Finite Group Databasebeta

← #121829

Subgroups — #121829

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13954C211yesmaximal, characteristic
2#10847C211yesmaximal, characteristic
3C2×#1303C212yesmaximal
4C2×#1286C212yesmaximal
5#7908C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×#389—21no—
9C2×#389—21no—
10C2×#382C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13C2×#343—21no—
14C2×#343—21no—
15#1444—22no—
16C4⋊#95C412yes—
17C4⋊#95C412yes—
18#1303—24no—
19#1286—24no—
20#1280—22no—
21C22×M16—22no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42—21no—
29C22×C42C2×C412yes—
30C22×C42C2311yescharacteristic
31#431—22no—
32#431—22no—
33#431—22no—
34#431Q814yes—
35#423—22no—
36#423—22no—
37#423—22no—
38#423D414yes—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C4×C8—22no—
44#389—44no—
45#389—42no—
46#389—42no—
47#382—22no—
48#382—22no—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C412yes—
59C2×#95D412yes—
60#343—42no—
61#343—42no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4D812yes—
97C2×C4⋊C4SD1612yes—
98C2×C4⋊C4SD1612yes—
99C2×C4⋊C4Q1612yes—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—42no—
106C2×C42—82no—
107C2×C42C2×D412yes—
108C2×C42C2×Q812yes—
109C2×C42#4112yes—
110C2×C42#4112yes—
111C2×C42#4112yes—
112C2×C42#4112yes—
113C4⋊C8—24no—
114C4⋊C8—24no—
115C4⋊C8—24no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C4×C8—24no—
122C4×C8—42no—
123#95—24no—
124#95—22no—
125#95—22no—
126#95—24no—
127#95—44no—
128#95—44no—
129#95—41no—
130#95—41no—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
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—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#10214yes—
166C22×C4#10314yes—
167C22×C4#11711yescharacteristic
168C22×C4#11711yescharacteristic
169C22×C4C4×D412yes—
170C22×C4C4×Q812yes—
171C22×C4#12112yes—
172C22×C4C22⋊Q812yes—
173C22×C4#12312yes—
174C22×C4#12312yes—
175C22×C4#12411yescharacteristic
176C22×C4#12411yescharacteristic
177C22×C4#12711yescharacteristic
178C22×C4C4⋊Q811yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4C2×SD1611yescharacteristic
182C22×C4C2×Q1611yescharacteristic
183C22×C4AGL(1,ℤ/8)12yes—
184C22×C4#13712yes—
185M16—44no—
186M16—44no—
187M16—82no—
188M16—82no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—88no—
202C4⋊C4—88no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C42—82no—
206C42—82no—
207C42—82no—
208C42—82no—
209C42C22⋊Q812yes—
210C42C22⋊Q812yes—
211C42#12312yes—
212C42#12312yes—
213C42#12512yes—
214C42#12512yes—
215C42#12612yes—
216C42#12612yes—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23#38111yescharacteristic
222C23#39011yescharacteristic
223C23#39111yescharacteristic
224C23C2×#10211yescharacteristic
225C23C2×#10311yescharacteristic
226C23#41811yescharacteristic
227C23#41911yescharacteristic
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#38712yes—
271C2×C4#38712yes—
272C2×C4#39712yes—
273C2×C4#39812yes—
274C2×C4#39912yes—
275C2×C4#40012yes—
276C2×C4#45912yes—
277C2×C4#46012yes—
278C2×C4#46112yes—
279C2×C4#46212yes—
280C2×C4#48012yes—
281C2×C4#48112yes—
282C2×C4#48312yes—
283C2×C4#48412yes—
284C2×C4#48611yescharacteristic
285C2×C4#48711yescharacteristic
286C2×C4#48912yes—
287C2×C4#49811yescharacteristic
288C2×C4#50011yescharacteristic
289C2×C4#50112yes—
290C8—84no—
291C8—84no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22#144511yescharacteristic
305C22#182311yescharacteristic
306C22#182411yescharacteristic
307C22#182711yescharacteristic
308C22#182711yescharacteristic
309C22#182911yescharacteristic
310C22#183011yescharacteristic
311C22#198111yescharacteristic
312C22#198211yescharacteristic
313C22#199011yescharacteristic
314C22#199311yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—44no—
320C4—44no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4#207112yes—
325C4#207212yes—
326C4#208312yes—
327C4#208412yes—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#876411yescharacteristic
333C2#876511yescharacteristic
334C2#876611yescharacteristic
335C2#876711yescharacteristic
336C2#1134811yescharacteristic
337C2#1135211yescharacteristic
338C2#1135711yescharacteristic
subgroupG/H
center, socleC23#381
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
#139542
#108472
C2×#13032
C2×#12862
#79082