Finite Group Databasebeta

← #101014

Subgroups — #101014

Order 512.

602 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 112 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#21902C211yesmaximal, characteristic
2#15195C211yesmaximal, characteristic
3#8724C212yesmaximal, complemented
4#8557C211yesmaximal, characteristic
5#7039C212yesmaximal, complemented
6C2×#384C2211yescharacteristic
7C2×#380—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1923—24no—
10#1797C2212yes—
11#1797C2212yes—
12#1779—21no—
13C4×C22⋊C8—21no—
14C2×#323—21no—
15#1442—21no—
16#1442C412yes—
17#1439—21no—
18#1439C412yes—
19C4⋊#95—21no—
20C4⋊#95C2211yescharacteristic
21#1296—24no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C412yes—
28C22×C22⋊C4—22no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C4⋊C8D414yes—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—22no—
40#384—22no—
41#384—22no—
42#380—44no—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—22no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C412yes—
51C2×#95D411yescharacteristic
52#336—22no—
53#336—22no—
54#336—22no—
55#323—42no—
56C25C22⋊C411yescharacteristic
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—21no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—24no—
72C22×C8—42no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—82no—
83C2×C4⋊C4—82no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—82no—
91C2×C22⋊C4—82no—
92C2×C22⋊C4SD1612yes—
93C2×C22⋊C4Q1612yes—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—42no—
97C2×C42—42no—
98C2×C42C2×D412yes—
99C2×C42#4112yes—
100C4⋊C8—44no—
101C4⋊C8—44no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—82no—
107C22⋊C8—82no—
108C4×C8—82no—
109#95—22no—
110#95—22no—
111#95—42no—
112#95—44no—
113#95—42no—
114#95—82no—
115#95—82no—
116C24—21no—
117C24—21no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—41no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24#10312yes—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—48no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
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—81no—
169C22×C4—81no—
170C22×C4—84no—
171C22×C4#10112yes—
172C22×C4#10312yes—
173C22×C4C4×D412yes—
174C22×C4#12011yescharacteristic
175C22×C4#12112yes—
176C22×C4#12311yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178C22×C4C2×Q1611yescharacteristic
179C22×C4#13512yes—
180C22×C4#13512yes—
181C22×C4AGL(1,ℤ/8)12yes—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C4⋊C4—84no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—44no—
196C22⋊C4—88no—
197C42—24no—
198C42—42no—
199C42—42no—
200C42—81no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23#38611yescharacteristic
229C23C2×#10111yescharacteristic
230C23#41311yescharacteristic
231C23C2×#10311yescharacteristic
232C23#41611yescharacteristic
233C23#41611yescharacteristic
234C23#41811yescharacteristic
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—84no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4#44912yes—
281C2×C4#45512yes—
282C2×C4#45812yes—
283C2×C4#46512yes—
284C2×C4#46712yes—
285C2×C4#48212yes—
286C8—84no—
287C8—162no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22#154211yescharacteristic
312C22#154411yescharacteristic
313C22#154911yescharacteristic
314C22#155311yescharacteristic
315C22#187511yescharacteristic
316C22#188511yescharacteristic
317C22#189011yescharacteristic
318C22#189211yescharacteristic
319C22#190111yescharacteristic
320C22#191011yescharacteristic
321C22#191411yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—162no—
330C4—162no—
331C4#161214yes—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—82no—
337C2#902811yescharacteristic
338C2#913211yescharacteristic
339C2#919811yescharacteristic
340C2#939111yescharacteristic
341C2#945311yescharacteristic
342C2#946911yescharacteristic
343C2#951011yescharacteristic
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. #8724C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219022
#151952
#87242
#85572
#70392