Finite Group Databasebeta

← #120048

Subgroups — #120048

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20753C211yesmaximal, characteristic
2#14277C211yesmaximal, characteristic
3#8043C212yesmaximal
4C2×#1286C212yesmaximal
5#7906C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×#389—21no—
9C2×#384C2211yescharacteristic
10C2×#382C2211yescharacteristic
11C2×#376C2211yescharacteristic
12#1832—24no—
13#1728—21no—
14#1728—21no—
15#1444—21no—
16#1444C412yes—
17#1442—21no—
18#1442C412yes—
19C4⋊#95—21no—
20#1296—24no—
21#1286—24no—
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#431Q814yes—
37#423—22no—
38#423—22no—
39#423—22no—
40#423—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8D414yes—
44C2×C22⋊C8—22no—
45C2×C4×C8—22no—
46#389—44no—
47#384—22no—
48#384—22no—
49#382—21no—
50#382—21no—
51#382D412yes—
52#382Q812yes—
53#376—22no—
54#376—22no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C412yes—
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64#336—24no—
65#336—24no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—24no—
82C22×C8—42no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4—82no—
97C2×C4⋊C4—82no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D412yes—
108C2×C42C2×Q812yes—
109C2×C42C2×Q811yescharacteristic
110C2×C42#4112yes—
111C2×C42#4111yescharacteristic
112C2×C42#4112yes—
113C2×C42#4111yescharacteristic
114C4⋊C8—24no—
115C4⋊C8—24no—
116C4⋊C8—24no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C22⋊C8—24no—
121C22⋊C8—42no—
122C4×C8—24no—
123C4×C8—42no—
124#95—22no—
125#95—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—44no—
132#95—82no—
133#95—82no—
134#95C4⋊C414yes—
135C24C2×C22⋊C411yescharacteristic
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
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—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
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—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×C4C2×C4⋊C411yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4C4×D412yes—
177C22×C4C4×Q812yes—
178C22×C4#12011yescharacteristic
179C22×C4#12112yes—
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22⋊Q812yes—
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4#12311yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188C22×C4#13512yes—
189C22×C4#13512yes—
190C22×C4AGL(1,ℤ/8)12yes—
191C22×C4#13712yes—
192M16—82no—
193M16—82no—
194M16—82no—
195M16—82no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—82no—
201C2×C8—82no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—88no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C42—42no—
210C42—42no—
211C42—42no—
212C42—42no—
213C42#12112yes—
214C42C22⋊Q812yes—
215C42C22⋊Q812yes—
216C42#12312yes—
217C42#12512yes—
218C42#12512yes—
219C42#12612yes—
220C42C4⋊Q812yes—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23#38011yescharacteristic
226C23#38611yescharacteristic
227C23#39111yescharacteristic
228C23#41611yescharacteristic
229C23#41611yescharacteristic
230C23#41811yescharacteristic
231C23#41911yescharacteristic
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#38712yes—
277C2×C4#38912yes—
278C2×C4#39411yescharacteristic
279C2×C4#39511yescharacteristic
280C2×C4#39612yes—
281C2×C4#39812yes—
282C2×C4#39811yescharacteristic
283C2×C4#40011yescharacteristic
284C2×C4#44912yes—
285C2×C4#45212yes—
286C2×C4#46312yes—
287C2×C4#46412yes—
288C2×C4#47812yes—
289C2×C4#47912yes—
290C2×C4#48212yes—
291C2×C4#48512yes—
292C2×C4#48812yes—
293C2×C4#48912yes—
294C2×C4#49912yes—
295C2×C4#50112yes—
296C8—84no—
297C8—84no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22#144111yescharacteristic
311C22#180411yescharacteristic
312C22#180511yescharacteristic
313C22#187311yescharacteristic
314C22#187511yescharacteristic
315C22#190911yescharacteristic
316C22#191011yescharacteristic
317C22#191411yescharacteristic
318C22#191511yescharacteristic
319C22#199111yescharacteristic
320C22#199311yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—84no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—162no—
331C4#204212yes—
332C4#205912yes—
333C4#208512yes—
334C4#210112yes—
335C2—21no—
336C2—21no—
337C2—21no—
338C2—21no—
339C2#919211yescharacteristic
340C2#919411yescharacteristic
341C2#919611yescharacteristic
342C2#919811yescharacteristic
343C2#1120611yescharacteristic
344C2#1120811yescharacteristic
345C2#1122511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#207532
#142772
#80432
C2×#12862
#79062