Finite Group Databasebeta

← #239329

Subgroups — #239329

Order 512.

1043 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 182 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×#3189C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3#21753C212yesmaximal
4C2×#1508C212yesmaximal
5#10559C218yesmaximal, complemented
6#8008C211yesmaximal, characteristic
7C22×C22⋊Q8C2212yes—
8#3189—22no—
9#3189—24no—
10#3189C2214yes—
11C2×#484—22no—
12C2×#467—22no—
13C2×#451—22no—
14C2×#407C2212yes—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1901—24no—
19#1901C2218yes—
20#1851C2214yes—
21#1850—22no—
22#1821—22no—
23#1797—22no—
24#1784—22no—
25#1508—22no—
26#1508—22no—
27#1508C2218yes—
28C2×#328—24no—
29#1293C2214yes—
30C23×Q8D412yes—
31C24×C4C2311yescharacteristic
32C22×Q16—22no—
33C22×M16—22no—
34C22×M16C2312yes—
35C23×C8D412yes—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8D414yes—
47C2×C22⋊Q8C2314yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D412yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53#484—28no—
54#484—28no—
55#467—28no—
56#467—28no—
57#451—28no—
58#451—28no—
59C2×#107—22no—
60C2×C4⋊C8—24no—
61C2×#103—24no—
62C2×#101—28no—
63#407—22no—
64#407—22no—
65#407—22no—
66#407—22no—
67#407—22no—
68#407C2314yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8D412yes—
76C2×C22⋊C8C2312yes—
77C4×C22⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4D411yescharacteristic
80C4×C22⋊C4D411yescharacteristic
81C4×C22⋊C4C2312yes—
82C2×#95D412yes—
83#343—28no—
84#336—24no—
85#328—28no—
86#328—28no—
87C25#4111yescharacteristic
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—84no—
94C22×Q8C2×D414yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—82no—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4112yes—
107C23×C4#4112yes—
108C23×C4C2411yescharacteristic
109C2×Q16—28no—
110C2×Q16—84no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—42no—
114C2×M16—82no—
115C2×M16—84no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—42no—
120C22×C8—82no—
121C22×C8C2×D414yes—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—24no—
128C22⋊Q8—24no—
129C22⋊Q8—24no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×D414yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—82no—
164#107—28no—
165C4⋊C8—84no—
166#103—216no—
167#101—88no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C22⋊C8—42no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177#95—22no—
178#95C2×D412yes—
179#95C2×D412yes—
180C24—24no—
181C24—42no—
182C24—81no—
183C24#12111yescharacteristic
184C24#12111yescharacteristic
185C24#12312yes—
186C24#12312yes—
187C24C2×#4111yescharacteristic
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—48no—
191C2×Q8—48no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×Q8—84no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4#12012yes—
222C22×C4#12011yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12112yes—
225C22×C4#12112yes—
226C22×C4#12112yes—
227C22×C4#12112yes—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12312yes—
231C22×C4#12312yes—
232C22×C4#12411yescharacteristic
233C22×C4#12412yes—
234C22×C4#12411yescharacteristic
235C22×C4C22×D412yes—
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C2×#4112yes—
239Q16—88no—
240M16—48no—
241M16—48no—
242M16—84no—
243C2×C8—24no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8—82no—
250C2×C8—84no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C42—44no—
267C42—81no—
268C42—81no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—24no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—41no—
282C23—42no—
283C23—88no—
284C23#39411yescharacteristic
285C23#39411yescharacteristic
286C23#39412yes—
287C23#47314yes—
288C23C2×#12111yescharacteristic
289C23C2×#12312yes—
290Q8—48no—
291Q8—84no—
292Q8—84no—
293Q8—84no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—88no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4#39411yescharacteristic
328C2×C4#39412yes—
329C2×C4#39411yescharacteristic
330C2×C4C2×#12011yescharacteristic
331C2×C4C2×#12112yes—
332C2×C4C2×#12411yescharacteristic
333C8—44no—
334C8—84no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—21no—
348C22—44no—
349C22—44no—
350C22—48no—
351C22—82no—
352C22—84no—
353C22#204714yes—
354C22#205012yes—
355C22C2×#39411yescharacteristic
356C22C2×#47312yes—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—21no—
362C4—44no—
363C4—82no—
364C4—84no—
365C4—84no—
366C4#302612yes—
367C2—22no—
368C2—22no—
369C2—24no—
370C2—42no—
371C2—41no—
372C2#2351511yescharacteristic
373C2C2×#204711yescharacteristic
374C2C2×#205011yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10559C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#219322
#217532
C2×#15082
#105592
#80082