Finite Group Databasebeta

← #101038

Subgroups — #101038

Order 512.

636 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 104 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#21906C211yesmaximal, characteristic
2#15135C211yesmaximal, characteristic
3#8725C212yesmaximal, complemented
4#8557C211yesmaximal, characteristic
5#7039C212yesmaximal, complemented
6C2×#399C412yes—
7C2×#389—21no—
8C2×#384C2211yescharacteristic
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1924—24no—
12#1797C2212yes—
13#1797C2212yes—
14#1779—21no—
15#1749—21no—
16C2×#323—21no—
17#1445C412yes—
18#1444—21no—
19#1439—21no—
20C4⋊#95C2211yescharacteristic
21#1296—24no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—22no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C4⋊C8—22no—
35C2×C4⋊C8D414yes—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41#399—44no—
42#389—44no—
43#384—22no—
44#384—22no—
45#380—44no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C412yes—
53C2×#95D411yescharacteristic
54#336—22no—
55#336—22no—
56#336—22no—
57#323—42no—
58C25C22⋊C411yescharacteristic
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—24no—
74C22×C8—42no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4—82no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—82no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42C2×D412yes—
105C2×C42#4112yes—
106C4⋊C8—44no—
107C4⋊C8—44no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114#97—82no—
115#95—22no—
116#95—22no—
117#95—42no—
118#95—44no—
119#95—44no—
120#95—41no—
121#95—41no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24#10012yes—
133C24C2×C22⋊C411yescharacteristic
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—48no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—84no—
179C22×C4C4×D412yes—
180C22×C4#12011yescharacteristic
181C22×C4#12112yes—
182C22×C4#12311yescharacteristic
183C22×C4#13512yes—
184C22×C4AGL(1,ℤ/8)11yescharacteristic
185C22×C4#13712yes—
186C22×C4#13712yes—
187C22×C4#13711yescharacteristic
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C4⋊C4—48no—
195C4⋊C4—84no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C4⋊C4—88no—
199C4⋊C4—82no—
200C22⋊C4—44no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—88no—
204C42—24no—
205C42—42no—
206C42—42no—
207C42—81no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23#38611yescharacteristic
236C23C2×#10011yescharacteristic
237C23#41311yescharacteristic
238C23#41611yescharacteristic
239C23#41711yescharacteristic
240C23#41911yescharacteristic
241C23#41911yescharacteristic
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4#45212yes—
288C2×C4#45312yes—
289C2×C4#45812yes—
290C2×C4#46812yes—
291C2×C4#47012yes—
292C2×C4#48512yes—
293C8—84no—
294C8—162no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#155111yescharacteristic
319C22#155411yescharacteristic
320C22#155811yescharacteristic
321C22#155911yescharacteristic
322C22#187311yescharacteristic
323C22#188711yescharacteristic
324C22#189011yescharacteristic
325C22#189411yescharacteristic
326C22#190011yescharacteristic
327C22#191011yescharacteristic
328C22#191511yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—84no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—162no—
338C4#161814yes—
339C2—21no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—82no—
344C2#902411yescharacteristic
345C2#913611yescharacteristic
346C2#919411yescharacteristic
347C2#940311yescharacteristic
348C2#945711yescharacteristic
349C2#947311yescharacteristic
350C2#950711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8725C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#151352
#87252
#85572
#70392