×#13062