Finite Group Databasebeta

← #121830

Subgroups — #121830

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13955C211yesmaximal, characteristic
2#10847C211yesmaximal, characteristic
3C2×#1303C212yesmaximal
4C2×#1286C212yesmaximal
5#7908C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×#382C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11C2×#343—21no—
12C2×#343—21no—
13#1442—22no—
14C4⋊#95—21no—
15C4⋊#95—21no—
16#1434C412yes—
17#1434C412yes—
18#1303—24no—
19#1286—24no—
20#1280—22no—
21C22×M16—22no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C42—21no—
27C22×C42C2×C412yes—
28C22×C42C2311yescharacteristic
29#431—22no—
30#431—22no—
31#431—22no—
32#431Q814yes—
33#423—22no—
34#423—22no—
35#423—22no—
36#423D414yes—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C4×C8—22no—
42#382—22no—
43#382—22no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—21no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C412yes—
56C2×#95D412yes—
57#343—42no—
58#343—42no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C412yes—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C2×M16—24no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C22×C8—42no—
74C22×C8—42no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42—82no—
98C2×C42C2×D412yes—
99C2×C42C2×Q812yes—
100C2×C42#4112yes—
101C2×C42#4112yes—
102C2×C42#4112yes—
103C2×C42#4112yes—
104C4⋊C8—24no—
105C4⋊C8—24no—
106C4⋊C8—24no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C4×C8—24no—
113C4×C8—42no—
114#95—24no—
115#95—22no—
116#95—22no—
117#95—24no—
118#95—44no—
119#95—44no—
120#95—41no—
121#95—41no—
122#95—82no—
123#95—82no—
124C24C2×C22⋊C411yescharacteristic
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
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#11711yescharacteristic
161C22×C4#11711yescharacteristic
162C22×C4C4×D412yes—
163C22×C4C4×Q812yes—
164C22×C4#12112yes—
165C22×C4C22⋊Q812yes—
166C22×C4#12312yes—
167C22×C4#12312yes—
168C22×C4#12411yescharacteristic
169C22×C4#12411yescharacteristic
170C22×C4#12711yescharacteristic
171C22×C4C4⋊Q811yescharacteristic
172C22×C4#13512yes—
173C22×C4#13512yes—
174C22×C4AGL(1,ℤ/8)11yescharacteristic
175C22×C4AGL(1,ℤ/8)11yescharacteristic
176C22×C4#13711yescharacteristic
177C22×C4#13711yescharacteristic
178M16—44no—
179M16—44no—
180M16—82no—
181M16—82no—
182C2×C8—44no—
183C2×C8—44no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C42—82no—
197C42—82no—
198C42—82no—
199C42—82no—
200C42C22⋊Q812yes—
201C42C22⋊Q812yes—
202C42#12312yes—
203C42#12312yes—
204C42#12512yes—
205C42#12512yes—
206C42#12612yes—
207C42#12612yes—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23#38111yescharacteristic
213C23#39011yescharacteristic
214C23#39111yescharacteristic
215C23#41611yescharacteristic
216C23#41611yescharacteristic
217C23#41711yescharacteristic
218C23#41711yescharacteristic
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#38712yes—
262C2×C4#38712yes—
263C2×C4#39712yes—
264C2×C4#39812yes—
265C2×C4#39912yes—
266C2×C4#40012yes—
267C2×C4#46312yes—
268C2×C4#46312yes—
269C2×C4#46412yes—
270C2×C4#46412yes—
271C2×C4#48212yes—
272C2×C4#48212yes—
273C2×C4#48512yes—
274C2×C4#48512yes—
275C2×C4#48812yes—
276C2×C4#49011yescharacteristic
277C2×C4#49111yescharacteristic
278C2×C4#49912yes—
279C2×C4#50111yescharacteristic
280C2×C4#50111yescharacteristic
281C8—84no—
282C8—84no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22#144511yescharacteristic
296C22#182511yescharacteristic
297C22#182511yescharacteristic
298C22#182711yescharacteristic
299C22#182711yescharacteristic
300C22#182911yescharacteristic
301C22#183011yescharacteristic
302C22#198311yescharacteristic
303C22#198311yescharacteristic
304C22#199111yescharacteristic
305C22#199211yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—82no—
311C4—82no—
312C4—82no—
313C4—82no—
314C4—162no—
315C4#207312yes—
316C4#207412yes—
317C4#208512yes—
318C4#208512yes—
319C2—21no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2#876811yescharacteristic
324C2#876911yescharacteristic
325C2#877011yescharacteristic
326C2#877111yescharacteristic
327C2#1134811yescharacteristic
328C2#1135211yescharacteristic
329C2#1135811yescharacteristic
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
#139552
#108472
C2×#13032
C2×#12862
#79082