Finite Group Databasebeta

← #109903

Subgroups — #109903

Order 512.

678 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20536C211yesmaximal, 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#1770—24no—
13#1749—22no—
14C2×#343—21no—
15C2×#343—21no—
16#1444C412yes—
17#1442C412yes—
18#1436—22no—
19C4⋊#95—21no—
20#1286C418yescomplemented
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#431C2×C414yes—
35#431Q814yes—
36#423—22no—
37#423—22no—
38#423—22no—
39#423C2×C414yes—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×#97—24no—
45C2×C4×C8D414yes—
46#389—44no—
47#382—22no—
48#382C2×C414yes—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C2×#95—21no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95C2×C412yes—
59C2×#95C2×C412yes—
60#343—42no—
61#343—42no—
62#336—24no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—21no—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C412yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—24no—
78C22×C8—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—82no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—42no—
101C2×C42—42no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—82no—
105C2×C42C22×C412yes—
106C2×C42C2×D412yes—
107C2×C42C2×Q812yes—
108C2×C42#4112yes—
109C2×C42#4112yes—
110C2×C42#4112yes—
111C4⋊C8—24no—
112C4⋊C8—24no—
113C4⋊C8—24no—
114C4⋊C8—24no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119#97—84no—
120C4×C8—24no—
121C4×C8C4⋊C418yescomplemented
122#95—22no—
123#95—24no—
124#95—24no—
125#95—44no—
126#95—44no—
127#95—41no—
128#95—41no—
129#95—82no—
130#95C4214yes—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4C2×C4211yescharacteristic
168C22×C4#11711yescharacteristic
169C22×C4C4×D412yes—
170C22×C4C4×D412yes—
171C22×C4C4×D412yes—
172C22×C4C4×Q812yes—
173C22×C4#12011yescharacteristic
174C22×C4#12112yes—
175C22×C4C22⋊Q812yes—
176C22×C4#12311yescharacteristic
177C22×C4#12411yescharacteristic
178C22×C4C4⋊Q811yescharacteristic
179C22×C4AGL(1,ℤ/8)11yescharacteristic
180C22×C4AGL(1,ℤ/8)12yes—
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4#13711yescharacteristic
183C22×C4#13712yes—
184C22×C4#13711yescharacteristic
185M16—44no—
186M16—44no—
187M16—82no—
188M16—82no—
189C2×C8—28no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—88no—
206C42—44no—
207C42—82no—
208C42—82no—
209C42—82no—
210C42C2×C4⋊C412yes—
211C42#11712yes—
212C42C4×D412yes—
213C42C4×Q812yes—
214C42C22⋊Q812yes—
215C42#12312yes—
216C42#12512yes—
217C42#12612yes—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23C4×C22⋊C411yescharacteristic
223C23#38611yescharacteristic
224C23#39111yescharacteristic
225C23#41711yescharacteristic
226C23#41711yescharacteristic
227C23#41811yescharacteristic
228C23#41911yescharacteristic
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4C4×C4⋊C412yes—
272C2×C4#38512yes—
273C2×C4#38712yes—
274C2×C4#38712yes—
275C2×C4#38812yes—
276C2×C4#38912yes—
277C2×C4#44012yes—
278C2×C4#44012yes—
279C2×C4#44112yes—
280C2×C4#44212yes—
281C2×C4#46812yes—
282C2×C4#46812yes—
283C2×C4#46912yes—
284C2×C4#47012yes—
285C2×C4#48912yes—
286C2×C4#49011yescharacteristic
287C2×C4#49111yescharacteristic
288C2×C4#50111yescharacteristic
289C2×C4#50112yes—
290C2×C4#50111yescharacteristic
291C8—48no—
292C8—84no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#143611yescharacteristic
306C22#176211yescharacteristic
307C22#176311yescharacteristic
308C22#189111yescharacteristic
309C22#189111yescharacteristic
310C22#189411yescharacteristic
311C22#189511yescharacteristic
312C22#191111yescharacteristic
313C22#191211yescharacteristic
314C22#199211yescharacteristic
315C22#199311yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—84no—
321C4—82no—
322C4—82no—
323C4—82no—
324C4—82no—
325C4#193512yes—
326C4#193612yes—
327C4#194312yes—
328C4#194412yes—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2#948711yescharacteristic
334C2#948811yescharacteristic
335C2#948911yescharacteristic
336C2#949011yescharacteristic
337C2#1088111yescharacteristic
338C2#1088811yescharacteristic
339C2#1091911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
20. #1286C4semidirect
121. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205362
#141502
#108472
C2×#12862
#79012