Finite Group Databasebeta

← #120163

Subgroups — #120163

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 359 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—42no—
47#389—42no—
48#382—21no—
49#382—21no—
50#382D412yes—
51#382Q812yes—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C4×C4⋊C4—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63#343—42no—
64#343—42no—
65#336—24no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C412yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×M16—24no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C22×C8—24no—
81C22×C8—42no—
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—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—82no—
101C2×C4⋊C4D812yes—
102C2×C4⋊C4SD1612yes—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—42no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42—42no—
109C2×C42—82no—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×Q812yes—
113C2×C42C2×Q811yescharacteristic
114C2×C42#4112yes—
115C2×C42#4111yescharacteristic
116C2×C42#4112yes—
117C2×C42#4111yescharacteristic
118C4⋊C8—24no—
119C4⋊C8—24no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—84no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C4×C8—24no—
128C4×C8—24no—
129#95—22no—
130#95—24no—
131#95—22no—
132#95—22no—
133#95—44no—
134#95—44no—
135#95—41no—
136#95—41no—
137#95—82no—
138#95C4⋊C414yes—
139C24C2×C22⋊C411yescharacteristic
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4#10214yes—
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4C4×D412yes—
178C22×C4C4×Q812yes—
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12112yes—
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q812yes—
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4C4⋊Q811yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4C2×SD1611yescharacteristic
191C22×C4#13512yes—
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4#13711yescharacteristic
194C22×C4#13712yes—
195M16—44no—
196M16—44no—
197M16—82no—
198M16—82no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—82no—
204C2×C8—82no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—88no—
218C42—44no—
219C42—82no—
220C42—82no—
221C42—82no—
222C42#12112yes—
223C42C22⋊Q812yes—
224C42C22⋊Q812yes—
225C42#12312yes—
226C42#12512yes—
227C42#12512yes—
228C42#12612yes—
229C42C4⋊Q812yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23#38011yescharacteristic
235C23#38611yescharacteristic
236C23#39111yescharacteristic
237C23C2×#10211yescharacteristic
238C23#41611yescharacteristic
239C23#41711yescharacteristic
240C23#41911yescharacteristic
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4#38712yes—
286C2×C4#38912yes—
287C2×C4#39411yescharacteristic
288C2×C4#39511yescharacteristic
289C2×C4#39612yes—
290C2×C4#39811yescharacteristic
291C2×C4#39812yes—
292C2×C4#40011yescharacteristic
293C2×C4#45911yescharacteristic
294C2×C4#46111yescharacteristic
295C2×C4#46412yes—
296C2×C4#46512yes—
297C2×C4#46612yes—
298C2×C4#46812yes—
299C2×C4#47012yes—
300C2×C4#47412yes—
301C2×C4#47411yescharacteristic
302C2×C4#47512yes—
303C2×C4#47611yescharacteristic
304C2×C4#47812yes—
305C2×C4#47912yes—
306C2×C4#47912yes—
307C8—84no—
308C8—84no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22#144111yescharacteristic
322C22#180211yescharacteristic
323C22#180511yescharacteristic
324C22#189111yescharacteristic
325C22#189111yescharacteristic
326C22#189311yescharacteristic
327C22#189411yescharacteristic
328C22#191111yescharacteristic
329C22#191211yescharacteristic
330C22#198811yescharacteristic
331C22#198911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—44no—
339C4—84no—
340C4—82no—
341C4—82no—
342C4—82no—
343C4#206612yes—
344C4#206712yes—
345C4#208612yes—
346C4#208812yes—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2#948511yescharacteristic
352C2#948611yescharacteristic
353C2#948711yescharacteristic
354C2#948811yescharacteristic
355C2#1122311yescharacteristic
356C2#1122411yescharacteristic
357C2#1122711yescharacteristic
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