Finite Group Databasebeta

← #120164

Subgroups — #120164

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21951C211yesmaximal, characteristic
2#14150C211yesmaximal, characteristic
3#10847C211yesmaximal, characteristic
4C2×#1286C212yesmaximal
5#7901C212yesmaximal
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×#389—21no—
9C2×#382C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1939—24no—
13#1797—22no—
14C2×#343—21no—
15C2×#343—21no—
16#1444C412yes—
17#1442C412yes—
18#1436—22no—
19C4⋊#95—21no—
20#1286—24no—
21#1279—24no—
22C22×M16—22no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42D412yes—
31C22×C42C2311yescharacteristic
32#431—22no—
33#431—22no—
34#431—22no—
35#431—22no—
36#423—22no—
37#423—22no—
38#423—22no—
39#423Q814yes—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—24no—
44C2×C22⋊C8—22no—
45C2×C4×C8D414yes—
46#389—44no—
47#382—21no—
48#382—21no—
49#382D412yes—
50#382Q812yes—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95C2×C412yes—
61C2×#95C2×C412yes—
62#343—42no—
63#343—42no—
64#336—24no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C412yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—24no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C22×C8—24no—
80C22×C8—42no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4SD1612yes—
99C2×C4⋊C4Q1612yes—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—82no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D412yes—
109C2×C42C2×Q812yes—
110C2×C42C2×Q811yescharacteristic
111C2×C42#4112yes—
112C2×C42#4111yescharacteristic
113C2×C42#4112yes—
114C2×C42#4111yescharacteristic
115C4⋊C8—24no—
116C4⋊C8—24no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—84no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C4×C8—24no—
125C4×C8—24no—
126#95—22no—
127#95—24no—
128#95—22no—
129#95—22no—
130#95—44no—
131#95—44no—
132#95—41no—
133#95—41no—
134#95—82no—
135#95C4⋊C414yes—
136C24C2×C22⋊C411yescharacteristic
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
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—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4#10314yes—
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C4×Q812yes—
176C22×C4#12011yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12112yes—
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q812yes—
182C22×C4C22⋊Q811yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12411yescharacteristic
185C22×C4C4⋊Q811yescharacteristic
186C22×C4C2×SD1611yescharacteristic
187C22×C4C2×Q1611yescharacteristic
188C22×C4#13512yes—
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4AGL(1,ℤ/8)12yes—
191C22×C4#13711yescharacteristic
192M16—44no—
193M16—44no—
194M16—82no—
195M16—82no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—82no—
201C2×C8—82no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—88no—
213C42—44no—
214C42—82no—
215C42—82no—
216C42—82no—
217C42#12112yes—
218C42C22⋊Q812yes—
219C42C22⋊Q812yes—
220C42#12312yes—
221C42#12512yes—
222C42#12512yes—
223C42#12612yes—
224C42C4⋊Q812yes—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23#38011yescharacteristic
230C23#38611yescharacteristic
231C23#39111yescharacteristic
232C23C2×#10311yescharacteristic
233C23#41611yescharacteristic
234C23#41711yescharacteristic
235C23#41811yescharacteristic
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4#38712yes—
281C2×C4#38912yes—
282C2×C4#39411yescharacteristic
283C2×C4#39511yescharacteristic
284C2×C4#39612yes—
285C2×C4#39811yescharacteristic
286C2×C4#39812yes—
287C2×C4#40011yescharacteristic
288C2×C4#46011yescharacteristic
289C2×C4#46211yescharacteristic
290C2×C4#46312yes—
291C2×C4#46512yes—
292C2×C4#46712yes—
293C2×C4#46812yes—
294C2×C4#46912yes—
295C2×C4#47511yescharacteristic
296C2×C4#47612yes—
297C2×C4#47712yes—
298C2×C4#47711yescharacteristic
299C2×C4#47812yes—
300C2×C4#47812yes—
301C2×C4#47912yes—
302C8—84no—
303C8—84no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22#144111yescharacteristic
317C22#180311yescharacteristic
318C22#180411yescharacteristic
319C22#189111yescharacteristic
320C22#189111yescharacteristic
321C22#189211yescharacteristic
322C22#189511yescharacteristic
323C22#191111yescharacteristic
324C22#191211yescharacteristic
325C22#198711yescharacteristic
326C22#198911yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—84no—
335C4—82no—
336C4—82no—
337C4—82no—
338C4#206512yes—
339C4#206812yes—
340C4#208712yes—
341C4#208812yes—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2#948311yescharacteristic
347C2#948411yescharacteristic
348C2#948911yescharacteristic
349C2#949011yescharacteristic
350C2#1122311yescharacteristic
351C2#1122411yescharacteristic
352C2#1122811yescharacteristic
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
#219512
#141502
#108472
C2×#12862
#79012