Finite Group Databasebeta

← #109697

Subgroups — #109697

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20429C211yesmaximal, characteristic
2#14277C211yesmaximal, characteristic
3#8043C212yesmaximal
4C2×#1286C212yesmaximal
5#7906C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×C4×M16—21no—
9C2×#389—21no—
10C2×#384C2211yescharacteristic
11C2×#382C2211yescharacteristic
12C2×#376C2211yescharacteristic
13#1768—24no—
14#1728—21no—
15#1444—21no—
16#1444C412yes—
17#1442—21no—
18#1442C412yes—
19C4⋊#95—21no—
20#1296—24no—
21#1286C418yescomplemented
22#1280—21no—
23#1280—21no—
24C22×M16—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C412yes—
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42C2311yescharacteristic
33#431—22no—
34#431—22no—
35#431—22no—
36#431C2×C414yes—
37#423—22no—
38#423—22no—
39#423C2×C414yes—
40#423Q814yes—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8D414yes—
44C2×C22⋊C8—22no—
45C4×M16—42no—
46C4×M16—42no—
47C2×#97—22no—
48C2×C4×C8—22no—
49C2×C4×C8—22no—
50#389—42no—
51#389—42no—
52#384—22no—
53#384—22no—
54#382—22no—
55#382C2×C414yes—
56#376—22no—
57#376—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C412yes—
65C2×#95D411yescharacteristic
66C2×#95D411yescharacteristic
67#336—24no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—24no—
84C22×C8—42no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—82no—
100C2×C4⋊C4—82no—
101C2×C4⋊C4—82no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42D812yes—
111C2×C42SD1612yes—
112C2×C42C22×C412yes—
113C2×C42C2×D412yes—
114C2×C42C2×Q812yes—
115C2×C42#4112yes—
116C2×C42#4112yes—
117C2×C42#4112yes—
118C4⋊C8—24no—
119C4⋊C8—24no—
120C4⋊C8—24no—
121C4⋊C8—24no—
122C4⋊C8—42no—
123C4⋊C8C4⋊C418yescomplemented
124C22⋊C8—24no—
125C22⋊C8—42no—
126#97—44no—
127C4×C8—24no—
128C4×C8—44no—
129C4×C8—42no—
130#95—22no—
131#95—24no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—44no—
137#95—82no—
138#95—82no—
139#95C4214yes—
140C24C2×C22⋊C411yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
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—42no—
177C22×C4#10212yes—
178C22×C4#10212yes—
179C22×C4C2×C4211yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4C4×D412yes—
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4C4×Q812yes—
185C22×C4#12011yescharacteristic
186C22×C4#12112yes—
187C22×C4C22⋊Q812yes—
188C22×C4#12311yescharacteristic
189C22×C4#12411yescharacteristic
190C22×C4C4⋊Q811yescharacteristic
191C22×C4C2×D811yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4#13512yes—
194C22×C4AGL(1,ℤ/8)11yescharacteristic
195C22×C4#13712yes—
196C22×C4#13711yescharacteristic
197M16—44no—
198M16—44no—
199M16—44no—
200M16—44no—
201C2×C8—28no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C4⋊C4—88no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—22no—
220C42—42no—
221C42—42no—
222C42C2×C4⋊C412yes—
223C42#11712yes—
224C42C4×D412yes—
225C42C4×Q812yes—
226C42C22⋊Q812yes—
227C42#12312yes—
228C42#12512yes—
229C42#12612yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23C4×C22⋊C411yescharacteristic
235C23#38611yescharacteristic
236C23#39111yescharacteristic
237C23C2×#10211yescharacteristic
238C23#41611yescharacteristic
239C23#41711yescharacteristic
240C23#41911yescharacteristic
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4C4×C4⋊C412yes—
286C2×C4#38512yes—
287C2×C4#38712yes—
288C2×C4#38712yes—
289C2×C4#38812yes—
290C2×C4#38912yes—
291C2×C4C4×D812yes—
292C2×C4C4×SD1612yes—
293C2×C4#44012yes—
294C2×C4#44112yes—
295C2×C4#44711yescharacteristic
296C2×C4#44811yescharacteristic
297C2×C4#45212yes—
298C2×C4#47412yes—
299C2×C4#47512yes—
300C2×C4#47812yes—
301C2×C4#47912yes—
302C2×C4#48011yescharacteristic
303C2×C4#48111yescharacteristic
304C2×C4#48512yes—
305C8—48no—
306C8—84no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22#143611yescharacteristic
320C22C4×#10211yescharacteristic
321C22#176311yescharacteristic
322C22#187311yescharacteristic
323C22#187411yescharacteristic
324C22#190911yescharacteristic
325C22#191011yescharacteristic
326C22#191411yescharacteristic
327C22#191511yescharacteristic
328C22#198811yescharacteristic
329C22#198911yescharacteristic
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—84no—
337C4—81no—
338C4—81no—
339C4—162no—
340C4#186212yes—
341C4#186512yes—
342C4#191812yes—
343C4#191912yes—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#919211yescharacteristic
349C2#919411yescharacteristic
350C2#919511yescharacteristic
351C2#919711yescharacteristic
352C2#1088411yescharacteristic
353C2#1088911yescharacteristic
354C2#1091611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1286C4semidirect
123. C4⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#204292
#142772
#80432
C2×#12862
#79062